mintOC - User contributions [en]

Diels-Alder Reaction Experimental Design

2016-02-01T13:04:07Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x^i,\ G^i,\ F^i,\ Tc^i,\ n_{a1}^i,\ n_{a2}^i,\ n_{a4}^i,\ c_{kat}^i,\ \vartheta(t)^i} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}^i(t) & = & f(x^i(t), u^i(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}^i(t) & = & f_x(x^i(t),u^i(t),p)G^i(t) \ + \ f_p(x^i(t),u^i(t),p) \\
\\
\dot{F}(t) & = & \sum\limits_{i=1}^{4} w^i(t) (h^i_x(x^i(t),u^i(t),p)G^i(t))^T (h^i_x(x^i(t),u(t),p)G^i(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\
\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\
\vartheta_{up} + 273 & t \in [8,t_{end}]
\end{array} \right. \\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P \\
\dot{z}^i(t) & = & w^i(t) \\
z(0) & = & 0 \\
w^i(t) &\in& [0,1] \\
0 & \le & 4 - z^i(t_f). \\
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|Initial value Exp 1
|Initial value Exp 2
|Initial value Exp 3
|Initial value Exp 4
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0,10.0]
|1.0
|1.0
|1.0
|1.0
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0,10.0]
|1.0
|1.0
|1.0
|1.0
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|2.0
|2.0
|2.0
|2.0
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0,10.0]
|0.0
|1.0
|2.0
|3.0
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value Exp 1
|Initial value Exp 2
|Initial value Exp 3
|Initial value Exp 4
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|60.0
|40.0
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|60.0
|40.0
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|60.0
|40.0
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== Source Code ==

* The VPLAN code using [[:Category: VPLAN | VPLAN code]] can be found in: [[Diels-Alder Reaction Experimental Design (VPLAN)]]

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2016-02-01T13:00:57Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x^i,\ G^i,\ F^i,\ Tc^i,\ n_{a1}^i,\ n_{a2}^i,\ n_{a4}^i,\ c_{kat}^i,\ \vartheta(t)^i} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}^i(t) & = & f(x^i(t), u^i(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}^i(t) & = & f_x(x^i(t),u^i(t),p)G^i(t) \ + \ f_p(x^i(t),u^i(t),p) \\
\\
\dot{F}(t) & = & \sum_{i=1}^{4} w^i(t) (h^i_x(x^i(t),u^i(t),p)G^i(t))^T (h^i_x(x^i(t),u(t),p)G^i(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\
\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\
\vartheta_{up} + 273 & t \in [8,t_{end}]
\end{array} \right. \\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P \\
\dot{z}^i(t) & = & w^i(t) \\
z(0) & = & 0 \\
w^i(t) &\in& [0,1] \\
0 & \le & 4 - z^i(t_f). \\
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|Initial value Exp 1
|Initial value Exp 2
|Initial value Exp 3
|Initial value Exp 4
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0,10.0]
|1.0
|1.0
|1.0
|1.0
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0,10.0]
|1.0
|1.0
|1.0
|1.0
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|2.0
|2.0
|2.0
|2.0
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0,10.0]
|0.0
|1.0
|2.0
|3.0
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value Exp 1
|Initial value Exp 2
|Initial value Exp 3
|Initial value Exp 4
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|60.0
|40.0
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|60.0
|40.0
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|60.0
|40.0
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== Source Code ==

* The VPLAN code using [[:Category: VPLAN | VPLAN code]] can be found in: [[Diels-Alder Reaction Experimental Design (VPLAN)]]

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2016-02-01T13:00:16Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x^i,\ G^i,\ F^i,\ Tc^i,\ n_{a1}^i,\ n_{a2}^i,\ n_{a4}^i,\ c_{kat}^i,\ \vartheta(t)^i} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}^i(t) & = & f(x^i(t), u^i(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}^i(t) & = & f_x(x^i(t),u^i(t),p)G^i(t) \ + \ f_p(x^i(t),u^i(t),p) \\
\\
\dot{F}(t) & = & \sum_{i=1}^{4} w^i(t) (h^i_x(x^i(t),u^i(t),p)G^i(t))^T (h^i_x(x^i(t),u(t),p)G^i(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\
\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\
\vartheta_{up} + 273 & t \in [8,t_{end}]
\end{array} \right. \\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P \\
\dot{z}^i(t) & = & w^i(t) \\
z(0) & = & 0 \\
w^i(t) &\in& [0,1] \\
0 & \le & 4 - z^i(t_f). \\
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|Initial value (Exp 1)
|Initial value (Exp 2)
|Initial value (Exp 3)
|Initial value (Exp 4)
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0,10.0]
|1.0
|1.0
|1.0
|1.0
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0,10.0]
|1.0
|1.0
|1.0
|1.0
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|2.0
|2.0
|2.0
|2.0
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0,10.0]
|0.0
|1.0
|2.0
|3.0
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value (Exp 1)
|Initial value (Exp 2)
|Initial value (Exp 3)
|Initial value (Exp 4)
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|60.0
|40.0
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|60.0
|40.0
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|60.0
|40.0
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== Source Code ==

* The VPLAN code using [[:Category: VPLAN | VPLAN code]] can be found in: [[Diels-Alder Reaction Experimental Design (VPLAN)]]

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2016-02-01T12:57:10Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x^i,\ G^i,\ F^i,\ Tc^i,\ n_{a1}^i,\ n_{a2}^i,\ n_{a4}^i,\ c_{kat}^i,\ \vartheta(t)^i} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}^i(t) & = & f(x^i(t), u^i(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}^i(t) & = & f_x(x^i(t),u^i(t),p)G^i(t) \ + \ f_p(x^i(t),u^i(t),p) \\
\\
\dot{F}(t) & = & \sum_{i=1}^{4} w^i(t) (h^i_x(x^i(t),u^i(t),p)G^i(t))^T (h^i_x(x^i(t),u(t),p)G^i(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\
\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\
\vartheta_{up} + 273 & t \in [8,t_{end}]
\end{array} \right. \\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P \\
\dot{z}^i(t) & = & w^i(t) \\
z(0) & = & 0 \\
w^i(t) &\in& [0,1] \\
0 & \le & 4 - z^i(t_f). \\
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|Initial value Exp 1
|Initial value Exp 2
|Initial value Exp 3
|Initial value Exp 4
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0,10.0]
|1.0
|1.0
|1.0
|1.0
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0,10.0]
|1.0
|1.0
|1.0
|1.0
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|2.0
|2.0
|2.0
|2.0
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0,10.0]
|0.0
|1.0
|2.0
|3.0
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value (Exp 1)
|Initial value (Exp 2)
|Initial value (Exp 3)
|Initial value (Exp 4)
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|60.0
|40.0
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|60.0
|40.0
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|60.0
|40.0
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== Source Code ==

* The VPLAN code using [[:Category: VPLAN | VPLAN code]] can be found in: [[Diels-Alder Reaction Experimental Design (VPLAN)]]

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2016-02-01T12:47:56Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x^i,\ G^i,\ F^i,\ Tc^i,\ n_{a1}^i,\ n_{a2}^i,\ n_{a4}^i,\ c_{kat}^i,\ \vartheta(t)^i} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}^i(t) & = & f(x^i(t), u^i(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}^i(t) & = & f_x(x^i(t),u^i(t),p)G^i(t) \ + \ f_p(x^i(t),u^i(t),p) \\
\\
\dot{F}(t) & = & \sum_{i=1}^{4} w^i(t) (h^i_x(x^i(t),u^i(t),p)G^i(t))^T (h^i_x(x^i(t),u(t),p)G^i(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\
\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\
\vartheta_{up} + 273 & t \in [8,t_{end}]
\end{array} \right. \\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P \\
\dot{z}^i(t) & = & w^i(t) \\
z(0) & = & 0 \\
w^i(t) &\in& [0,1] \\
0 & \le & 4 - z^i(t_f). \\
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== Source Code ==

* The VPLAN code using [[:Category: VPLAN | VPLAN code]] can be found in: [[Diels-Alder Reaction Experimental Design (VPLAN)]]

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2016-02-01T12:47:38Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x^i,\ G^i,\ F^i,\ Tc^i,\ n_{a1}^i,\ n_{a2}^i,\ n_{a4}^i,\ c_{kat}^i,\ \vartheta(t)^i} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}^i(t) & = & f(x^i(t), u^i(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}^i(t) & = & f_x(x^i(t),u^i(t),p)G^i(t) \ + \ f_p(x^i(t),u^i(t),p) \\
\\
\dot{F}(t) & = & \sum_{i=1}^{4} w^i(t) (h^i_x(x^i(t),u^i(t),p)G^i(t))^T (h^i_x(x^i(t),u(t),p)G^i(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\
\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\
\vartheta_{up} + 273 & t \in [8,t_{end}]
\end{array} \right. \\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P \\
\dot{z}^i(t) & = & w^i(t) \\
z(0) & = & 0 \\
w^i(t) &\in& [0,1] \\
0 & \le & 4 - z^i(t_f) \\.
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== Source Code ==

* The VPLAN code using [[:Category: VPLAN | VPLAN code]] can be found in: [[Diels-Alder Reaction Experimental Design (VPLAN)]]

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2016-02-01T12:46:43Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x^i,\ G^i,\ F^i,\ Tc^i,\ n_{a1}^i,\ n_{a2}^i,\ n_{a4}^i,\ c_{kat}^i,\ \vartheta(t)^i} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}^i(t) & = & f(x^i(t), u^i(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}^i(t) & = & f_x(x^i(t),u^i(t),p)G^i(t) \ + \ f_p(x^i(t),u^i(t),p) \\
\\
\dot{F}(t) & = & \sum_{i=1}^{4} w^i(t) (h^i_x(x^i(t),u^i(t),p)G^i(t))^T (h^i_x(x^i(t),u(t),p)G^i(t)) \\
\\
\dot{z}^i(t) & = & w^i(t) \\
z(0) & = & 0 \\
w^i(t) &\in& [0,1] \\
0 & \ge & 4 - z^i(t_f) \\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\
\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\
\vartheta_{up} + 273 & t \in [8,t_{end}]
\end{array} \right. \\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== Source Code ==

* The VPLAN code using [[:Category: VPLAN | VPLAN code]] can be found in: [[Diels-Alder Reaction Experimental Design (VPLAN)]]

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design (VPLAN)

2016-02-01T12:36:10Z

FelixJost:

== VPLAN ==

Differential equations:

<source lang="fortran">

c RHS of the differential equations

subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), f(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

real*8 n1, n2, n3, n4
real*8 na1, na2, na4
real*8 fg, Temp, E , Rg , T1, Tc
real*8 r1, mR
real*8 kr1, kkat, Ckat, Ekat
real*8 k1, lambda
real*8 M1, M2, M3, M4
real*8 dm

c State variables

n1 = x(1)
n2 = x(2)
n3 = x(3)
n4 = x(4)

c Control variables

na1 = q(1)
na2 = q(2)
na4 = q(3)
Ckat = q(4)

c Control function

c DISCRETIZE1( Tc, rwh, iwh )

c Parameters

kr1 = p(1) * 1.0d-2
E = p(2) * 60000.0d+0
k1 = p(3) * 0.10d+0
Ekat = p(4) * 40000.0d0
lambda = p(5) * 0.25d+0

c Molar masses (in kg/mol)

M1 = 0.1362d+0
M2 = 0.09806d+0
M3 = M1 + M2
M4 = 0.236d+0

Temp = Tc + 273.0d+0
Rg = 8.314d+0
T1 = 293.0d+0

c Reaction rates

mR = n1*M1 + n2*M2 +n3*M3 + n4*M4

kkat = kr1 * dexp( -E/Rg * ( 1.0d+0/Temp - 1.0d+0/T1 ) )
& + k1 * dexp( -Ekat/Rg *( 1.0d+0/Temp - 1.0d+0/T1 ) )
& * Ckat * dexp( -lambda * t )

r1 = kkat * n1 * n2 / mR

f(1) = -r1
f(2) = -r1
f(3) = r1
f(4) = 0.0d0

end

</source>

Algebraic equations:

<source lang="fortran">

c Dummyfunction for RHS of algebraic equations

subroutine gfcn( t, x, g, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), g(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

iflag=0

end

</source>

Measurement function:

<source lang="fortran">

c Messfunktion

subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )
implicit none

real*8 t, x(*), h, p(*), q(*), rwh(*)
integer*4 iwh(*), iflag

real*8 M1, M2, M3, M4, mR

c Berechnung der Reaktormasse

M1 = 0.1362d+0
M2 = 0.09806d+0
M3 = M1 + M2
M4 = 0.236d+0

mR = M1*x(1) + M2*x(2) + M3*x(3) + M4*x(4)

c Messwert: Massenprozent

h = M3*x(3) * 100.0d+0/mR

iflag = 0

end

</source>

Standard deviation of measurement function:

<source lang="fortran">

c Standardabweichung der Messfunktion

subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )
implicit none

real*8 rwh(*)
integer*4 iwh(*), iflag

real*8 t, x(*), p(*), q(*)
real*8 s
real*8 h

s = 1.0d+0

iflag = 0

end

</source>

VPLAN specific experimental setup:
Experiment 1
<source lang="optimica">

[Flags]
switch=1

[Kosten]
costs=0 -1e+10 1e+10

[Integrationsintervall]
t0=0
tend=10

[Modellfunktionen]
ffcn=ffcn
gfcn=gfcn

[Zustandsvariablen]
yAnzahl=4
y1=n1 na1 -1e+10 1e+10
y2=n2 na2 -1e+10 1e+10
y3=n3 0 -1e+10 1e+10
y4=n4 na4 -1e+10 1e+10

zAnzahl=0

[Mehrzielknoten]
tAnzahl=0
t1=0
t2=0.33
t3=0.66
t4=1
t5=1.33
t6=1.66
t7=2
t8=2.33
t9=2.66
t10=3
t11=3.33
t12=3.66
t13=4
t14=4.33
t15=4.66
t16=5
t17=6
t18=7
t19=9
t20=10

[DynamischeNebenbedingungen]
bAnzahl=0

[GitterUeberpruefungNebenbedingungen]
tAnzahl=0

[Steuergroessen]
qAnzahl=4
q1=na1 1.0 0 10 0 -1
q2=na2 1.0 0 10 0 -1
q3=na4 2.0 0.4 9 0 -1
q4=Ckat 0.0 0 6 0 -1

[Steuerfunktionen]
uAnzahl=1
u1=Tc 3 20 100
u1tAnzahl=3
u1t0=t0
u1t1q=20 20 100 0 0 0
u1t1=2
u1t2q=20 20 100 0.0 -1e+10 1e+10
u1t2=8
u1t3q=20 20 100 0 0 0
u1t3=tend

[NebenbedingungenSteuergroessen]
cAnzahl=2
c1=cfcn1 1
c1bnd1=0.1 0.7
c2=cfcn2 1
c2bnd1=0.1 10

[Messverfahren]
mAnzahl=1
m1=mfcn1 1 0 1e+10 0
m1f1=mess3 sigma3 1
mminmaxges=0 6

[Messungen]
tAnzahl=20

t1=0.33
t1Anzahl=1
t1m1=mfcn1 0.3 1e-06 1
t1minmax=0 1e+10

t2=0.66
t2Anzahl=1
t2m1=mfcn1 0.3 1e-06 1
t2minmax=0 1e+10

t3=1
t3Anzahl=1
t3m1=mfcn1 0.3 1e-06 1
t3minmax=0 1e+10

t4=1.33
t4Anzahl=1
t4m1=mfcn1 0.3 1e-06 1
t4minmax=0 1e+10

t5=1.66
t5Anzahl=1
t5m1=mfcn1 0.3 1e-06 1
t5minmax=0 1e+10

t6=2
t6Anzahl=1
t6m1=mfcn1 0.3 1e-06 1
t6minmax=0 1e+10

t7=2.33
t7Anzahl=1
t7m1=mfcn1 0.3 1e-06 1
t7minmax=0 1e+10

t8=2.66
t8Anzahl=1
t8m1=mfcn1 0.3 1e-06 1
t8minmax=0 1e+10

t9=3
t9Anzahl=1
t9m1=mfcn1 0.3 1e-06 1
t9minmax=0 1e+10

t10=3.33
t10Anzahl=1
t10m1=mfcn1 0.3 1e-06 1
t10minmax=0 1e+10

t11=3.66
t11Anzahl=1
t11m1=mfcn1 0.3 1e-06 1
t11minmax=0 1e+10

t12=4
t12Anzahl=1
t12m1=mfcn1 0.3 1e-06 1
t12minmax=0 1e+10

t13=4.33
t13Anzahl=1
t13m1=mfcn1 0.3 1e-06 1
t13minmax=0 1e+10

t14=4.66
t14Anzahl=1
t14m1=mfcn1 0.3 1e-06 1
t14minmax=0 1e+10

t15=5
t15Anzahl=1
t15m1=mfcn1 0.3 1e-06 1
t15minmax=0 1e+10

t16=6
t16Anzahl=1
t16m1=mfcn1 0.3 1e-06 1
t16minmax=0 1e+10

t17=7
t17Anzahl=1
t17m1=mfcn1 0.3 1e-06 1
t17minmax=0 1e+10

t18=8
t18Anzahl=1
t18m1=mfcn1 0.3 1e-06 1
t18minmax=0 1e+10

t19=9
t19Anzahl=1
t19m1=mfcn1 0.3 1e-06 1
t19minmax=0 1e+10

t20=10
t20Anzahl=1
t20m1=mfcn1 0.3 1e-06 1
t20minmax=0 1e+10

[OptionenIntegration]
teps=1e-08
rtol=1e-08
atol=1e-07
stepsize=0.0001
maxorder=6
maxstepnumber=4000
minstepsize=-1
maxstepsize=-1
maxitNewton=3
noerrorcontrol=0
realworkspace=170000
integerworkspace=500
printlevel=0
mcnonlinearflag=0
mcDAEflag=0
mctol=1e-07
mcmaxit=50
mclinesearch=1
mcalpha0=1
rndmethod=-1
rndeps=1e-05
rndverbose=0

</source>

Experiment 2
<source lang="optimica">

[Flags]
switch=1

[Kosten]
costs=0 -1e+10 1e+10

[Integrationsintervall]
t0=0
tend=10

[Modellfunktionen]
ffcn=ffcn
gfcn=gfcn

[Zustandsvariablen]
yAnzahl=4
y1=n1 na1 -1e+10 1e+10
y2=n2 na2 -1e+10 1e+10
y3=n3 0 -1e+10 1e+10
y4=n4 na4 -1e+10 1e+10

zAnzahl=0

[Mehrzielknoten]
tAnzahl=0
t1=0
t2=0.33
t3=0.66
t4=1
t5=1.33
t6=1.66
t7=2
t8=2.33
t9=2.66
t10=3
t11=3.33
t12=3.66
t13=4
t14=4.33
t15=4.66
t16=5
t17=6
t18=7
t19=9
t20=10

[DynamischeNebenbedingungen]
bAnzahl=0

[GitterUeberpruefungNebenbedingungen]
tAnzahl=0

[Steuergroessen]
qAnzahl=4
q1=na1 1.0 0 10 0 -1
q2=na2 1.0 0 10 0 -1
q3=na4 2.0 0.4 9 0 -1
q4=Ckat 1.0 0 6 0 -1

[Steuerfunktionen]
uAnzahl=1
u1=Tc 3 20 100
u1tAnzahl=3
u1t0=t0
u1t1q=60 20 100 0 0 0 0
u1t1=2
u1t2q=60 20 100 0.0 -1e+10 1e+10
u1t2=8
u1t3q=60 20 100 0 0 0
u1t3=tend

[NebenbedingungenSteuergroessen]
cAnzahl=2
c1=cfcn1 1
c1bnd1=0.1 0.7
c2=cfcn2 1
c2bnd1=0.1 10

[Messverfahren]
mAnzahl=1
m1=mfcn1 1 0 1e+10 0
m1f1=mess3 sigma3 1
mminmaxges=0 6

[Messungen]
tAnzahl=20

t1=0.33
t1Anzahl=1
t1m1=mfcn1 0.3 1e-06 1
t1minmax=0 1e+10

t2=0.66
t2Anzahl=1
t2m1=mfcn1 0.3 1e-06 1
t2minmax=0 1e+10

t3=1
t3Anzahl=1
t3m1=mfcn1 0.3 1e-06 1
t3minmax=0 1e+10

t4=1.33
t4Anzahl=1
t4m1=mfcn1 0.3 1e-06 1
t4minmax=0 1e+10

t5=1.66
t5Anzahl=1
t5m1=mfcn1 0.3 1e-06 1
t5minmax=0 1e+10

t6=2
t6Anzahl=1
t6m1=mfcn1 0.3 1e-06 1
t6minmax=0 1e+10

t7=2.33
t7Anzahl=1
t7m1=mfcn1 0.3 1e-06 1
t7minmax=0 1e+10

t8=2.66
t8Anzahl=1
t8m1=mfcn1 0.3 1e-06 1
t8minmax=0 1e+10

t9=3
t9Anzahl=1
t9m1=mfcn1 0.3 1e-06 1
t9minmax=0 1e+10

t10=3.33
t10Anzahl=1
t10m1=mfcn1 0.3 1e-06 1
t10minmax=0 1e+10

t11=3.66
t11Anzahl=1
t11m1=mfcn1 0.3 1e-06 1
t11minmax=0 1e+10

t12=4
t12Anzahl=1
t12m1=mfcn1 0.3 1e-06 1
t12minmax=0 1e+10

t13=4.33
t13Anzahl=1
t13m1=mfcn1 0.3 1e-06 1
t13minmax=0 1e+10

t14=4.66
t14Anzahl=1
t14m1=mfcn1 0.3 1e-06 1
t14minmax=0 1e+10

t15=5
t15Anzahl=1
t15m1=mfcn1 0.3 1e-06 1
t15minmax=0 1e+10

t16=6
t16Anzahl=1
t16m1=mfcn1 0.3 1e-06 1
t16minmax=0 1e+10

t17=7
t17Anzahl=1
t17m1=mfcn1 0.3 1e-06 1
t17minmax=0 1e+10

t18=8
t18Anzahl=1
t18m1=mfcn1 0.3 1e-06 1
t18minmax=0 1e+10

t19=9
t19Anzahl=1
t19m1=mfcn1 0.3 1e-06 1
t19minmax=0 1e+10

t20=10
t20Anzahl=1
t20m1=mfcn1 0.3 1e-06 1
t20minmax=0 1e+10

[OptionenIntegration]
teps=1e-08
rtol=1e-08
atol=1e-07
stepsize=0.0001
maxorder=6
maxstepnumber=4000
minstepsize=-1
maxstepsize=-1
maxitNewton=3
noerrorcontrol=0
realworkspace=170000
integerworkspace=500
printlevel=0
mcnonlinearflag=0
mcDAEflag=0
mctol=1e-07
mcmaxit=50
mclinesearch=1
mcalpha0=1
rndmethod=-1
rndeps=1e-05
rndverbose=0

</source>

Experiment 3
<source lang="optimica">

[Flags]
switch=1

[Kosten]
costs=0 -1e+10 1e+10

[Integrationsintervall]
t0=0
tend=10

[Modellfunktionen]
ffcn=ffcn
gfcn=gfcn

[Zustandsvariablen]
yAnzahl=4
y1=n1 na1 -1e+10 1e+10
y2=n2 na2 -1e+10 1e+10
y3=n3 0 -1e+10 1e+10
y4=n4 na4 -1e+10 1e+10

zAnzahl=0

[Mehrzielknoten]
tAnzahl=0
t1=0
t2=0.33
t3=0.66
t4=1
t5=1.33
t6=1.66
t7=2
t8=2.33
t9=2.66
t10=3
t11=3.33
t12=3.66
t13=4
t14=4.33
t15=4.66
t16=5
t17=6
t18=7
t19=9
t20=10

[DynamischeNebenbedingungen]
bAnzahl=0

[GitterUeberpruefungNebenbedingungen]
tAnzahl=0

[Steuergroessen]
qAnzahl=4
q1=na1 1.0 0 10 0 -1
q2=na2 1.0 0 10 0 -1
q3=na4 2.0 0.4 9 0 -1
q4=Ckat 2.0 0 6 0 -1

[Steuerfunktionen]
uAnzahl=1
u1=Tc 3 20 100
u1tAnzahl=3
u1t0=t0
u1t1q=40 20 100 0 0 0 0
u1t1=2
u1t2q=40 20 100 0.0 -1e+10 1e+10
u1t2=8
u1t3q=40 20 100 0 0 0
u1t3=tend

[NebenbedingungenSteuergroessen]
cAnzahl=2
c1=cfcn1 1
c1bnd1=0.1 0.7
c2=cfcn2 1
c2bnd1=0.1 10

[Messverfahren]
mAnzahl=1
m1=mfcn1 1 0 1e+10 0
m1f1=mess3 sigma3 1
mminmaxges=0 6

[Messungen]
tAnzahl=20

t1=0.33
t1Anzahl=1
t1m1=mfcn1 0.3 1e-06 1
t1minmax=0 1e+10

t2=0.66
t2Anzahl=1
t2m1=mfcn1 0.3 1e-06 1
t2minmax=0 1e+10

t3=1
t3Anzahl=1
t3m1=mfcn1 0.3 1e-06 1
t3minmax=0 1e+10

t4=1.33
t4Anzahl=1
t4m1=mfcn1 0.3 1e-06 1
t4minmax=0 1e+10

t5=1.66
t5Anzahl=1
t5m1=mfcn1 0.3 1e-06 1
t5minmax=0 1e+10

t6=2
t6Anzahl=1
t6m1=mfcn1 0.3 1e-06 1
t6minmax=0 1e+10

t7=2.33
t7Anzahl=1
t7m1=mfcn1 0.3 1e-06 1
t7minmax=0 1e+10

t8=2.66
t8Anzahl=1
t8m1=mfcn1 0.3 1e-06 1
t8minmax=0 1e+10

t9=3
t9Anzahl=1
t9m1=mfcn1 0.3 1e-06 1
t9minmax=0 1e+10

t10=3.33
t10Anzahl=1
t10m1=mfcn1 0.3 1e-06 1
t10minmax=0 1e+10

t11=3.66
t11Anzahl=1
t11m1=mfcn1 0.3 1e-06 1
t11minmax=0 1e+10

t12=4
t12Anzahl=1
t12m1=mfcn1 0.3 1e-06 1
t12minmax=0 1e+10

t13=4.33
t13Anzahl=1
t13m1=mfcn1 0.3 1e-06 1
t13minmax=0 1e+10

t14=4.66
t14Anzahl=1
t14m1=mfcn1 0.3 1e-06 1
t14minmax=0 1e+10

t15=5
t15Anzahl=1
t15m1=mfcn1 0.3 1e-06 1
t15minmax=0 1e+10

t16=6
t16Anzahl=1
t16m1=mfcn1 0.3 1e-06 1
t16minmax=0 1e+10

t17=7
t17Anzahl=1
t17m1=mfcn1 0.3 1e-06 1
t17minmax=0 1e+10

t18=8
t18Anzahl=1
t18m1=mfcn1 0.3 1e-06 1
t18minmax=0 1e+10

t19=9
t19Anzahl=1
t19m1=mfcn1 0.3 1e-06 1
t19minmax=0 1e+10

t20=10
t20Anzahl=1
t20m1=mfcn1 0.3 1e-06 1
t20minmax=0 1e+10

[OptionenIntegration]
teps=1e-08
rtol=1e-08
atol=1e-07
stepsize=0.0001
maxorder=6
maxstepnumber=4000
minstepsize=-1
maxstepsize=-1
maxitNewton=3
noerrorcontrol=0
realworkspace=170000
integerworkspace=500
printlevel=0
mcnonlinearflag=0
mcDAEflag=0
mctol=1e-07
mcmaxit=50
mclinesearch=1
mcalpha0=1
rndmethod=-1
rndeps=1e-05
rndverbose=0

</source>

Experiment 4
<source lang="optimica">

[Flags]
switch=1

[Kosten]
costs=0 -1e+10 1e+10

[Integrationsintervall]
t0=0
tend=10

[Modellfunktionen]
ffcn=ffcn
gfcn=gfcn

[Zustandsvariablen]
yAnzahl=4
y1=n1 na1 -1e+10 1e+10
y2=n2 na2 -1e+10 1e+10
y3=n3 0 -1e+10 1e+10
y4=n4 na4 -1e+10 1e+10

zAnzahl=0

[Mehrzielknoten]
tAnzahl=0
t1=0
t2=0.33
t3=0.66
t4=1
t5=1.33
t6=1.66
t7=2
t8=2.33
t9=2.66
t10=3
t11=3.33
t12=3.66
t13=4
t14=4.33
t15=4.66
t16=5
t17=6
t18=7
t19=9
t20=10

[DynamischeNebenbedingungen]
bAnzahl=0

[GitterUeberpruefungNebenbedingungen]
tAnzahl=0

[Steuergroessen]
qAnzahl=4
q1=na1 1.0 0 10 0 -1
q2=na2 1.0 0 10 0 -1
q3=na4 2.0 0.4 9 0 -1
q4=Ckat 3.0 0 6 0 -1

[Steuerfunktionen]
uAnzahl=1
u1=Tc 3 20 100
u1tAnzahl=3
u1t0=t0
u1t1q=20 20 100 0 0 0 0
u1t1=2
u1t2q=20 20 100 0.0 -1e+10 1e+10
u1t2=8
u1t3q=20 20 100 0 0 0
u1t3=tend

[NebenbedingungenSteuergroessen]
cAnzahl=2
c1=cfcn1 1
c1bnd1=0.1 0.7
c2=cfcn2 1
c2bnd1=0.1 10

[Messverfahren]
mAnzahl=1
m1=mfcn1 1 0 1e+10 0
m1f1=mess3 sigma3 1
mminmaxges=0 6

[Messungen]
tAnzahl=20

t1=0.33
t1Anzahl=1
t1m1=mfcn1 0.3 1e-06 1
t1minmax=0 1e+10

t2=0.66
t2Anzahl=1
t2m1=mfcn1 0.3 1e-06 1
t2minmax=0 1e+10

t3=1
t3Anzahl=1
t3m1=mfcn1 0.3 1e-06 1
t3minmax=0 1e+10

t4=1.33
t4Anzahl=1
t4m1=mfcn1 0.3 1e-06 1
t4minmax=0 1e+10

t5=1.66
t5Anzahl=1
t5m1=mfcn1 0.3 1e-06 1
t5minmax=0 1e+10

t6=2
t6Anzahl=1
t6m1=mfcn1 0.3 1e-06 1
t6minmax=0 1e+10

t7=2.33
t7Anzahl=1
t7m1=mfcn1 0.3 1e-06 1
t7minmax=0 1e+10

t8=2.66
t8Anzahl=1
t8m1=mfcn1 0.3 1e-06 1
t8minmax=0 1e+10

t9=3
t9Anzahl=1
t9m1=mfcn1 0.3 1e-06 1
t9minmax=0 1e+10

t10=3.33
t10Anzahl=1
t10m1=mfcn1 0.3 1e-06 1
t10minmax=0 1e+10

t11=3.66
t11Anzahl=1
t11m1=mfcn1 0.3 1e-06 1
t11minmax=0 1e+10

t12=4
t12Anzahl=1
t12m1=mfcn1 0.3 1e-06 1
t12minmax=0 1e+10

t13=4.33
t13Anzahl=1
t13m1=mfcn1 0.3 1e-06 1
t13minmax=0 1e+10

t14=4.66
t14Anzahl=1
t14m1=mfcn1 0.3 1e-06 1
t14minmax=0 1e+10

t15=5
t15Anzahl=1
t15m1=mfcn1 0.3 1e-06 1
t15minmax=0 1e+10

t16=6
t16Anzahl=1
t16m1=mfcn1 0.3 1e-06 1
t16minmax=0 1e+10

t17=7
t17Anzahl=1
t17m1=mfcn1 0.3 1e-06 1
t17minmax=0 1e+10

t18=8
t18Anzahl=1
t18m1=mfcn1 0.3 1e-06 1
t18minmax=0 1e+10

t19=9
t19Anzahl=1
t19m1=mfcn1 0.3 1e-06 1
t19minmax=0 1e+10

t20=10
t20Anzahl=1
t20m1=mfcn1 0.3 1e-06 1
t20minmax=0 1e+10

[OptionenIntegration]
teps=1e-08
rtol=1e-08
atol=1e-07
stepsize=0.0001
maxorder=6
maxstepnumber=4000
minstepsize=-1
maxstepsize=-1
maxitNewton=3
noerrorcontrol=0
realworkspace=170000
integerworkspace=500
printlevel=0
mcnonlinearflag=0
mcDAEflag=0
mctol=1e-07
mcmaxit=50
mclinesearch=1
mcalpha0=1
rndmethod=-1
rndeps=1e-05
rndverbose=0

</source>

ini-file for running VPLAN:
<source lang="optimica">

[Aktion]
;aktion=Integration
;aktion=Simulationsumgebung
;aktion=Parameterschaetzung
;aktion=Versuchsplanung
;aktion=ObjectiveTest
;aktion=DerivativeTest
;aktion={CS}
aktion={CSVCS}

[Pfade]
problempath=vpbimolkat_origin
inpath=in
outpath=out
messpath=mess
plotpath=plot
fortranpath=fortran

[Parameter]
pAnzahl=5
p1=kr1 1 -1e+10 1e+10 0
p2=e 1 -1e+10 1e+10 0
p3=k1 1 -1e+10 1e+10 0
p4=ekat 1 -1e+10 1e+10 0
p5=lambda 1 -1e+10 1e+10 0

[Versuchsplan]
expAnzahl=4
exp1=exp1.ini exp1.ini
exp2=exp2.ini exp2.ini
exp3=exp3.ini exp3.ini
exp4=exp4.ini exp4.ini

[Guetekriterium]
Optimierungskriterium=A
AKriterium=-1
DKriterium=-1
EKriterium=-1
MKriterium=-1
covmat=covmat.m
jacmat=jacmat.m
status=undefiniert

[Residuum]
res=0

[Kosten]
costs=0 -1e+10 1e+10

[Messdatenfiles]
mess1=mess1.dat
mess2=mess2.dat
mess3=mess3.dat
mess4=mess4.dat

[Outputfiles]
out1=plot 0.05 integ.plt.1
out2=plot 0.05 integ.plt.2
out3=plot 0.05 integ.plt.3
out4=plot 0.05 integ.plt.4

[Residuenfiles]
rsd1=res1.txt
rsd2=res2.txt
rsd3=res3.txt
rsd4=res4.txt

[ExtensionFlags]
experimenttype=0
integrator=0
dmode=0
pdeFlag=0

[OptionenAllgemein]
visflag=0
messfileflag=0
seed=-1
numberofthreads=1
robustflag=0
epsmach=0
infinity=1e+10
epsilon=1e-08
conflevel=0.95
hrobust=1e-05
computesigma=0
exitonFPE=1
iniprecision=6
clipboardflag=0
printxi=0
printconstr=0
printcolorful=-1

[OptionenParameterschaetzung]
eps=0.001
itmax=50
cond=10000
condflag=1
boundcheck=0
startflag=0
index1=1e-08
fashort=0.8
fa0=0.01
farel=0.1
famax=1.0
realworkspace=10000
integerworkspace=1000
printlevel=2
method=3

[OptionenVersuchsplanung]
maxit=300
opttol=1e-06
funcprec=1e-07
linfeas=1e-07
nlinfeas=0.01
maxitQP=300
maxitgesQP=10000
opttolQP=1e-06
pivottolQP=3.7e-11
steplimitLS=2
tolLS=0.9
crashtol=0.0001
elasticweight=100
superbasics=1
scaling=1
sconstraints=0
realworkspace=300000
integerworkspace=300000
charworkspace=500
printlevel=10
method=1

</source>

Diels-Alder Reaction Experimental Design (VPLAN)

2016-02-01T12:32:17Z

FelixJost: /* VPLAN */

== VPLAN ==

Differential equations:

<source lang="fortran">

c RHS of the differential equations

subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), f(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

real*8 n1, n2, n3, n4
real*8 na1, na2, na4
real*8 fg, Temp, E , Rg , T1, Tc
real*8 r1, mR
real*8 kr1, kkat, Ckat, Ekat
real*8 k1, lambda
real*8 M1, M2, M3, M4
real*8 dm

c State variables

n1 = x(1)
n2 = x(2)
n3 = x(3)
n4 = x(4)

c Control variables

na1 = q(1)
na2 = q(2)
na4 = q(3)
Ckat = q(4)

c Control function

c DISCRETIZE1( Tc, rwh, iwh )

c Parameters

kr1 = p(1) * 1.0d-2
E = p(2) * 60000.0d+0
k1 = p(3) * 0.10d+0
Ekat = p(4) * 40000.0d0
lambda = p(5) * 0.25d+0

c Molar masses (in kg/mol)

M1 = 0.1362d+0
M2 = 0.09806d+0
M3 = M1 + M2
M4 = 0.236d+0

Temp = Tc + 273.0d+0
Rg = 8.314d+0
T1 = 293.0d+0

c Reaction rates

mR = n1*M1 + n2*M2 +n3*M3 + n4*M4

kkat = kr1 * dexp( -E/Rg * ( 1.0d+0/Temp - 1.0d+0/T1 ) )
& + k1 * dexp( -Ekat/Rg *( 1.0d+0/Temp - 1.0d+0/T1 ) )
& * Ckat * dexp( -lambda * t )

r1 = kkat * n1 * n2 / mR

f(1) = -r1
f(2) = -r1
f(3) = r1
f(4) = 0.0d0

end

</source>

Algebraic equations:

<source lang="fortran">

c Dummyfunction for RHS of algebraic equations

subroutine gfcn( t, x, g, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), g(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

iflag=0

end

</source>

Measurement function:

<source lang="fortran">

c Messfunktion

subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )
implicit none

real*8 t, x(*), h, p(*), q(*), rwh(*)
integer*4 iwh(*), iflag

real*8 M1, M2, M3, M4, mR

c Berechnung der Reaktormasse

M1 = 0.1362d+0
M2 = 0.09806d+0
M3 = M1 + M2
M4 = 0.236d+0

mR = M1*x(1) + M2*x(2) + M3*x(3) + M4*x(4)

c Messwert: Massenprozent

h = M3*x(3) * 100.0d+0/mR

iflag = 0

end

</source>

Standard deviation of measurement function:

<source lang="fortran">

c Standardabweichung der Messfunktion

subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )
implicit none

real*8 rwh(*)
integer*4 iwh(*), iflag

real*8 t, x(*), p(*), q(*)
real*8 s
real*8 h

s = 1.0d+0

iflag = 0

end

</source>

VPLAN specific experimental setup:
Experiment 1
<source lang="optimica">

[Flags]
switch=1

[Kosten]
costs=0 -1e+10 1e+10

[Integrationsintervall]
t0=0
tend=10

[Modellfunktionen]
ffcn=ffcn
gfcn=gfcn

[Zustandsvariablen]
yAnzahl=4
y1=n1 na1 -1e+10 1e+10
y2=n2 na2 -1e+10 1e+10
y3=n3 0 -1e+10 1e+10
y4=n4 na4 -1e+10 1e+10

zAnzahl=0

[Mehrzielknoten]
tAnzahl=0
t1=0
t2=0.33
t3=0.66
t4=1
t5=1.33
t6=1.66
t7=2
t8=2.33
t9=2.66
t10=3
t11=3.33
t12=3.66
t13=4
t14=4.33
t15=4.66
t16=5
t17=6
t18=7
t19=9
t20=10

[DynamischeNebenbedingungen]
bAnzahl=0

[GitterUeberpruefungNebenbedingungen]
tAnzahl=0

[Steuergroessen]
qAnzahl=4
q1=na1 1.0 0 10 0 -1
q2=na2 1.0 0 10 0 -1
q3=na4 2.0 0.4 9 0 -1
q4=Ckat 0.0 0 6 0 -1

[Steuerfunktionen]
uAnzahl=1
u1=Tc 3 20 100
u1tAnzahl=3
u1t0=t0
u1t1q=20 20 100 0 0 0
u1t1=2
u1t2q=20 20 100 0.0 -1e+10 1e+10
u1t2=8
u1t3q=20 20 100 0 0 0
u1t3=tend

[NebenbedingungenSteuergroessen]
cAnzahl=2
c1=cfcn1 1
c1bnd1=0.1 0.7
c2=cfcn2 1
c2bnd1=0.1 10

[Messverfahren]
mAnzahl=1
m1=mfcn1 1 0 1e+10 0
m1f1=mess3 sigma3 1
mminmaxges=0 6

[Messungen]
tAnzahl=20

t1=0.33
t1Anzahl=1
t1m1=mfcn1 0.3 1e-06 1
t1minmax=0 1e+10

t2=0.66
t2Anzahl=1
t2m1=mfcn1 0.3 1e-06 1
t2minmax=0 1e+10

t3=1
t3Anzahl=1
t3m1=mfcn1 0.3 1e-06 1
t3minmax=0 1e+10

t4=1.33
t4Anzahl=1
t4m1=mfcn1 0.3 1e-06 1
t4minmax=0 1e+10

t5=1.66
t5Anzahl=1
t5m1=mfcn1 0.3 1e-06 1
t5minmax=0 1e+10

t6=2
t6Anzahl=1
t6m1=mfcn1 0.3 1e-06 1
t6minmax=0 1e+10

t7=2.33
t7Anzahl=1
t7m1=mfcn1 0.3 1e-06 1
t7minmax=0 1e+10

t8=2.66
t8Anzahl=1
t8m1=mfcn1 0.3 1e-06 1
t8minmax=0 1e+10

t9=3
t9Anzahl=1
t9m1=mfcn1 0.3 1e-06 1
t9minmax=0 1e+10

t10=3.33
t10Anzahl=1
t10m1=mfcn1 0.3 1e-06 1
t10minmax=0 1e+10

t11=3.66
t11Anzahl=1
t11m1=mfcn1 0.3 1e-06 1
t11minmax=0 1e+10

t12=4
t12Anzahl=1
t12m1=mfcn1 0.3 1e-06 1
t12minmax=0 1e+10

t13=4.33
t13Anzahl=1
t13m1=mfcn1 0.3 1e-06 1
t13minmax=0 1e+10

t14=4.66
t14Anzahl=1
t14m1=mfcn1 0.3 1e-06 1
t14minmax=0 1e+10

t15=5
t15Anzahl=1
t15m1=mfcn1 0.3 1e-06 1
t15minmax=0 1e+10

t16=6
t16Anzahl=1
t16m1=mfcn1 0.3 1e-06 1
t16minmax=0 1e+10

t17=7
t17Anzahl=1
t17m1=mfcn1 0.3 1e-06 1
t17minmax=0 1e+10

t18=8
t18Anzahl=1
t18m1=mfcn1 0.3 1e-06 1
t18minmax=0 1e+10

t19=9
t19Anzahl=1
t19m1=mfcn1 0.3 1e-06 1
t19minmax=0 1e+10

t20=10
t20Anzahl=1
t20m1=mfcn1 0.3 1e-06 1
t20minmax=0 1e+10

[OptionenIntegration]
teps=1e-08
rtol=1e-08
atol=1e-07
stepsize=0.0001
maxorder=6
maxstepnumber=4000
minstepsize=-1
maxstepsize=-1
maxitNewton=3
noerrorcontrol=0
realworkspace=170000
integerworkspace=500
printlevel=0
mcnonlinearflag=0
mcDAEflag=0
mctol=1e-07
mcmaxit=50
mclinesearch=1
mcalpha0=1
rndmethod=-1
rndeps=1e-05
rndverbose=0

</source>

Experiment 2
<source lang="optimica">

[Flags]
switch=1

[Kosten]
costs=0 -1e+10 1e+10

[Integrationsintervall]
t0=0
tend=10

[Modellfunktionen]
ffcn=ffcn
gfcn=gfcn

[Zustandsvariablen]
yAnzahl=4
y1=n1 na1 -1e+10 1e+10
y2=n2 na2 -1e+10 1e+10
y3=n3 0 -1e+10 1e+10
y4=n4 na4 -1e+10 1e+10

zAnzahl=0

[Mehrzielknoten]
tAnzahl=0
t1=0
t2=0.33
t3=0.66
t4=1
t5=1.33
t6=1.66
t7=2
t8=2.33
t9=2.66
t10=3
t11=3.33
t12=3.66
t13=4
t14=4.33
t15=4.66
t16=5
t17=6
t18=7
t19=9
t20=10

[DynamischeNebenbedingungen]
bAnzahl=0

[GitterUeberpruefungNebenbedingungen]
tAnzahl=0

[Steuergroessen]
qAnzahl=4
q1=na1 1.0 0 10 0 -1
q2=na2 1.0 0 10 0 -1
q3=na4 2.0 0.4 9 0 -1
q4=Ckat 1.0 0 6 0 -1

[Steuerfunktionen]
uAnzahl=1
u1=Tc 3 20 100
u1tAnzahl=3
u1t0=t0
u1t1q=60 20 100 0 0 0 0
u1t1=2
u1t2q=60 20 100 0.0 -1e+10 1e+10
u1t2=8
u1t3q=60 20 100 0 0 0
u1t3=tend

[NebenbedingungenSteuergroessen]
cAnzahl=2
c1=cfcn1 1
c1bnd1=0.1 0.7
c2=cfcn2 1
c2bnd1=0.1 10

[Messverfahren]
mAnzahl=1
m1=mfcn1 1 0 1e+10 0
m1f1=mess3 sigma3 1
mminmaxges=0 6

[Messungen]
tAnzahl=20

t1=0.33
t1Anzahl=1
t1m1=mfcn1 0.3 1e-06 1
t1minmax=0 1e+10

t2=0.66
t2Anzahl=1
t2m1=mfcn1 0.3 1e-06 1
t2minmax=0 1e+10

t3=1
t3Anzahl=1
t3m1=mfcn1 0.3 1e-06 1
t3minmax=0 1e+10

t4=1.33
t4Anzahl=1
t4m1=mfcn1 0.3 1e-06 1
t4minmax=0 1e+10

t5=1.66
t5Anzahl=1
t5m1=mfcn1 0.3 1e-06 1
t5minmax=0 1e+10

t6=2
t6Anzahl=1
t6m1=mfcn1 0.3 1e-06 1
t6minmax=0 1e+10

t7=2.33
t7Anzahl=1
t7m1=mfcn1 0.3 1e-06 1
t7minmax=0 1e+10

t8=2.66
t8Anzahl=1
t8m1=mfcn1 0.3 1e-06 1
t8minmax=0 1e+10

t9=3
t9Anzahl=1
t9m1=mfcn1 0.3 1e-06 1
t9minmax=0 1e+10

t10=3.33
t10Anzahl=1
t10m1=mfcn1 0.3 1e-06 1
t10minmax=0 1e+10

t11=3.66
t11Anzahl=1
t11m1=mfcn1 0.3 1e-06 1
t11minmax=0 1e+10

t12=4
t12Anzahl=1
t12m1=mfcn1 0.3 1e-06 1
t12minmax=0 1e+10

t13=4.33
t13Anzahl=1
t13m1=mfcn1 0.3 1e-06 1
t13minmax=0 1e+10

t14=4.66
t14Anzahl=1
t14m1=mfcn1 0.3 1e-06 1
t14minmax=0 1e+10

t15=5
t15Anzahl=1
t15m1=mfcn1 0.3 1e-06 1
t15minmax=0 1e+10

t16=6
t16Anzahl=1
t16m1=mfcn1 0.3 1e-06 1
t16minmax=0 1e+10

t17=7
t17Anzahl=1
t17m1=mfcn1 0.3 1e-06 1
t17minmax=0 1e+10

t18=8
t18Anzahl=1
t18m1=mfcn1 0.3 1e-06 1
t18minmax=0 1e+10

t19=9
t19Anzahl=1
t19m1=mfcn1 0.3 1e-06 1
t19minmax=0 1e+10

t20=10
t20Anzahl=1
t20m1=mfcn1 0.3 1e-06 1
t20minmax=0 1e+10

[OptionenIntegration]
teps=1e-08
rtol=1e-08
atol=1e-07
stepsize=0.0001
maxorder=6
maxstepnumber=4000
minstepsize=-1
maxstepsize=-1
maxitNewton=3
noerrorcontrol=0
realworkspace=170000
integerworkspace=500
printlevel=0
mcnonlinearflag=0
mcDAEflag=0
mctol=1e-07
mcmaxit=50
mclinesearch=1
mcalpha0=1
rndmethod=-1
rndeps=1e-05
rndverbose=0

</source>

Experiment 3
<source lang="optimica">

</source>

[Flags]
switch=1

[Kosten]
costs=0 -1e+10 1e+10

[Integrationsintervall]
t0=0
tend=10

[Modellfunktionen]
ffcn=ffcn
gfcn=gfcn

[Zustandsvariablen]
yAnzahl=4
y1=n1 na1 -1e+10 1e+10
y2=n2 na2 -1e+10 1e+10
y3=n3 0 -1e+10 1e+10
y4=n4 na4 -1e+10 1e+10

zAnzahl=0

[Mehrzielknoten]
tAnzahl=0
t1=0
t2=0.33
t3=0.66
t4=1
t5=1.33
t6=1.66
t7=2
t8=2.33
t9=2.66
t10=3
t11=3.33
t12=3.66
t13=4
t14=4.33
t15=4.66
t16=5
t17=6
t18=7
t19=9
t20=10

[DynamischeNebenbedingungen]
bAnzahl=0

[GitterUeberpruefungNebenbedingungen]
tAnzahl=0

[Steuergroessen]
qAnzahl=4
q1=na1 1.0 0 10 0 -1
q2=na2 1.0 0 10 0 -1
q3=na4 2.0 0.4 9 0 -1
q4=Ckat 2.0 0 6 0 -1

[Steuerfunktionen]
uAnzahl=1
u1=Tc 3 20 100
u1tAnzahl=3
u1t0=t0
u1t1q=40 20 100 0 0 0 0
u1t1=2
u1t2q=40 20 100 0.0 -1e+10 1e+10
u1t2=8
u1t3q=40 20 100 0 0 0
u1t3=tend

[NebenbedingungenSteuergroessen]
cAnzahl=2
c1=cfcn1 1
c1bnd1=0.1 0.7
c2=cfcn2 1
c2bnd1=0.1 10

[Messverfahren]
mAnzahl=1
m1=mfcn1 1 0 1e+10 0
m1f1=mess3 sigma3 1
mminmaxges=0 6

[Messungen]
tAnzahl=20

t1=0.33
t1Anzahl=1
t1m1=mfcn1 0.3 1e-06 1
t1minmax=0 1e+10

t2=0.66
t2Anzahl=1
t2m1=mfcn1 0.3 1e-06 1
t2minmax=0 1e+10

t3=1
t3Anzahl=1
t3m1=mfcn1 0.3 1e-06 1
t3minmax=0 1e+10

t4=1.33
t4Anzahl=1
t4m1=mfcn1 0.3 1e-06 1
t4minmax=0 1e+10

t5=1.66
t5Anzahl=1
t5m1=mfcn1 0.3 1e-06 1
t5minmax=0 1e+10

t6=2
t6Anzahl=1
t6m1=mfcn1 0.3 1e-06 1
t6minmax=0 1e+10

t7=2.33
t7Anzahl=1
t7m1=mfcn1 0.3 1e-06 1
t7minmax=0 1e+10

t8=2.66
t8Anzahl=1
t8m1=mfcn1 0.3 1e-06 1
t8minmax=0 1e+10

t9=3
t9Anzahl=1
t9m1=mfcn1 0.3 1e-06 1
t9minmax=0 1e+10

t10=3.33
t10Anzahl=1
t10m1=mfcn1 0.3 1e-06 1
t10minmax=0 1e+10

t11=3.66
t11Anzahl=1
t11m1=mfcn1 0.3 1e-06 1
t11minmax=0 1e+10

t12=4
t12Anzahl=1
t12m1=mfcn1 0.3 1e-06 1
t12minmax=0 1e+10

t13=4.33
t13Anzahl=1
t13m1=mfcn1 0.3 1e-06 1
t13minmax=0 1e+10

t14=4.66
t14Anzahl=1
t14m1=mfcn1 0.3 1e-06 1
t14minmax=0 1e+10

t15=5
t15Anzahl=1
t15m1=mfcn1 0.3 1e-06 1
t15minmax=0 1e+10

t16=6
t16Anzahl=1
t16m1=mfcn1 0.3 1e-06 1
t16minmax=0 1e+10

t17=7
t17Anzahl=1
t17m1=mfcn1 0.3 1e-06 1
t17minmax=0 1e+10

t18=8
t18Anzahl=1
t18m1=mfcn1 0.3 1e-06 1
t18minmax=0 1e+10

t19=9
t19Anzahl=1
t19m1=mfcn1 0.3 1e-06 1
t19minmax=0 1e+10

t20=10
t20Anzahl=1
t20m1=mfcn1 0.3 1e-06 1
t20minmax=0 1e+10

[OptionenIntegration]
teps=1e-08
rtol=1e-08
atol=1e-07
stepsize=0.0001
maxorder=6
maxstepnumber=4000
minstepsize=-1
maxstepsize=-1
maxitNewton=3
noerrorcontrol=0
realworkspace=170000
integerworkspace=500
printlevel=0
mcnonlinearflag=0
mcDAEflag=0
mctol=1e-07
mcmaxit=50
mclinesearch=1
mcalpha0=1
rndmethod=-1
rndeps=1e-05
rndverbose=0

</source>

Experiment 4
<source lang="optimica">

[Flags]
switch=1

[Kosten]
costs=0 -1e+10 1e+10

[Integrationsintervall]
t0=0
tend=10

[Modellfunktionen]
ffcn=ffcn
gfcn=gfcn

[Zustandsvariablen]
yAnzahl=4
y1=n1 na1 -1e+10 1e+10
y2=n2 na2 -1e+10 1e+10
y3=n3 0 -1e+10 1e+10
y4=n4 na4 -1e+10 1e+10

zAnzahl=0

[Mehrzielknoten]
tAnzahl=0
t1=0
t2=0.33
t3=0.66
t4=1
t5=1.33
t6=1.66
t7=2
t8=2.33
t9=2.66
t10=3
t11=3.33
t12=3.66
t13=4
t14=4.33
t15=4.66
t16=5
t17=6
t18=7
t19=9
t20=10

[DynamischeNebenbedingungen]
bAnzahl=0

[GitterUeberpruefungNebenbedingungen]
tAnzahl=0

[Steuergroessen]
qAnzahl=4
q1=na1 1.0 0 10 0 -1
q2=na2 1.0 0 10 0 -1
q3=na4 2.0 0.4 9 0 -1
q4=Ckat 3.0 0 6 0 -1

[Steuerfunktionen]
uAnzahl=1
u1=Tc 3 20 100
u1tAnzahl=3
u1t0=t0
u1t1q=20 20 100 0 0 0 0
u1t1=2
u1t2q=20 20 100 0.0 -1e+10 1e+10
u1t2=8
u1t3q=20 20 100 0 0 0
u1t3=tend

[NebenbedingungenSteuergroessen]
cAnzahl=2
c1=cfcn1 1
c1bnd1=0.1 0.7
c2=cfcn2 1
c2bnd1=0.1 10

[Messverfahren]
mAnzahl=1
m1=mfcn1 1 0 1e+10 0
m1f1=mess3 sigma3 1
mminmaxges=0 6

[Messungen]
tAnzahl=20

t1=0.33
t1Anzahl=1
t1m1=mfcn1 0.3 1e-06 1
t1minmax=0 1e+10

t2=0.66
t2Anzahl=1
t2m1=mfcn1 0.3 1e-06 1
t2minmax=0 1e+10

t3=1
t3Anzahl=1
t3m1=mfcn1 0.3 1e-06 1
t3minmax=0 1e+10

t4=1.33
t4Anzahl=1
t4m1=mfcn1 0.3 1e-06 1
t4minmax=0 1e+10

t5=1.66
t5Anzahl=1
t5m1=mfcn1 0.3 1e-06 1
t5minmax=0 1e+10

t6=2
t6Anzahl=1
t6m1=mfcn1 0.3 1e-06 1
t6minmax=0 1e+10

t7=2.33
t7Anzahl=1
t7m1=mfcn1 0.3 1e-06 1
t7minmax=0 1e+10

t8=2.66
t8Anzahl=1
t8m1=mfcn1 0.3 1e-06 1
t8minmax=0 1e+10

t9=3
t9Anzahl=1
t9m1=mfcn1 0.3 1e-06 1
t9minmax=0 1e+10

t10=3.33
t10Anzahl=1
t10m1=mfcn1 0.3 1e-06 1
t10minmax=0 1e+10

t11=3.66
t11Anzahl=1
t11m1=mfcn1 0.3 1e-06 1
t11minmax=0 1e+10

t12=4
t12Anzahl=1
t12m1=mfcn1 0.3 1e-06 1
t12minmax=0 1e+10

t13=4.33
t13Anzahl=1
t13m1=mfcn1 0.3 1e-06 1
t13minmax=0 1e+10

t14=4.66
t14Anzahl=1
t14m1=mfcn1 0.3 1e-06 1
t14minmax=0 1e+10

t15=5
t15Anzahl=1
t15m1=mfcn1 0.3 1e-06 1
t15minmax=0 1e+10

t16=6
t16Anzahl=1
t16m1=mfcn1 0.3 1e-06 1
t16minmax=0 1e+10

t17=7
t17Anzahl=1
t17m1=mfcn1 0.3 1e-06 1
t17minmax=0 1e+10

t18=8
t18Anzahl=1
t18m1=mfcn1 0.3 1e-06 1
t18minmax=0 1e+10

t19=9
t19Anzahl=1
t19m1=mfcn1 0.3 1e-06 1
t19minmax=0 1e+10

t20=10
t20Anzahl=1
t20m1=mfcn1 0.3 1e-06 1
t20minmax=0 1e+10

[OptionenIntegration]
teps=1e-08
rtol=1e-08
atol=1e-07
stepsize=0.0001
maxorder=6
maxstepnumber=4000
minstepsize=-1
maxstepsize=-1
maxitNewton=3
noerrorcontrol=0
realworkspace=170000
integerworkspace=500
printlevel=0
mcnonlinearflag=0
mcDAEflag=0
mctol=1e-07
mcmaxit=50
mclinesearch=1
mcalpha0=1
rndmethod=-1
rndeps=1e-05
rndverbose=0

</source>

ini-file for running VPLAN:
<source lang="optimica">

[Aktion]
;aktion=Integration
;aktion=Simulationsumgebung
;aktion=Parameterschaetzung
;aktion=Versuchsplanung
;aktion=ObjectiveTest
;aktion=DerivativeTest
;aktion={CS}
aktion={CSVCS}

[Pfade]
problempath=vpbimolkat_origin
inpath=in
outpath=out
messpath=mess
plotpath=plot
fortranpath=fortran

[Parameter]
pAnzahl=5
p1=kr1 1 -1e+10 1e+10 0
p2=e 1 -1e+10 1e+10 0
p3=k1 1 -1e+10 1e+10 0
p4=ekat 1 -1e+10 1e+10 0
p5=lambda 1 -1e+10 1e+10 0

[Versuchsplan]
expAnzahl=4
exp1=exp1.ini exp1.ini
exp2=exp2.ini exp2.ini
exp3=exp3.ini exp3.ini
exp4=exp4.ini exp4.ini

[Guetekriterium]
Optimierungskriterium=A
AKriterium=-1
DKriterium=-1
EKriterium=-1
MKriterium=-1
covmat=covmat.m
jacmat=jacmat.m
status=undefiniert

[Residuum]
res=0

[Kosten]
costs=0 -1e+10 1e+10

[Messdatenfiles]
mess1=mess1.dat
mess2=mess2.dat
mess3=mess3.dat
mess4=mess4.dat

[Outputfiles]
out1=plot 0.05 integ.plt.1
out2=plot 0.05 integ.plt.2
out3=plot 0.05 integ.plt.3
out4=plot 0.05 integ.plt.4

[Residuenfiles]
rsd1=res1.txt
rsd2=res2.txt
rsd3=res3.txt
rsd4=res4.txt

[ExtensionFlags]
experimenttype=0
integrator=0
dmode=0
pdeFlag=0

[OptionenAllgemein]
visflag=0
messfileflag=0
seed=-1
numberofthreads=1
robustflag=0
epsmach=0
infinity=1e+10
epsilon=1e-08
conflevel=0.95
hrobust=1e-05
computesigma=0
exitonFPE=1
iniprecision=6
clipboardflag=0
printxi=0
printconstr=0
printcolorful=-1

[OptionenParameterschaetzung]
eps=0.001
itmax=50
cond=10000
condflag=1
boundcheck=0
startflag=0
index1=1e-08
fashort=0.8
fa0=0.01
farel=0.1
famax=1.0
realworkspace=10000
integerworkspace=1000
printlevel=2
method=3

[OptionenVersuchsplanung]
maxit=300
opttol=1e-06
funcprec=1e-07
linfeas=1e-07
nlinfeas=0.01
maxitQP=300
maxitgesQP=10000
opttolQP=1e-06
pivottolQP=3.7e-11
steplimitLS=2
tolLS=0.9
crashtol=0.0001
elasticweight=100
superbasics=1
scaling=1
sconstraints=0
realworkspace=300000
integerworkspace=300000
charworkspace=500
printlevel=10
method=1

</source>

Diels-Alder Reaction Experimental Design (VPLAN)

2016-02-01T12:30:58Z

FelixJost: /* VPLAN */

== VPLAN ==

Differential equations:

<source lang="fortran">

c RHS of the differential equations

subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), f(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

real*8 n1, n2, n3, n4
real*8 na1, na2, na4
real*8 fg, Temp, E , Rg , T1, Tc
real*8 r1, mR
real*8 kr1, kkat, Ckat, Ekat
real*8 k1, lambda
real*8 M1, M2, M3, M4
real*8 dm

c State variables

n1 = x(1)
n2 = x(2)
n3 = x(3)
n4 = x(4)

c Control variables

na1 = q(1)
na2 = q(2)
na4 = q(3)
Ckat = q(4)

c Control function

c DISCRETIZE1( Tc, rwh, iwh )

c Parameters

kr1 = p(1) * 1.0d-2
E = p(2) * 60000.0d+0
k1 = p(3) * 0.10d+0
Ekat = p(4) * 40000.0d0
lambda = p(5) * 0.25d+0

c Molar masses (in kg/mol)

M1 = 0.1362d+0
M2 = 0.09806d+0
M3 = M1 + M2
M4 = 0.236d+0

Temp = Tc + 273.0d+0
Rg = 8.314d+0
T1 = 293.0d+0

c Reaction rates

mR = n1*M1 + n2*M2 +n3*M3 + n4*M4

kkat = kr1 * dexp( -E/Rg * ( 1.0d+0/Temp - 1.0d+0/T1 ) )
& + k1 * dexp( -Ekat/Rg *( 1.0d+0/Temp - 1.0d+0/T1 ) )
& * Ckat * dexp( -lambda * t )

r1 = kkat * n1 * n2 / mR

f(1) = -r1
f(2) = -r1
f(3) = r1
f(4) = 0.0d0

end

</source>

Algebraic equations:

<source lang="fortran">

c Dummyfunction for RHS of algebraic equations

subroutine gfcn( t, x, g, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), g(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

iflag=0

end

</source>

Measurement function:

<source lang="fortran">

c Messfunktion

subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )
implicit none

real*8 t, x(*), h, p(*), q(*), rwh(*)
integer*4 iwh(*), iflag

real*8 M1, M2, M3, M4, mR

c Berechnung der Reaktormasse

M1 = 0.1362d+0
M2 = 0.09806d+0
M3 = M1 + M2
M4 = 0.236d+0

mR = M1*x(1) + M2*x(2) + M3*x(3) + M4*x(4)

c Messwert: Massenprozent

h = M3*x(3) * 100.0d+0/mR

iflag = 0

end

</source>

Standard deviation of measurement function:

<source lang="fortran">

c Standardabweichung der Messfunktion

subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )
implicit none

real*8 rwh(*)
integer*4 iwh(*), iflag

real*8 t, x(*), p(*), q(*)
real*8 s
real*8 h

s = 1.0d+0

iflag = 0

end

</source>

VPLAN specific experimental setup:
Experiment 1
<source lang="optimica">

[Flags]
switch=1

[Kosten]
costs=0 -1e+10 1e+10

[Integrationsintervall]
t0=0
tend=10

[Modellfunktionen]
ffcn=ffcn
gfcn=gfcn

[Zustandsvariablen]
yAnzahl=4
y1=n1 na1 -1e+10 1e+10
y2=n2 na2 -1e+10 1e+10
y3=n3 0 -1e+10 1e+10
y4=n4 na4 -1e+10 1e+10

zAnzahl=0

[Mehrzielknoten]
tAnzahl=0
t1=0
t2=0.33
t3=0.66
t4=1
t5=1.33
t6=1.66
t7=2
t8=2.33
t9=2.66
t10=3
t11=3.33
t12=3.66
t13=4
t14=4.33
t15=4.66
t16=5
t17=6
t18=7
t19=9
t20=10

[DynamischeNebenbedingungen]
bAnzahl=0

[GitterUeberpruefungNebenbedingungen]
tAnzahl=0

[Steuergroessen]
qAnzahl=4
q1=na1 1.0 0 10 0 -1
q2=na2 1.0 0 10 0 -1
q3=na4 2.0 0.4 9 0 -1
q4=Ckat 0.0 0 6 0 -1

[Steuerfunktionen]
uAnzahl=1
u1=Tc 3 20 100
u1tAnzahl=3
u1t0=t0
u1t1q=20 20 100 0 0 0
u1t1=2
u1t2q=20 20 100 0.0 -1e+10 1e+10
u1t2=8
u1t3q=20 20 100 0 0 0
u1t3=tend

[NebenbedingungenSteuergroessen]
cAnzahl=2
c1=cfcn1 1
c1bnd1=0.1 0.7
c2=cfcn2 1
c2bnd1=0.1 10

[Messverfahren]
mAnzahl=1
m1=mfcn1 1 0 1e+10 0
m1f1=mess3 sigma3 1
mminmaxges=0 6

[Messungen]
tAnzahl=20

t1=0.33
t1Anzahl=1
t1m1=mfcn1 0.3 1e-06 1
t1minmax=0 1e+10

t2=0.66
t2Anzahl=1
t2m1=mfcn1 0.3 1e-06 1
t2minmax=0 1e+10

t3=1
t3Anzahl=1
t3m1=mfcn1 0.3 1e-06 1
t3minmax=0 1e+10

t4=1.33
t4Anzahl=1
t4m1=mfcn1 0.3 1e-06 1
t4minmax=0 1e+10

t5=1.66
t5Anzahl=1
t5m1=mfcn1 0.3 1e-06 1
t5minmax=0 1e+10

t6=2
t6Anzahl=1
t6m1=mfcn1 0.3 1e-06 1
t6minmax=0 1e+10

t7=2.33
t7Anzahl=1
t7m1=mfcn1 0.3 1e-06 1
t7minmax=0 1e+10

t8=2.66
t8Anzahl=1
t8m1=mfcn1 0.3 1e-06 1
t8minmax=0 1e+10

t9=3
t9Anzahl=1
t9m1=mfcn1 0.3 1e-06 1
t9minmax=0 1e+10

t10=3.33
t10Anzahl=1
t10m1=mfcn1 0.3 1e-06 1
t10minmax=0 1e+10

t11=3.66
t11Anzahl=1
t11m1=mfcn1 0.3 1e-06 1
t11minmax=0 1e+10

t12=4
t12Anzahl=1
t12m1=mfcn1 0.3 1e-06 1
t12minmax=0 1e+10

t13=4.33
t13Anzahl=1
t13m1=mfcn1 0.3 1e-06 1
t13minmax=0 1e+10

t14=4.66
t14Anzahl=1
t14m1=mfcn1 0.3 1e-06 1
t14minmax=0 1e+10

t15=5
t15Anzahl=1
t15m1=mfcn1 0.3 1e-06 1
t15minmax=0 1e+10

t16=6
t16Anzahl=1
t16m1=mfcn1 0.3 1e-06 1
t16minmax=0 1e+10

t17=7
t17Anzahl=1
t17m1=mfcn1 0.3 1e-06 1
t17minmax=0 1e+10

t18=8
t18Anzahl=1
t18m1=mfcn1 0.3 1e-06 1
t18minmax=0 1e+10

t19=9
t19Anzahl=1
t19m1=mfcn1 0.3 1e-06 1
t19minmax=0 1e+10

t20=10
t20Anzahl=1
t20m1=mfcn1 0.3 1e-06 1
t20minmax=0 1e+10

[OptionenIntegration]
teps=1e-08
rtol=1e-08
atol=1e-07
stepsize=0.0001
maxorder=6
maxstepnumber=4000
minstepsize=-1
maxstepsize=-1
maxitNewton=3
noerrorcontrol=0
realworkspace=170000
integerworkspace=500
printlevel=0
mcnonlinearflag=0
mcDAEflag=0
mctol=1e-07
mcmaxit=50
mclinesearch=1
mcalpha0=1
rndmethod=-1
rndeps=1e-05
rndverbose=0

</source>

Experiment 2
<source lang="optimica">

[Flags]
switch=1

[Kosten]
costs=0 -1e+10 1e+10

[Integrationsintervall]
t0=0
tend=10

[Modellfunktionen]
ffcn=ffcn
gfcn=gfcn

[Zustandsvariablen]
yAnzahl=4
y1=n1 na1 -1e+10 1e+10
y2=n2 na2 -1e+10 1e+10
y3=n3 0 -1e+10 1e+10
y4=n4 na4 -1e+10 1e+10

zAnzahl=0

[Mehrzielknoten]
tAnzahl=0
t1=0
t2=0.33
t3=0.66
t4=1
t5=1.33
t6=1.66
t7=2
t8=2.33
t9=2.66
t10=3
t11=3.33
t12=3.66
t13=4
t14=4.33
t15=4.66
t16=5
t17=6
t18=7
t19=9
t20=10

[DynamischeNebenbedingungen]
bAnzahl=0

[GitterUeberpruefungNebenbedingungen]
tAnzahl=0

[Steuergroessen]
qAnzahl=4
q1=na1 1.0 0 10 0 -1
q2=na2 1.0 0 10 0 -1
q3=na4 2.0 0.4 9 0 -1
q4=Ckat 1.0 0 6 0 -1

[Steuerfunktionen]
uAnzahl=1
u1=Tc 3 20 100
u1tAnzahl=3
u1t0=t0
u1t1q=60 20 100 0 0 0 0
u1t1=2
u1t2q=60 20 100 0.0 -1e+10 1e+10
u1t2=8
u1t3q=60 20 100 0 0 0
u1t3=tend

[NebenbedingungenSteuergroessen]
cAnzahl=2
c1=cfcn1 1
c1bnd1=0.1 0.7
c2=cfcn2 1
c2bnd1=0.1 10

[Messverfahren]
mAnzahl=1
m1=mfcn1 1 0 1e+10 0
m1f1=mess3 sigma3 1
mminmaxges=0 6

[Messungen]
tAnzahl=20

t1=0.33
t1Anzahl=1
t1m1=mfcn1 0.3 1e-06 1
t1minmax=0 1e+10

t2=0.66
t2Anzahl=1
t2m1=mfcn1 0.3 1e-06 1
t2minmax=0 1e+10

t3=1
t3Anzahl=1
t3m1=mfcn1 0.3 1e-06 1
t3minmax=0 1e+10

t4=1.33
t4Anzahl=1
t4m1=mfcn1 0.3 1e-06 1
t4minmax=0 1e+10

t5=1.66
t5Anzahl=1
t5m1=mfcn1 0.3 1e-06 1
t5minmax=0 1e+10

t6=2
t6Anzahl=1
t6m1=mfcn1 0.3 1e-06 1
t6minmax=0 1e+10

t7=2.33
t7Anzahl=1
t7m1=mfcn1 0.3 1e-06 1
t7minmax=0 1e+10

t8=2.66
t8Anzahl=1
t8m1=mfcn1 0.3 1e-06 1
t8minmax=0 1e+10

t9=3
t9Anzahl=1
t9m1=mfcn1 0.3 1e-06 1
t9minmax=0 1e+10

t10=3.33
t10Anzahl=1
t10m1=mfcn1 0.3 1e-06 1
t10minmax=0 1e+10

t11=3.66
t11Anzahl=1
t11m1=mfcn1 0.3 1e-06 1
t11minmax=0 1e+10

t12=4
t12Anzahl=1
t12m1=mfcn1 0.3 1e-06 1
t12minmax=0 1e+10

t13=4.33
t13Anzahl=1
t13m1=mfcn1 0.3 1e-06 1
t13minmax=0 1e+10

t14=4.66
t14Anzahl=1
t14m1=mfcn1 0.3 1e-06 1
t14minmax=0 1e+10

t15=5
t15Anzahl=1
t15m1=mfcn1 0.3 1e-06 1
t15minmax=0 1e+10

t16=6
t16Anzahl=1
t16m1=mfcn1 0.3 1e-06 1
t16minmax=0 1e+10

t17=7
t17Anzahl=1
t17m1=mfcn1 0.3 1e-06 1
t17minmax=0 1e+10

t18=8
t18Anzahl=1
t18m1=mfcn1 0.3 1e-06 1
t18minmax=0 1e+10

t19=9
t19Anzahl=1
t19m1=mfcn1 0.3 1e-06 1
t19minmax=0 1e+10

t20=10
t20Anzahl=1
t20m1=mfcn1 0.3 1e-06 1
t20minmax=0 1e+10

[OptionenIntegration]
teps=1e-08
rtol=1e-08
atol=1e-07
stepsize=0.0001
maxorder=6
maxstepnumber=4000
minstepsize=-1
maxstepsize=-1
maxitNewton=3
noerrorcontrol=0
realworkspace=170000
integerworkspace=500
printlevel=0
mcnonlinearflag=0
mcDAEflag=0
mctol=1e-07
mcmaxit=50
mclinesearch=1
mcalpha0=1
rndmethod=-1
rndeps=1e-05
rndverbose=0

</source>

Experiment 3
<source lang="optimica">

</source>

[Flags]
switch=1

[Kosten]
costs=0 -1e+10 1e+10

[Integrationsintervall]
t0=0
tend=10

[Modellfunktionen]
ffcn=ffcn
gfcn=gfcn

[Zustandsvariablen]
yAnzahl=4
y1=n1 na1 -1e+10 1e+10
y2=n2 na2 -1e+10 1e+10
y3=n3 0 -1e+10 1e+10
y4=n4 na4 -1e+10 1e+10

zAnzahl=0

[Mehrzielknoten]
tAnzahl=0
t1=0
t2=0.33
t3=0.66
t4=1
t5=1.33
t6=1.66
t7=2
t8=2.33
t9=2.66
t10=3
t11=3.33
t12=3.66
t13=4
t14=4.33
t15=4.66
t16=5
t17=6
t18=7
t19=9
t20=10

[DynamischeNebenbedingungen]
bAnzahl=0

[GitterUeberpruefungNebenbedingungen]
tAnzahl=0

[Steuergroessen]
qAnzahl=4
q1=na1 1.0 0 10 0 -1
q2=na2 1.0 0 10 0 -1
q3=na4 2.0 0.4 9 0 -1
q4=Ckat 2.0 0 6 0 -1

[Steuerfunktionen]
uAnzahl=1
u1=Tc 3 20 100
u1tAnzahl=3
u1t0=t0
u1t1q=40 20 100 0 0 0 0
u1t1=2
u1t2q=40 20 100 0.0 -1e+10 1e+10
u1t2=8
u1t3q=40 20 100 0 0 0
u1t3=tend

[NebenbedingungenSteuergroessen]
cAnzahl=2
c1=cfcn1 1
c1bnd1=0.1 0.7
c2=cfcn2 1
c2bnd1=0.1 10

[Messverfahren]
mAnzahl=1
m1=mfcn1 1 0 1e+10 0
m1f1=mess3 sigma3 1
mminmaxges=0 6

[Messungen]
tAnzahl=20

t1=0.33
t1Anzahl=1
t1m1=mfcn1 0.3 1e-06 1
t1minmax=0 1e+10

t2=0.66
t2Anzahl=1
t2m1=mfcn1 0.3 1e-06 1
t2minmax=0 1e+10

t3=1
t3Anzahl=1
t3m1=mfcn1 0.3 1e-06 1
t3minmax=0 1e+10

t4=1.33
t4Anzahl=1
t4m1=mfcn1 0.3 1e-06 1
t4minmax=0 1e+10

t5=1.66
t5Anzahl=1
t5m1=mfcn1 0.3 1e-06 1
t5minmax=0 1e+10

t6=2
t6Anzahl=1
t6m1=mfcn1 0.3 1e-06 1
t6minmax=0 1e+10

t7=2.33
t7Anzahl=1
t7m1=mfcn1 0.3 1e-06 1
t7minmax=0 1e+10

t8=2.66
t8Anzahl=1
t8m1=mfcn1 0.3 1e-06 1
t8minmax=0 1e+10

t9=3
t9Anzahl=1
t9m1=mfcn1 0.3 1e-06 1
t9minmax=0 1e+10

t10=3.33
t10Anzahl=1
t10m1=mfcn1 0.3 1e-06 1
t10minmax=0 1e+10

t11=3.66
t11Anzahl=1
t11m1=mfcn1 0.3 1e-06 1
t11minmax=0 1e+10

t12=4
t12Anzahl=1
t12m1=mfcn1 0.3 1e-06 1
t12minmax=0 1e+10

t13=4.33
t13Anzahl=1
t13m1=mfcn1 0.3 1e-06 1
t13minmax=0 1e+10

t14=4.66
t14Anzahl=1
t14m1=mfcn1 0.3 1e-06 1
t14minmax=0 1e+10

t15=5
t15Anzahl=1
t15m1=mfcn1 0.3 1e-06 1
t15minmax=0 1e+10

t16=6
t16Anzahl=1
t16m1=mfcn1 0.3 1e-06 1
t16minmax=0 1e+10

t17=7
t17Anzahl=1
t17m1=mfcn1 0.3 1e-06 1
t17minmax=0 1e+10

t18=8
t18Anzahl=1
t18m1=mfcn1 0.3 1e-06 1
t18minmax=0 1e+10

t19=9
t19Anzahl=1
t19m1=mfcn1 0.3 1e-06 1
t19minmax=0 1e+10

t20=10
t20Anzahl=1
t20m1=mfcn1 0.3 1e-06 1
t20minmax=0 1e+10

[OptionenIntegration]
teps=1e-08
rtol=1e-08
atol=1e-07
stepsize=0.0001
maxorder=6
maxstepnumber=4000
minstepsize=-1
maxstepsize=-1
maxitNewton=3
noerrorcontrol=0
realworkspace=170000
integerworkspace=500
printlevel=0
mcnonlinearflag=0
mcDAEflag=0
mctol=1e-07
mcmaxit=50
mclinesearch=1
mcalpha0=1
rndmethod=-1
rndeps=1e-05
rndverbose=0

</source>

Experiment 4
<source lang="optimica">

[Flags]
switch=1

[Kosten]
costs=0 -1e+10 1e+10

[Integrationsintervall]
t0=0
tend=10

[Modellfunktionen]
ffcn=ffcn
gfcn=gfcn

[Zustandsvariablen]
yAnzahl=4
y1=n1 na1 -1e+10 1e+10
y2=n2 na2 -1e+10 1e+10
y3=n3 0 -1e+10 1e+10
y4=n4 na4 -1e+10 1e+10

zAnzahl=0

[Mehrzielknoten]
tAnzahl=0
t1=0
t2=0.33
t3=0.66
t4=1
t5=1.33
t6=1.66
t7=2
t8=2.33
t9=2.66
t10=3
t11=3.33
t12=3.66
t13=4
t14=4.33
t15=4.66
t16=5
t17=6
t18=7
t19=9
t20=10

[DynamischeNebenbedingungen]
bAnzahl=0

[GitterUeberpruefungNebenbedingungen]
tAnzahl=0

[Steuergroessen]
qAnzahl=4
q1=na1 1.0 0 10 0 -1
q2=na2 1.0 0 10 0 -1
q3=na4 2.0 0.4 9 0 -1
q4=Ckat 3.0 0 6 0 -1

[Steuerfunktionen]
uAnzahl=1
u1=Tc 3 20 100
u1tAnzahl=3
u1t0=t0
u1t1q=20 20 100 0 0 0 0
u1t1=2
u1t2q=20 20 100 0.0 -1e+10 1e+10
u1t2=8
u1t3q=20 20 100 0 0 0
u1t3=tend

[NebenbedingungenSteuergroessen]
cAnzahl=2
c1=cfcn1 1
c1bnd1=0.1 0.7
c2=cfcn2 1
c2bnd1=0.1 10

[Messverfahren]
mAnzahl=1
m1=mfcn1 1 0 1e+10 0
m1f1=mess3 sigma3 1
mminmaxges=0 6

[Messungen]
tAnzahl=20

t1=0.33
t1Anzahl=1
t1m1=mfcn1 0.3 1e-06 1
t1minmax=0 1e+10

t2=0.66
t2Anzahl=1
t2m1=mfcn1 0.3 1e-06 1
t2minmax=0 1e+10

t3=1
t3Anzahl=1
t3m1=mfcn1 0.3 1e-06 1
t3minmax=0 1e+10

t4=1.33
t4Anzahl=1
t4m1=mfcn1 0.3 1e-06 1
t4minmax=0 1e+10

t5=1.66
t5Anzahl=1
t5m1=mfcn1 0.3 1e-06 1
t5minmax=0 1e+10

t6=2
t6Anzahl=1
t6m1=mfcn1 0.3 1e-06 1
t6minmax=0 1e+10

t7=2.33
t7Anzahl=1
t7m1=mfcn1 0.3 1e-06 1
t7minmax=0 1e+10

t8=2.66
t8Anzahl=1
t8m1=mfcn1 0.3 1e-06 1
t8minmax=0 1e+10

t9=3
t9Anzahl=1
t9m1=mfcn1 0.3 1e-06 1
t9minmax=0 1e+10

t10=3.33
t10Anzahl=1
t10m1=mfcn1 0.3 1e-06 1
t10minmax=0 1e+10

t11=3.66
t11Anzahl=1
t11m1=mfcn1 0.3 1e-06 1
t11minmax=0 1e+10

t12=4
t12Anzahl=1
t12m1=mfcn1 0.3 1e-06 1
t12minmax=0 1e+10

t13=4.33
t13Anzahl=1
t13m1=mfcn1 0.3 1e-06 1
t13minmax=0 1e+10

t14=4.66
t14Anzahl=1
t14m1=mfcn1 0.3 1e-06 1
t14minmax=0 1e+10

t15=5
t15Anzahl=1
t15m1=mfcn1 0.3 1e-06 1
t15minmax=0 1e+10

t16=6
t16Anzahl=1
t16m1=mfcn1 0.3 1e-06 1
t16minmax=0 1e+10

t17=7
t17Anzahl=1
t17m1=mfcn1 0.3 1e-06 1
t17minmax=0 1e+10

t18=8
t18Anzahl=1
t18m1=mfcn1 0.3 1e-06 1
t18minmax=0 1e+10

t19=9
t19Anzahl=1
t19m1=mfcn1 0.3 1e-06 1
t19minmax=0 1e+10

t20=10
t20Anzahl=1
t20m1=mfcn1 0.3 1e-06 1
t20minmax=0 1e+10

[OptionenIntegration]
teps=1e-08
rtol=1e-08
atol=1e-07
stepsize=0.0001
maxorder=6
maxstepnumber=4000
minstepsize=-1
maxstepsize=-1
maxitNewton=3
noerrorcontrol=0
realworkspace=170000
integerworkspace=500
printlevel=0
mcnonlinearflag=0
mcDAEflag=0
mctol=1e-07
mcmaxit=50
mclinesearch=1
mcalpha0=1
rndmethod=-1
rndeps=1e-05
rndverbose=0

</source>

Diels-Alder Reaction Experimental Design (VPLAN)

2016-02-01T12:29:50Z

FelixJost: /* VPLAN */

== VPLAN ==

Differential equations:

<source lang="fortran">

c RHS of the differential equations

subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), f(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

real*8 n1, n2, n3, n4
real*8 na1, na2, na4
real*8 fg, Temp, E , Rg , T1, Tc
real*8 r1, mR
real*8 kr1, kkat, Ckat, Ekat
real*8 k1, lambda
real*8 M1, M2, M3, M4
real*8 dm

c State variables

n1 = x(1)
n2 = x(2)
n3 = x(3)
n4 = x(4)

c Control variables

na1 = q(1)
na2 = q(2)
na4 = q(3)
Ckat = q(4)

c Control function

c DISCRETIZE1( Tc, rwh, iwh )

c Parameters

kr1 = p(1) * 1.0d-2
E = p(2) * 60000.0d+0
k1 = p(3) * 0.10d+0
Ekat = p(4) * 40000.0d0
lambda = p(5) * 0.25d+0

c Molar masses (in kg/mol)

M1 = 0.1362d+0
M2 = 0.09806d+0
M3 = M1 + M2
M4 = 0.236d+0

Temp = Tc + 273.0d+0
Rg = 8.314d+0
T1 = 293.0d+0

c Reaction rates

mR = n1*M1 + n2*M2 +n3*M3 + n4*M4

kkat = kr1 * dexp( -E/Rg * ( 1.0d+0/Temp - 1.0d+0/T1 ) )
& + k1 * dexp( -Ekat/Rg *( 1.0d+0/Temp - 1.0d+0/T1 ) )
& * Ckat * dexp( -lambda * t )

r1 = kkat * n1 * n2 / mR

f(1) = -r1
f(2) = -r1
f(3) = r1
f(4) = 0.0d0

end

</source>

Algebraic equations:

<source lang="fortran">

c Dummyfunction for RHS of algebraic equations

subroutine gfcn( t, x, g, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), g(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

iflag=0

end

</source>

Measurement function:

<source lang="fortran">

c Messfunktion

subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )
implicit none

real*8 t, x(*), h, p(*), q(*), rwh(*)
integer*4 iwh(*), iflag

real*8 M1, M2, M3, M4, mR

c Berechnung der Reaktormasse

M1 = 0.1362d+0
M2 = 0.09806d+0
M3 = M1 + M2
M4 = 0.236d+0

mR = M1*x(1) + M2*x(2) + M3*x(3) + M4*x(4)

c Messwert: Massenprozent

h = M3*x(3) * 100.0d+0/mR

iflag = 0

end

</source>

Standard deviation of measurement function:

<source lang="fortran">

c Standardabweichung der Messfunktion

subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )
implicit none

real*8 rwh(*)
integer*4 iwh(*), iflag

real*8 t, x(*), p(*), q(*)
real*8 s
real*8 h

s = 1.0d+0

iflag = 0

end

</source>

VPLAN specific experimental setup:
Experiment 1
<source lang="optimica">

[Flags]
switch=1

[Kosten]
costs=0 -1e+10 1e+10

[Integrationsintervall]
t0=0
tend=10

[Modellfunktionen]
ffcn=ffcn
gfcn=gfcn

[Zustandsvariablen]
yAnzahl=4
y1=n1 na1 -1e+10 1e+10
y2=n2 na2 -1e+10 1e+10
y3=n3 0 -1e+10 1e+10
y4=n4 na4 -1e+10 1e+10

zAnzahl=0

[Mehrzielknoten]
tAnzahl=0
t1=0
t2=0.33
t3=0.66
t4=1
t5=1.33
t6=1.66
t7=2
t8=2.33
t9=2.66
t10=3
t11=3.33
t12=3.66
t13=4
t14=4.33
t15=4.66
t16=5
t17=6
t18=7
t19=9
t20=10

[DynamischeNebenbedingungen]
bAnzahl=0

[GitterUeberpruefungNebenbedingungen]
tAnzahl=0

[Steuergroessen]
qAnzahl=4
q1=na1 1.0 0 10 0 -1
q2=na2 1.0 0 10 0 -1
q3=na4 2.0 0.4 9 0 -1
q4=Ckat 0.0 0 6 0 -1

[Steuerfunktionen]
uAnzahl=1
u1=Tc 3 20 100
u1tAnzahl=3
u1t0=t0
u1t1q=20 20 100 0 0 0
u1t1=2
u1t2q=20 20 100 0.0 -1e+10 1e+10
u1t2=8
u1t3q=20 20 100 0 0 0
u1t3=tend

[NebenbedingungenSteuergroessen]
cAnzahl=2
c1=cfcn1 1
c1bnd1=0.1 0.7
c2=cfcn2 1
c2bnd1=0.1 10

[Messverfahren]
mAnzahl=1
m1=mfcn1 1 0 1e+10 0
m1f1=mess3 sigma3 1
mminmaxges=0 6

[Messungen]
tAnzahl=20

t1=0.33
t1Anzahl=1
t1m1=mfcn1 0.3 1e-06 1
t1minmax=0 1e+10

t2=0.66
t2Anzahl=1
t2m1=mfcn1 0.3 1e-06 1
t2minmax=0 1e+10

t3=1
t3Anzahl=1
t3m1=mfcn1 0.3 1e-06 1
t3minmax=0 1e+10

t4=1.33
t4Anzahl=1
t4m1=mfcn1 0.3 1e-06 1
t4minmax=0 1e+10

t5=1.66
t5Anzahl=1
t5m1=mfcn1 0.3 1e-06 1
t5minmax=0 1e+10

t6=2
t6Anzahl=1
t6m1=mfcn1 0.3 1e-06 1
t6minmax=0 1e+10

t7=2.33
t7Anzahl=1
t7m1=mfcn1 0.3 1e-06 1
t7minmax=0 1e+10

t8=2.66
t8Anzahl=1
t8m1=mfcn1 0.3 1e-06 1
t8minmax=0 1e+10

t9=3
t9Anzahl=1
t9m1=mfcn1 0.3 1e-06 1
t9minmax=0 1e+10

t10=3.33
t10Anzahl=1
t10m1=mfcn1 0.3 1e-06 1
t10minmax=0 1e+10

t11=3.66
t11Anzahl=1
t11m1=mfcn1 0.3 1e-06 1
t11minmax=0 1e+10

t12=4
t12Anzahl=1
t12m1=mfcn1 0.3 1e-06 1
t12minmax=0 1e+10

t13=4.33
t13Anzahl=1
t13m1=mfcn1 0.3 1e-06 1
t13minmax=0 1e+10

t14=4.66
t14Anzahl=1
t14m1=mfcn1 0.3 1e-06 1
t14minmax=0 1e+10

t15=5
t15Anzahl=1
t15m1=mfcn1 0.3 1e-06 1
t15minmax=0 1e+10

t16=6
t16Anzahl=1
t16m1=mfcn1 0.3 1e-06 1
t16minmax=0 1e+10

t17=7
t17Anzahl=1
t17m1=mfcn1 0.3 1e-06 1
t17minmax=0 1e+10

t18=8
t18Anzahl=1
t18m1=mfcn1 0.3 1e-06 1
t18minmax=0 1e+10

t19=9
t19Anzahl=1
t19m1=mfcn1 0.3 1e-06 1
t19minmax=0 1e+10

t20=10
t20Anzahl=1
t20m1=mfcn1 0.3 1e-06 1
t20minmax=0 1e+10

[OptionenIntegration]
teps=1e-08
rtol=1e-08
atol=1e-07
stepsize=0.0001
maxorder=6
maxstepnumber=4000
minstepsize=-1
maxstepsize=-1
maxitNewton=3
noerrorcontrol=0
realworkspace=170000
integerworkspace=500
printlevel=0
mcnonlinearflag=0
mcDAEflag=0
mctol=1e-07
mcmaxit=50
mclinesearch=1
mcalpha0=1
rndmethod=-1
rndeps=1e-05
rndverbose=0

</source>

Experiment 2
<source lang="optimica">

[Flags]
switch=1

[Kosten]
costs=0 -1e+10 1e+10

[Integrationsintervall]
t0=0
tend=10

[Modellfunktionen]
ffcn=ffcn
gfcn=gfcn

[Zustandsvariablen]
yAnzahl=4
y1=n1 na1 -1e+10 1e+10
y2=n2 na2 -1e+10 1e+10
y3=n3 0 -1e+10 1e+10
y4=n4 na4 -1e+10 1e+10

zAnzahl=0

[Mehrzielknoten]
tAnzahl=0
t1=0
t2=0.33
t3=0.66
t4=1
t5=1.33
t6=1.66
t7=2
t8=2.33
t9=2.66
t10=3
t11=3.33
t12=3.66
t13=4
t14=4.33
t15=4.66
t16=5
t17=6
t18=7
t19=9
t20=10

[DynamischeNebenbedingungen]
bAnzahl=0

[GitterUeberpruefungNebenbedingungen]
tAnzahl=0

[Steuergroessen]
qAnzahl=4
q1=na1 1.0 0 10 0 -1
q2=na2 1.0 0 10 0 -1
q3=na4 2.0 0.4 9 0 -1
q4=Ckat 1.0 0 6 0 -1

[Steuerfunktionen]
uAnzahl=1
u1=Tc 3 20 100
u1tAnzahl=3
u1t0=t0
u1t1q=60 20 100 0 0 0 0
u1t1=2
u1t2q=60 20 100 0.0 -1e+10 1e+10
u1t2=8
u1t3q=60 20 100 0 0 0
u1t3=tend

[NebenbedingungenSteuergroessen]
cAnzahl=2
c1=cfcn1 1
c1bnd1=0.1 0.7
c2=cfcn2 1
c2bnd1=0.1 10

[Messverfahren]
mAnzahl=1
m1=mfcn1 1 0 1e+10 0
m1f1=mess3 sigma3 1
mminmaxges=0 6

[Messungen]
tAnzahl=20

t1=0.33
t1Anzahl=1
t1m1=mfcn1 0.3 1e-06 1
t1minmax=0 1e+10

t2=0.66
t2Anzahl=1
t2m1=mfcn1 0.3 1e-06 1
t2minmax=0 1e+10

t3=1
t3Anzahl=1
t3m1=mfcn1 0.3 1e-06 1
t3minmax=0 1e+10

t4=1.33
t4Anzahl=1
t4m1=mfcn1 0.3 1e-06 1
t4minmax=0 1e+10

t5=1.66
t5Anzahl=1
t5m1=mfcn1 0.3 1e-06 1
t5minmax=0 1e+10

t6=2
t6Anzahl=1
t6m1=mfcn1 0.3 1e-06 1
t6minmax=0 1e+10

t7=2.33
t7Anzahl=1
t7m1=mfcn1 0.3 1e-06 1
t7minmax=0 1e+10

t8=2.66
t8Anzahl=1
t8m1=mfcn1 0.3 1e-06 1
t8minmax=0 1e+10

t9=3
t9Anzahl=1
t9m1=mfcn1 0.3 1e-06 1
t9minmax=0 1e+10

t10=3.33
t10Anzahl=1
t10m1=mfcn1 0.3 1e-06 1
t10minmax=0 1e+10

t11=3.66
t11Anzahl=1
t11m1=mfcn1 0.3 1e-06 1
t11minmax=0 1e+10

t12=4
t12Anzahl=1
t12m1=mfcn1 0.3 1e-06 1
t12minmax=0 1e+10

t13=4.33
t13Anzahl=1
t13m1=mfcn1 0.3 1e-06 1
t13minmax=0 1e+10

t14=4.66
t14Anzahl=1
t14m1=mfcn1 0.3 1e-06 1
t14minmax=0 1e+10

t15=5
t15Anzahl=1
t15m1=mfcn1 0.3 1e-06 1
t15minmax=0 1e+10

t16=6
t16Anzahl=1
t16m1=mfcn1 0.3 1e-06 1
t16minmax=0 1e+10

t17=7
t17Anzahl=1
t17m1=mfcn1 0.3 1e-06 1
t17minmax=0 1e+10

t18=8
t18Anzahl=1
t18m1=mfcn1 0.3 1e-06 1
t18minmax=0 1e+10

t19=9
t19Anzahl=1
t19m1=mfcn1 0.3 1e-06 1
t19minmax=0 1e+10

t20=10
t20Anzahl=1
t20m1=mfcn1 0.3 1e-06 1
t20minmax=0 1e+10

[OptionenIntegration]
teps=1e-08
rtol=1e-08
atol=1e-07
stepsize=0.0001
maxorder=6
maxstepnumber=4000
minstepsize=-1
maxstepsize=-1
maxitNewton=3
noerrorcontrol=0
realworkspace=170000
integerworkspace=500
printlevel=0
mcnonlinearflag=0
mcDAEflag=0
mctol=1e-07
mcmaxit=50
mclinesearch=1
mcalpha0=1
rndmethod=-1
rndeps=1e-05
rndverbose=0

</source>

Experiment 3
<source lang="optimica">

</source>

[Flags]
switch=1

[Kosten]
costs=0 -1e+10 1e+10

[Integrationsintervall]
t0=0
tend=10

[Modellfunktionen]
ffcn=ffcn
gfcn=gfcn

[Zustandsvariablen]
yAnzahl=4
y1=n1 na1 -1e+10 1e+10
y2=n2 na2 -1e+10 1e+10
y3=n3 0 -1e+10 1e+10
y4=n4 na4 -1e+10 1e+10

zAnzahl=0

[Mehrzielknoten]
tAnzahl=0
t1=0
t2=0.33
t3=0.66
t4=1
t5=1.33
t6=1.66
t7=2
t8=2.33
t9=2.66
t10=3
t11=3.33
t12=3.66
t13=4
t14=4.33
t15=4.66
t16=5
t17=6
t18=7
t19=9
t20=10

[DynamischeNebenbedingungen]
bAnzahl=0

[GitterUeberpruefungNebenbedingungen]
tAnzahl=0

[Steuergroessen]
qAnzahl=4
q1=na1 1.0 0 10 0 -1
q2=na2 1.0 0 10 0 -1
q3=na4 2.0 0.4 9 0 -1
q4=Ckat 2.0 0 6 0 -1

[Steuerfunktionen]
uAnzahl=1
u1=Tc 3 20 100
u1tAnzahl=3
u1t0=t0
u1t1q=40 20 100 0 0 0 0
u1t1=2
u1t2q=40 20 100 0.0 -1e+10 1e+10
u1t2=8
u1t3q=40 20 100 0 0 0
u1t3=tend

[NebenbedingungenSteuergroessen]
cAnzahl=2
c1=cfcn1 1
c1bnd1=0.1 0.7
c2=cfcn2 1
c2bnd1=0.1 10

[Messverfahren]
mAnzahl=1
m1=mfcn1 1 0 1e+10 0
m1f1=mess3 sigma3 1
mminmaxges=0 6

[Messungen]
tAnzahl=20

t1=0.33
t1Anzahl=1
t1m1=mfcn1 0.3 1e-06 1
t1minmax=0 1e+10

t2=0.66
t2Anzahl=1
t2m1=mfcn1 0.3 1e-06 1
t2minmax=0 1e+10

t3=1
t3Anzahl=1
t3m1=mfcn1 0.3 1e-06 1
t3minmax=0 1e+10

t4=1.33
t4Anzahl=1
t4m1=mfcn1 0.3 1e-06 1
t4minmax=0 1e+10

t5=1.66
t5Anzahl=1
t5m1=mfcn1 0.3 1e-06 1
t5minmax=0 1e+10

t6=2
t6Anzahl=1
t6m1=mfcn1 0.3 1e-06 1
t6minmax=0 1e+10

t7=2.33
t7Anzahl=1
t7m1=mfcn1 0.3 1e-06 1
t7minmax=0 1e+10

t8=2.66
t8Anzahl=1
t8m1=mfcn1 0.3 1e-06 1
t8minmax=0 1e+10

t9=3
t9Anzahl=1
t9m1=mfcn1 0.3 1e-06 1
t9minmax=0 1e+10

t10=3.33
t10Anzahl=1
t10m1=mfcn1 0.3 1e-06 1
t10minmax=0 1e+10

t11=3.66
t11Anzahl=1
t11m1=mfcn1 0.3 1e-06 1
t11minmax=0 1e+10

t12=4
t12Anzahl=1
t12m1=mfcn1 0.3 1e-06 1
t12minmax=0 1e+10

t13=4.33
t13Anzahl=1
t13m1=mfcn1 0.3 1e-06 1
t13minmax=0 1e+10

t14=4.66
t14Anzahl=1
t14m1=mfcn1 0.3 1e-06 1
t14minmax=0 1e+10

t15=5
t15Anzahl=1
t15m1=mfcn1 0.3 1e-06 1
t15minmax=0 1e+10

t16=6
t16Anzahl=1
t16m1=mfcn1 0.3 1e-06 1
t16minmax=0 1e+10

t17=7
t17Anzahl=1
t17m1=mfcn1 0.3 1e-06 1
t17minmax=0 1e+10

t18=8
t18Anzahl=1
t18m1=mfcn1 0.3 1e-06 1
t18minmax=0 1e+10

t19=9
t19Anzahl=1
t19m1=mfcn1 0.3 1e-06 1
t19minmax=0 1e+10

t20=10
t20Anzahl=1
t20m1=mfcn1 0.3 1e-06 1
t20minmax=0 1e+10

[OptionenIntegration]
teps=1e-08
rtol=1e-08
atol=1e-07
stepsize=0.0001
maxorder=6
maxstepnumber=4000
minstepsize=-1
maxstepsize=-1
maxitNewton=3
noerrorcontrol=0
realworkspace=170000
integerworkspace=500
printlevel=0
mcnonlinearflag=0
mcDAEflag=0
mctol=1e-07
mcmaxit=50
mclinesearch=1
mcalpha0=1
rndmethod=-1
rndeps=1e-05
rndverbose=0

Experiment 4
<source lang="optimica">

[Flags]
switch=1

[Kosten]
costs=0 -1e+10 1e+10

[Integrationsintervall]
t0=0
tend=10

[Modellfunktionen]
ffcn=ffcn
gfcn=gfcn

[Zustandsvariablen]
yAnzahl=4
y1=n1 na1 -1e+10 1e+10
y2=n2 na2 -1e+10 1e+10
y3=n3 0 -1e+10 1e+10
y4=n4 na4 -1e+10 1e+10

zAnzahl=0

[Mehrzielknoten]
tAnzahl=0
t1=0
t2=0.33
t3=0.66
t4=1
t5=1.33
t6=1.66
t7=2
t8=2.33
t9=2.66
t10=3
t11=3.33
t12=3.66
t13=4
t14=4.33
t15=4.66
t16=5
t17=6
t18=7
t19=9
t20=10

[DynamischeNebenbedingungen]
bAnzahl=0

[GitterUeberpruefungNebenbedingungen]
tAnzahl=0

[Steuergroessen]
qAnzahl=4
q1=na1 1.0 0 10 0 -1
q2=na2 1.0 0 10 0 -1
q3=na4 2.0 0.4 9 0 -1
q4=Ckat 3.0 0 6 0 -1

[Steuerfunktionen]
uAnzahl=1
u1=Tc 3 20 100
u1tAnzahl=3
u1t0=t0
u1t1q=20 20 100 0 0 0 0
u1t1=2
u1t2q=20 20 100 0.0 -1e+10 1e+10
u1t2=8
u1t3q=20 20 100 0 0 0
u1t3=tend

[NebenbedingungenSteuergroessen]
cAnzahl=2
c1=cfcn1 1
c1bnd1=0.1 0.7
c2=cfcn2 1
c2bnd1=0.1 10

[Messverfahren]
mAnzahl=1
m1=mfcn1 1 0 1e+10 0
m1f1=mess3 sigma3 1
mminmaxges=0 6

[Messungen]
tAnzahl=20

t1=0.33
t1Anzahl=1
t1m1=mfcn1 0.3 1e-06 1
t1minmax=0 1e+10

t2=0.66
t2Anzahl=1
t2m1=mfcn1 0.3 1e-06 1
t2minmax=0 1e+10

t3=1
t3Anzahl=1
t3m1=mfcn1 0.3 1e-06 1
t3minmax=0 1e+10

t4=1.33
t4Anzahl=1
t4m1=mfcn1 0.3 1e-06 1
t4minmax=0 1e+10

t5=1.66
t5Anzahl=1
t5m1=mfcn1 0.3 1e-06 1
t5minmax=0 1e+10

t6=2
t6Anzahl=1
t6m1=mfcn1 0.3 1e-06 1
t6minmax=0 1e+10

t7=2.33
t7Anzahl=1
t7m1=mfcn1 0.3 1e-06 1
t7minmax=0 1e+10

t8=2.66
t8Anzahl=1
t8m1=mfcn1 0.3 1e-06 1
t8minmax=0 1e+10

t9=3
t9Anzahl=1
t9m1=mfcn1 0.3 1e-06 1
t9minmax=0 1e+10

t10=3.33
t10Anzahl=1
t10m1=mfcn1 0.3 1e-06 1
t10minmax=0 1e+10

t11=3.66
t11Anzahl=1
t11m1=mfcn1 0.3 1e-06 1
t11minmax=0 1e+10

t12=4
t12Anzahl=1
t12m1=mfcn1 0.3 1e-06 1
t12minmax=0 1e+10

t13=4.33
t13Anzahl=1
t13m1=mfcn1 0.3 1e-06 1
t13minmax=0 1e+10

t14=4.66
t14Anzahl=1
t14m1=mfcn1 0.3 1e-06 1
t14minmax=0 1e+10

t15=5
t15Anzahl=1
t15m1=mfcn1 0.3 1e-06 1
t15minmax=0 1e+10

t16=6
t16Anzahl=1
t16m1=mfcn1 0.3 1e-06 1
t16minmax=0 1e+10

t17=7
t17Anzahl=1
t17m1=mfcn1 0.3 1e-06 1
t17minmax=0 1e+10

t18=8
t18Anzahl=1
t18m1=mfcn1 0.3 1e-06 1
t18minmax=0 1e+10

t19=9
t19Anzahl=1
t19m1=mfcn1 0.3 1e-06 1
t19minmax=0 1e+10

t20=10
t20Anzahl=1
t20m1=mfcn1 0.3 1e-06 1
t20minmax=0 1e+10

[OptionenIntegration]
teps=1e-08
rtol=1e-08
atol=1e-07
stepsize=0.0001
maxorder=6
maxstepnumber=4000
minstepsize=-1
maxstepsize=-1
maxitNewton=3
noerrorcontrol=0
realworkspace=170000
integerworkspace=500
printlevel=0
mcnonlinearflag=0
mcDAEflag=0
mctol=1e-07
mcmaxit=50
mclinesearch=1
mcalpha0=1
rndmethod=-1
rndeps=1e-05
rndverbose=0

</source>

Diels-Alder Reaction Experimental Design (VPLAN)

2016-02-01T12:26:14Z

FelixJost: /* VPLAN */

== VPLAN ==

Differential equations:

<source lang="fortran">

c RHS of the differential equations

subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), f(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

real*8 n1, n2, n3, n4
real*8 na1, na2, na4
real*8 fg, Temp, E , Rg , T1, Tc
real*8 r1, mR
real*8 kr1, kkat, Ckat, Ekat
real*8 k1, lambda
real*8 M1, M2, M3, M4
real*8 dm

c State variables

n1 = x(1)
n2 = x(2)
n3 = x(3)
n4 = x(4)

c Control variables

na1 = q(1)
na2 = q(2)
na4 = q(3)
Ckat = q(4)

c Control function

c DISCRETIZE1( Tc, rwh, iwh )

c Parameters

kr1 = p(1) * 1.0d-2
E = p(2) * 60000.0d+0
k1 = p(3) * 0.10d+0
Ekat = p(4) * 40000.0d0
lambda = p(5) * 0.25d+0

c Molar masses (in kg/mol)

M1 = 0.1362d+0
M2 = 0.09806d+0
M3 = M1 + M2
M4 = 0.236d+0

Temp = Tc + 273.0d+0
Rg = 8.314d+0
T1 = 293.0d+0

c Reaction rates

mR = n1*M1 + n2*M2 +n3*M3 + n4*M4

kkat = kr1 * dexp( -E/Rg * ( 1.0d+0/Temp - 1.0d+0/T1 ) )
& + k1 * dexp( -Ekat/Rg *( 1.0d+0/Temp - 1.0d+0/T1 ) )
& * Ckat * dexp( -lambda * t )

r1 = kkat * n1 * n2 / mR

f(1) = -r1
f(2) = -r1
f(3) = r1
f(4) = 0.0d0

end

</source>

Algebraic equations:

<source lang="fortran">

c Dummyfunction for RHS of algebraic equations

subroutine gfcn( t, x, g, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), g(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

iflag=0

end

</source>

Measurement function:

<source lang="fortran">

c Messfunktion

subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )
implicit none

real*8 t, x(*), h, p(*), q(*), rwh(*)
integer*4 iwh(*), iflag

real*8 M1, M2, M3, M4, mR

c Berechnung der Reaktormasse

M1 = 0.1362d+0
M2 = 0.09806d+0
M3 = M1 + M2
M4 = 0.236d+0

mR = M1*x(1) + M2*x(2) + M3*x(3) + M4*x(4)

c Messwert: Massenprozent

h = M3*x(3) * 100.0d+0/mR

iflag = 0

end

</source>

Standard deviation of measurement function:

<source lang="fortran">

c Standardabweichung der Messfunktion

subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )
implicit none

real*8 rwh(*)
integer*4 iwh(*), iflag

real*8 t, x(*), p(*), q(*)
real*8 s
real*8 h

s = 1.0d+0

iflag = 0

end

</source>

Diels-Alder Reaction Experimental Design (VPLAN)

2016-02-01T12:23:38Z

FelixJost: /* VPLAN */

== VPLAN ==

Differential equations:

<source lang="fortran">

c RHS of the differential equations

subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), f(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

real*8 n1, n2, n3, n4
real*8 na1, na2, na4
real*8 fg, Temp, E , Rg , T1, Tc
real*8 r1, mR
real*8 kr1, kkat, Ckat, Ekat
real*8 k1, lambda
real*8 M1, M2, M3, M4
real*8 dm

c State variables

n1 = x(1)
n2 = x(2)
n3 = x(3)
n4 = x(4)

c Control variables

na1 = q(1)
na2 = q(2)
na4 = q(3)
Ckat = q(4)

c Control function

c DISCRETIZE1( Tc, rwh, iwh )

c Parameters

kr1 = p(1) * 1.0d-2
E = p(2) * 60000.0d+0
k1 = p(3) * 0.10d+0
Ekat = p(4) * 40000.0d0
lambda = p(5) * 0.25d+0

c Molar masses (in kg/mol)

M1 = 0.1362d+0
M2 = 0.09806d+0
M3 = M1 + M2
M4 = 0.236d+0

Temp = Tc + 273.0d+0
Rg = 8.314d+0
T1 = 293.0d+0

c Reaction rates

mR = n1*M1 + n2*M2 +n3*M3 + n4*M4

kkat = kr1 * dexp( -E/Rg * ( 1.0d+0/Temp - 1.0d+0/T1 ) )
& + k1 * dexp( -Ekat/Rg *( 1.0d+0/Temp - 1.0d+0/T1 ) )
& * Ckat * dexp( -lambda * t )

r1 = kkat * n1 * n2 / mR

f(1) = -r1
f(2) = -r1
f(3) = r1
f(4) = 0.0d0

end

</source>

Algebraic equations:

<source lang="fortran">

c Dummyfunction for RHS of algebraic equations

subroutine gfcn( t, x, g, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), g(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

iflag=0

end

</source>

First Measurement function:

<source lang="fortran">

c Messfunktion

subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )
implicit none

real*8 t, x(*), h, p(*), q(*), rwh(*)
integer*4 iwh(*), iflag

real*8 M1, M2, M3, M4, mR

c Berechnung der Reaktormasse

M1 = 0.1362d+0
M2 = 0.09806d+0
M3 = M1 + M2
M4 = 0.236d+0

mR = M1*x(1) + M2*x(2) + M3*x(3) + M4*x(4)

c Messwert: Massenprozent

h = M3*x(3) * 100.0d+0/mR

iflag = 0

end

</source>

Diels-Alder Reaction Experimental Design (VPLAN)

2016-02-01T12:22:07Z

FelixJost: Created page with "== VPLAN == Differential equations: <source lang="fortran"> c RHS of the differential equations subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag ) implic..."

== VPLAN ==

Differential equations:

<source lang="fortran">

c RHS of the differential equations

subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), f(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

real*8 n1, n2, n3, n4
real*8 na1, na2, na4
real*8 fg, Temp, E , Rg , T1, Tc
real*8 r1, mR
real*8 kr1, kkat, Ckat, Ekat
real*8 k1, lambda
real*8 M1, M2, M3, M4
real*8 dm

c State variables

n1 = x(1)
n2 = x(2)
n3 = x(3)
n4 = x(4)

c Control variables

na1 = q(1)
na2 = q(2)
na4 = q(3)
Ckat = q(4)

c Control function

c DISCRETIZE1( Tc, rwh, iwh )

c Parameters

kr1 = p(1) * 1.0d-2
E = p(2) * 60000.0d+0
k1 = p(3) * 0.10d+0
Ekat = p(4) * 40000.0d0
lambda = p(5) * 0.25d+0

c Molar masses (in kg/mol)

M1 = 0.1362d+0
M2 = 0.09806d+0
M3 = M1 + M2
M4 = 0.236d+0

Temp = Tc + 273.0d+0
Rg = 8.314d+0
T1 = 293.0d+0

c Reaction rates

mR = n1*M1 + n2*M2 +n3*M3 + n4*M4

kkat = kr1 * dexp( -E/Rg * ( 1.0d+0/Temp - 1.0d+0/T1 ) )
& + k1 * dexp( -Ekat/Rg *( 1.0d+0/Temp - 1.0d+0/T1 ) )
& * Ckat * dexp( -lambda * t )

r1 = kkat * n1 * n2 / mR

f(1) = -r1
f(2) = -r1
f(3) = r1
f(4) = 0.0d0

end

</source>

Diels-Alder Reaction Experimental Design

2016-02-01T12:19:14Z

FelixJost:

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\
\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\
\vartheta_{up} + 273 & t \in [8,t_{end}]
\end{array} \right. \\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== Source Code ==

* The VPLAN code using [[:Category: VPLAN | VPLAN code]] can be found in: [[Diels-Alder Reaction Experimental Design (VPLAN)]]

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2016-02-01T12:15:37Z

FelixJost:

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\
\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\
\vartheta_{up} + 273 & t \in [8,t_{end}]
\end{array} \right. \\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2016-02-01T12:11:59Z

FelixJost:

{{Dimensions
|nd = 1
|nx = 2
|nu = 1
|nre = 4
}}The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\
\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\
\vartheta_{up} + 273 & t \in [8,t_{end}]
\end{array} \right. \\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Lotka Experimental Design (VPLAN)

2016-01-19T16:29:14Z

FelixJost: /* VPLAN */

== VPLAN ==

Differential equations:

<source lang="fortran">

c RHS of the differential equations

subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), f(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

real*8 p1,p3,p5,p6, myu

c fixed parameters
p1 = 1.0
p3 = 1.0
p5 = 0.4
p6 = 0.2

c DISCRETIZE1( myu, rwh, iwh )

f(1) = p1*x(1) - p(1)*x(1)*x(2) - p5*myu*x(1)
f(2) = (-1.0)*p3*x(2) + p(2)*x(1)*x(2) - p6*myu*x(2)

end

</source>

Algebraic equations:

<source lang="fortran">

c Dummyfunction for RHS of algebraic equations

subroutine gfcn( t, x, g, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), g(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

iflag=0

end

</source>

First Measurement function:

<source lang="fortran">

c Messfunktion

subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )
implicit none

real*8 t, x(*), h, p(*), q(*), rwh(*)
integer*4 iwh(*), iflag

h = x(1)

iflag = 0

end

</source>

Second Measurement function:

<source lang="fortran">

c Messfunktion

subroutine mess4( t, x, h, p, q, rwh, iwh, iflag )
implicit none

real*8 t, x(*), h, p(*), q(*), rwh(*)
integer*4 iwh(*), iflag

h = x(2)

iflag = 0

end

</source>

Standard deviation of first measurement function:
<source lang="fortran">

c Standardabweichung der Messfunktion

subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )
implicit none

real*8 rwh(*)
integer*4 iwh(*), iflag

real*8 t, x(*), p(*), q(*)
real*8 s
real*8 h

s = 1.0d+0

iflag = 0

end

</source>

Standard deviation of second measurement function:
<source lang="fortran">
c Standardabweichung der Messfunktion

subroutine sigma4( t, x, s, p, q, rwh, iwh, iflag )
implicit none

real*8 rwh(*)
integer*4 iwh(*), iflag

real*8 t, x(*), p(*), q(*)
real*8 s(*)

s(1) = 1.0

iflag = 0

end

</source>

VPLAN specific experimental setup:
<source lang="optimica">

; ini-File fuer Experiment

[Flags]
switch=1

[Kosten]
costs=0 -1e+10 1e+10

[Integrationsintervall]
t0=0
tend=12

[Modellfunktionen]
ffcn=ffcn
gfcn=gfcn

[Zustandsvariablen]
yAnzahl=2
y1=x1 0.5 -1e+10 1e+10
y2=x2 0.7 -1e+10 1e+10

zAnzahl=0

[Mehrzielknoten]
tAnzahl=12
t1=1
t2=2
t3=3
t4=4
t5=5
t6=6
t7=7
t8=8
t9=9
t10=10
t11=11
t12=12

[DynamischeNebenbedingungen]
bAnzahl=0

[GitterUeberpruefungNebenbedingungen]
tAnzahl=0

[Steuergroessen]
qAnzahl=0

[Steuerfunktionen]
uAnzahl=1
u1=myu 0 0.0 1.0
u1tAnzahl=500
u1t0=t0
u1t1q=0.3 0.0 1.0 0 0
u1t1=0.024
u1t2q=0.3 0.0 1.0 0 0
u1t2=0.048
u1t3q=0.3 0.0 1.0 0 0
u1t3=0.072
u1t4q=0.3 0.0 1.0 0 0
u1t4=0.096
u1t5q=0.3 0.0 1.0 0 0
u1t5=0.12
u1t6q=0.3 0.0 1.0 0 0
u1t6=0.144
u1t7q=0.3 0.0 1.0 0 0
u1t7=0.168
u1t8q=0.3 0.0 1.0 0 0
u1t8=0.192
u1t9q=0.3 0.0 1.0 0 0
u1t9=0.216
u1t10q=0.3 0.0 1.0 0 0
u1t10=0.24
u1t11q=0.3 0.0 1.0 0 0
u1t11=0.264
u1t12q=0.3 0.0 1.0 0 0
u1t12=0.288
u1t13q=0.3 0.0 1.0 0 0
u1t13=0.312
u1t14q=0.3 0.0 1.0 0 0
u1t14=0.336
u1t15q=0.3 0.0 1.0 0 0
u1t15=0.36
u1t16q=0.3 0.0 1.0 0 0
u1t16=0.384
u1t17q=0.3 0.0 1.0 0 0
u1t17=0.408
u1t18q=0.3 0.0 1.0 0 0
u1t18=0.432
u1t19q=0.3 0.0 1.0 0 0
u1t19=0.456
u1t20q=0.3 0.0 1.0 0 0
u1t20=0.48
u1t21q=0.3 0.0 1.0 0 0
u1t21=0.504
u1t22q=0.3 0.0 1.0 0 0
u1t22=0.528
u1t23q=0.3 0.0 1.0 0 0
u1t23=0.552
u1t24q=0.3 0.0 1.0 0 0
u1t24=0.576
u1t25q=0.3 0.0 1.0 0 0
u1t25=0.6
u1t26q=0.3 0.0 1.0 0 0
u1t26=0.624
u1t27q=0.3 0.0 1.0 0 0
u1t27=0.648
u1t28q=0.3 0.0 1.0 0 0
u1t28=0.672
u1t29q=0.3 0.0 1.0 0 0
u1t29=0.696
u1t30q=0.3 0.0 1.0 0 0
u1t30=0.72
u1t31q=0.3 0.0 1.0 0 0
u1t31=0.744
u1t32q=0.3 0.0 1.0 0 0
u1t32=0.768
u1t33q=0.3 0.0 1.0 0 0
u1t33=0.792
u1t34q=0.3 0.0 1.0 0 0
u1t34=0.816
u1t35q=0.3 0.0 1.0 0 0
u1t35=0.84
u1t36q=0.3 0.0 1.0 0 0
u1t36=0.864
u1t37q=0.3 0.0 1.0 0 0
u1t37=0.888
u1t38q=0.3 0.0 1.0 0 0
u1t38=0.912
u1t39q=0.3 0.0 1.0 0 0
u1t39=0.936
u1t40q=0.3 0.0 1.0 0 0
u1t40=0.96
u1t41q=0.3 0.0 1.0 0 0
u1t41=0.984
u1t42q=0.3 0.0 1.0 0 0
u1t42=1.008
u1t43q=0.3 0.0 1.0 0 0
u1t43=1.032
u1t44q=0.3 0.0 1.0 0 0
u1t44=1.056
u1t45q=0.3 0.0 1.0 0 0
u1t45=1.08
u1t46q=0.3 0.0 1.0 0 0
u1t46=1.104
u1t47q=0.3 0.0 1.0 0 0
u1t47=1.128
u1t48q=0.3 0.0 1.0 0 0
u1t48=1.152
u1t49q=0.3 0.0 1.0 0 0
u1t49=1.176
u1t50q=0.3 0.0 1.0 0 0
u1t50=1.2
u1t51q=0.3 0.0 1.0 0 0
u1t51=1.224
u1t52q=0.3 0.0 1.0 0 0
u1t52=1.248
u1t53q=0.3 0.0 1.0 0 0
u1t53=1.272
u1t54q=0.3 0.0 1.0 0 0
u1t54=1.296
u1t55q=0.3 0.0 1.0 0 0
u1t55=1.32
u1t56q=0.3 0.0 1.0 0 0
u1t56=1.344
u1t57q=0.3 0.0 1.0 0 0
u1t57=1.368
u1t58q=0.3 0.0 1.0 0 0
u1t58=1.392
u1t59q=0.3 0.0 1.0 0 0
u1t59=1.416
u1t60q=0.3 0.0 1.0 0 0
u1t60=1.44
u1t61q=0.3 0.0 1.0 0 0
u1t61=1.464
u1t62q=0.3 0.0 1.0 0 0
u1t62=1.488
u1t63q=0.3 0.0 1.0 0 0
u1t63=1.512
u1t64q=0.3 0.0 1.0 0 0
u1t64=1.536
u1t65q=0.3 0.0 1.0 0 0
u1t65=1.56
u1t66q=0.3 0.0 1.0 0 0
u1t66=1.584
u1t67q=0.3 0.0 1.0 0 0
u1t67=1.608
u1t68q=0.3 0.0 1.0 0 0
u1t68=1.632
u1t69q=0.3 0.0 1.0 0 0
u1t69=1.656
u1t70q=0.3 0.0 1.0 0 0
u1t70=1.68
u1t71q=0.3 0.0 1.0 0 0
u1t71=1.704
u1t72q=0.3 0.0 1.0 0 0
u1t72=1.728
u1t73q=0.3 0.0 1.0 0 0
u1t73=1.752
u1t74q=0.3 0.0 1.0 0 0
u1t74=1.776
u1t75q=0.3 0.0 1.0 0 0
u1t75=1.8
u1t76q=0.3 0.0 1.0 0 0
u1t76=1.824
u1t77q=0.3 0.0 1.0 0 0
u1t77=1.848
u1t78q=0.3 0.0 1.0 0 0
u1t78=1.872
u1t79q=0.3 0.0 1.0 0 0
u1t79=1.896
u1t80q=0.3 0.0 1.0 0 0
u1t80=1.92
u1t81q=0.3 0.0 1.0 0 0
u1t81=1.944
u1t82q=0.3 0.0 1.0 0 0
u1t82=1.968
u1t83q=0.3 0.0 1.0 0 0
u1t83=1.992
u1t84q=0.3 0.0 1.0 0 0
u1t84=2.016
u1t85q=0.3 0.0 1.0 0 0
u1t85=2.04
u1t86q=0.3 0.0 1.0 0 0
u1t86=2.064
u1t87q=0.3 0.0 1.0 0 0
u1t87=2.088
u1t88q=0.3 0.0 1.0 0 0
u1t88=2.112
u1t89q=0.3 0.0 1.0 0 0
u1t89=2.136
u1t90q=0.3 0.0 1.0 0 0
u1t90=2.16
u1t91q=0.3 0.0 1.0 0 0
u1t91=2.184
u1t92q=0.3 0.0 1.0 0 0
u1t92=2.208
u1t93q=0.3 0.0 1.0 0 0
u1t93=2.232
u1t94q=0.3 0.0 1.0 0 0
u1t94=2.256
u1t95q=0.3 0.0 1.0 0 0
u1t95=2.28
u1t96q=0.3 0.0 1.0 0 0
u1t96=2.304
u1t97q=0.3 0.0 1.0 0 0
u1t97=2.328
u1t98q=0.3 0.0 1.0 0 0
u1t98=2.352
u1t99q=0.3 0.0 1.0 0 0
u1t99=2.376
u1t100q=0.3 0.0 1.0 0 0
u1t100=2.4
u1t101q=0.3 0.0 1.0 0 0
u1t101=2.424
u1t102q=0.3 0.0 1.0 0 0
u1t102=2.448
u1t103q=0.3 0.0 1.0 0 0
u1t103=2.472
u1t104q=0.3 0.0 1.0 0 0
u1t104=2.496
u1t105q=0.3 0.0 1.0 0 0
u1t105=2.52
u1t106q=0.3 0.0 1.0 0 0
u1t106=2.544
u1t107q=0.3 0.0 1.0 0 0
u1t107=2.568
u1t108q=0.3 0.0 1.0 0 0
u1t108=2.592
u1t109q=0.3 0.0 1.0 0 0
u1t109=2.616
u1t110q=0.3 0.0 1.0 0 0
u1t110=2.64
u1t111q=0.3 0.0 1.0 0 0
u1t111=2.664
u1t112q=0.3 0.0 1.0 0 0
u1t112=2.688
u1t113q=0.3 0.0 1.0 0 0
u1t113=2.712
u1t114q=0.3 0.0 1.0 0 0
u1t114=2.736
u1t115q=0.3 0.0 1.0 0 0
u1t115=2.76
u1t116q=0.3 0.0 1.0 0 0
u1t116=2.784
u1t117q=0.3 0.0 1.0 0 0
u1t117=2.808
u1t118q=0.3 0.0 1.0 0 0
u1t118=2.832
u1t119q=0.3 0.0 1.0 0 0
u1t119=2.856
u1t120q=0.3 0.0 1.0 0 0
u1t120=2.88
u1t121q=0.3 0.0 1.0 0 0
u1t121=2.904
u1t122q=0.3 0.0 1.0 0 0
u1t122=2.928
u1t123q=0.3 0.0 1.0 0 0
u1t123=2.952
u1t124q=0.3 0.0 1.0 0 0
u1t124=2.976
u1t125q=0.3 0.0 1.0 0 0
u1t125=3.0
u1t126q=0.3 0.0 1.0 0 0
u1t126=3.024
u1t127q=0.3 0.0 1.0 0 0
u1t127=3.048
u1t128q=0.3 0.0 1.0 0 0
u1t128=3.072
u1t129q=0.3 0.0 1.0 0 0
u1t129=3.096
u1t130q=0.3 0.0 1.0 0 0
u1t130=3.12
u1t131q=0.3 0.0 1.0 0 0
u1t131=3.144
u1t132q=0.3 0.0 1.0 0 0
u1t132=3.168
u1t133q=0.3 0.0 1.0 0 0
u1t133=3.192
u1t134q=0.3 0.0 1.0 0 0
u1t134=3.216
u1t135q=0.3 0.0 1.0 0 0
u1t135=3.24
u1t136q=0.3 0.0 1.0 0 0
u1t136=3.264
u1t137q=0.3 0.0 1.0 0 0
u1t137=3.288
u1t138q=0.3 0.0 1.0 0 0
u1t138=3.312
u1t139q=0.3 0.0 1.0 0 0
u1t139=3.336
u1t140q=0.3 0.0 1.0 0 0
u1t140=3.36
u1t141q=0.3 0.0 1.0 0 0
u1t141=3.384
u1t142q=0.3 0.0 1.0 0 0
u1t142=3.408
u1t143q=0.3 0.0 1.0 0 0
u1t143=3.432
u1t144q=0.3 0.0 1.0 0 0
u1t144=3.456
u1t145q=0.3 0.0 1.0 0 0
u1t145=3.48
u1t146q=0.3 0.0 1.0 0 0
u1t146=3.504
u1t147q=0.3 0.0 1.0 0 0
u1t147=3.528
u1t148q=0.3 0.0 1.0 0 0
u1t148=3.552
u1t149q=0.3 0.0 1.0 0 0
u1t149=3.576
u1t150q=0.3 0.0 1.0 0 0
u1t150=3.6
u1t151q=0.3 0.0 1.0 0 0
u1t151=3.624
u1t152q=0.3 0.0 1.0 0 0
u1t152=3.648
u1t153q=0.3 0.0 1.0 0 0
u1t153=3.672
u1t154q=0.3 0.0 1.0 0 0
u1t154=3.696
u1t155q=0.3 0.0 1.0 0 0
u1t155=3.72
u1t156q=0.3 0.0 1.0 0 0
u1t156=3.744
u1t157q=0.3 0.0 1.0 0 0
u1t157=3.768
u1t158q=0.3 0.0 1.0 0 0
u1t158=3.792
u1t159q=0.3 0.0 1.0 0 0
u1t159=3.816
u1t160q=0.3 0.0 1.0 0 0
u1t160=3.84
u1t161q=0.3 0.0 1.0 0 0
u1t161=3.864
u1t162q=0.3 0.0 1.0 0 0
u1t162=3.888
u1t163q=0.3 0.0 1.0 0 0
u1t163=3.912
u1t164q=0.3 0.0 1.0 0 0
u1t164=3.936
u1t165q=0.3 0.0 1.0 0 0
u1t165=3.96
u1t166q=0.3 0.0 1.0 0 0
u1t166=3.984
u1t167q=0.3 0.0 1.0 0 0
u1t167=4.008
u1t168q=0.3 0.0 1.0 0 0
u1t168=4.032
u1t169q=0.3 0.0 1.0 0 0
u1t169=4.056
u1t170q=0.3 0.0 1.0 0 0
u1t170=4.08
u1t171q=0.3 0.0 1.0 0 0
u1t171=4.104
u1t172q=0.3 0.0 1.0 0 0
u1t172=4.128
u1t173q=0.3 0.0 1.0 0 0
u1t173=4.152
u1t174q=0.3 0.0 1.0 0 0
u1t174=4.176
u1t175q=0.3 0.0 1.0 0 0
u1t175=4.2
u1t176q=0.3 0.0 1.0 0 0
u1t176=4.224
u1t177q=0.3 0.0 1.0 0 0
u1t177=4.248
u1t178q=0.3 0.0 1.0 0 0
u1t178=4.272
u1t179q=0.3 0.0 1.0 0 0
u1t179=4.296
u1t180q=0.3 0.0 1.0 0 0
u1t180=4.32
u1t181q=0.3 0.0 1.0 0 0
u1t181=4.344
u1t182q=0.3 0.0 1.0 0 0
u1t182=4.368
u1t183q=0.3 0.0 1.0 0 0
u1t183=4.392
u1t184q=0.3 0.0 1.0 0 0
u1t184=4.416
u1t185q=0.3 0.0 1.0 0 0
u1t185=4.44
u1t186q=0.3 0.0 1.0 0 0
u1t186=4.464
u1t187q=0.3 0.0 1.0 0 0
u1t187=4.488
u1t188q=0.3 0.0 1.0 0 0
u1t188=4.512
u1t189q=0.3 0.0 1.0 0 0
u1t189=4.536
u1t190q=0.3 0.0 1.0 0 0
u1t190=4.56
u1t191q=0.3 0.0 1.0 0 0
u1t191=4.584
u1t192q=0.3 0.0 1.0 0 0
u1t192=4.608
u1t193q=0.3 0.0 1.0 0 0
u1t193=4.632
u1t194q=0.3 0.0 1.0 0 0
u1t194=4.656
u1t195q=0.3 0.0 1.0 0 0
u1t195=4.68
u1t196q=0.3 0.0 1.0 0 0
u1t196=4.704
u1t197q=0.3 0.0 1.0 0 0
u1t197=4.728
u1t198q=0.3 0.0 1.0 0 0
u1t198=4.752
u1t199q=0.3 0.0 1.0 0 0
u1t199=4.776
u1t200q=0.3 0.0 1.0 0 0
u1t200=4.8
u1t201q=0.3 0.0 1.0 0 0
u1t201=4.824
u1t202q=0.3 0.0 1.0 0 0
u1t202=4.848
u1t203q=0.3 0.0 1.0 0 0
u1t203=4.872
u1t204q=0.3 0.0 1.0 0 0
u1t204=4.896
u1t205q=0.3 0.0 1.0 0 0
u1t205=4.92
u1t206q=0.3 0.0 1.0 0 0
u1t206=4.944
u1t207q=0.3 0.0 1.0 0 0
u1t207=4.968
u1t208q=0.3 0.0 1.0 0 0
u1t208=4.992
u1t209q=0.3 0.0 1.0 0 0
u1t209=5.016
u1t210q=0.3 0.0 1.0 0 0
u1t210=5.04
u1t211q=0.3 0.0 1.0 0 0
u1t211=5.064
u1t212q=0.3 0.0 1.0 0 0
u1t212=5.088
u1t213q=0.3 0.0 1.0 0 0
u1t213=5.112
u1t214q=0.3 0.0 1.0 0 0
u1t214=5.136
u1t215q=0.3 0.0 1.0 0 0
u1t215=5.16
u1t216q=0.3 0.0 1.0 0 0
u1t216=5.184
u1t217q=0.3 0.0 1.0 0 0
u1t217=5.208
u1t218q=0.3 0.0 1.0 0 0
u1t218=5.232
u1t219q=0.3 0.0 1.0 0 0
u1t219=5.256
u1t220q=0.3 0.0 1.0 0 0
u1t220=5.28
u1t221q=0.3 0.0 1.0 0 0
u1t221=5.304
u1t222q=0.3 0.0 1.0 0 0
u1t222=5.328
u1t223q=0.3 0.0 1.0 0 0
u1t223=5.352
u1t224q=0.3 0.0 1.0 0 0
u1t224=5.376
u1t225q=0.3 0.0 1.0 0 0
u1t225=5.4
u1t226q=0.3 0.0 1.0 0 0
u1t226=5.424
u1t227q=0.3 0.0 1.0 0 0
u1t227=5.448
u1t228q=0.3 0.0 1.0 0 0
u1t228=5.472
u1t229q=0.3 0.0 1.0 0 0
u1t229=5.496
u1t230q=0.3 0.0 1.0 0 0
u1t230=5.52
u1t231q=0.3 0.0 1.0 0 0
u1t231=5.544
u1t232q=0.3 0.0 1.0 0 0
u1t232=5.568
u1t233q=0.3 0.0 1.0 0 0
u1t233=5.592
u1t234q=0.3 0.0 1.0 0 0
u1t234=5.616
u1t235q=0.3 0.0 1.0 0 0
u1t235=5.64
u1t236q=0.3 0.0 1.0 0 0
u1t236=5.664
u1t237q=0.3 0.0 1.0 0 0
u1t237=5.688
u1t238q=0.3 0.0 1.0 0 0
u1t238=5.712
u1t239q=0.3 0.0 1.0 0 0
u1t239=5.736
u1t240q=0.3 0.0 1.0 0 0
u1t240=5.76
u1t241q=0.3 0.0 1.0 0 0
u1t241=5.784
u1t242q=0.3 0.0 1.0 0 0
u1t242=5.808
u1t243q=0.3 0.0 1.0 0 0
u1t243=5.832
u1t244q=0.3 0.0 1.0 0 0
u1t244=5.856
u1t245q=0.3 0.0 1.0 0 0
u1t245=5.88
u1t246q=0.3 0.0 1.0 0 0
u1t246=5.904
u1t247q=0.3 0.0 1.0 0 0
u1t247=5.928
u1t248q=0.3 0.0 1.0 0 0
u1t248=5.952
u1t249q=0.3 0.0 1.0 0 0
u1t249=5.976
u1t250q=0.3 0.0 1.0 0 0
u1t250=6.0
u1t251q=0.3 0.0 1.0 0 0
u1t251=6.024
u1t252q=0.3 0.0 1.0 0 0
u1t252=6.048
u1t253q=0.3 0.0 1.0 0 0
u1t253=6.072
u1t254q=0.3 0.0 1.0 0 0
u1t254=6.096
u1t255q=0.3 0.0 1.0 0 0
u1t255=6.12
u1t256q=0.3 0.0 1.0 0 0
u1t256=6.144
u1t257q=0.3 0.0 1.0 0 0
u1t257=6.168
u1t258q=0.3 0.0 1.0 0 0
u1t258=6.192
u1t259q=0.3 0.0 1.0 0 0
u1t259=6.216
u1t260q=0.3 0.0 1.0 0 0
u1t260=6.24
u1t261q=0.3 0.0 1.0 0 0
u1t261=6.264
u1t262q=0.3 0.0 1.0 0 0
u1t262=6.288
u1t263q=0.3 0.0 1.0 0 0
u1t263=6.312
u1t264q=0.3 0.0 1.0 0 0
u1t264=6.336
u1t265q=0.3 0.0 1.0 0 0
u1t265=6.36
u1t266q=0.3 0.0 1.0 0 0
u1t266=6.384
u1t267q=0.3 0.0 1.0 0 0
u1t267=6.408
u1t268q=0.3 0.0 1.0 0 0
u1t268=6.432
u1t269q=0.3 0.0 1.0 0 0
u1t269=6.456
u1t270q=0.3 0.0 1.0 0 0
u1t270=6.48
u1t271q=0.3 0.0 1.0 0 0
u1t271=6.504
u1t272q=0.3 0.0 1.0 0 0
u1t272=6.528
u1t273q=0.3 0.0 1.0 0 0
u1t273=6.552
u1t274q=0.3 0.0 1.0 0 0
u1t274=6.576
u1t275q=0.3 0.0 1.0 0 0
u1t275=6.6
u1t276q=0.3 0.0 1.0 0 0
u1t276=6.624
u1t277q=0.3 0.0 1.0 0 0
u1t277=6.648
u1t278q=0.3 0.0 1.0 0 0
u1t278=6.672
u1t279q=0.3 0.0 1.0 0 0
u1t279=6.696
u1t280q=0.3 0.0 1.0 0 0
u1t280=6.72
u1t281q=0.3 0.0 1.0 0 0
u1t281=6.744
u1t282q=0.3 0.0 1.0 0 0
u1t282=6.768
u1t283q=0.3 0.0 1.0 0 0
u1t283=6.792
u1t284q=0.3 0.0 1.0 0 0
u1t284=6.816
u1t285q=0.3 0.0 1.0 0 0
u1t285=6.84
u1t286q=0.3 0.0 1.0 0 0
u1t286=6.864
u1t287q=0.3 0.0 1.0 0 0
u1t287=6.888
u1t288q=0.3 0.0 1.0 0 0
u1t288=6.912
u1t289q=0.3 0.0 1.0 0 0
u1t289=6.936
u1t290q=0.3 0.0 1.0 0 0
u1t290=6.96
u1t291q=0.3 0.0 1.0 0 0
u1t291=6.984
u1t292q=0.3 0.0 1.0 0 0
u1t292=7.008
u1t293q=0.3 0.0 1.0 0 0
u1t293=7.032
u1t294q=0.3 0.0 1.0 0 0
u1t294=7.056
u1t295q=0.3 0.0 1.0 0 0
u1t295=7.08
u1t296q=0.3 0.0 1.0 0 0
u1t296=7.104
u1t297q=0.3 0.0 1.0 0 0
u1t297=7.128
u1t298q=0.3 0.0 1.0 0 0
u1t298=7.152
u1t299q=0.3 0.0 1.0 0 0
u1t299=7.176
u1t300q=0.3 0.0 1.0 0 0
u1t300=7.2
u1t301q=0.3 0.0 1.0 0 0
u1t301=7.224
u1t302q=0.3 0.0 1.0 0 0
u1t302=7.248
u1t303q=0.3 0.0 1.0 0 0
u1t303=7.272
u1t304q=0.3 0.0 1.0 0 0
u1t304=7.296
u1t305q=0.3 0.0 1.0 0 0
u1t305=7.32
u1t306q=0.3 0.0 1.0 0 0
u1t306=7.344
u1t307q=0.3 0.0 1.0 0 0
u1t307=7.368
u1t308q=0.3 0.0 1.0 0 0
u1t308=7.392
u1t309q=0.3 0.0 1.0 0 0
u1t309=7.416
u1t310q=0.3 0.0 1.0 0 0
u1t310=7.44
u1t311q=0.3 0.0 1.0 0 0
u1t311=7.464
u1t312q=0.3 0.0 1.0 0 0
u1t312=7.488
u1t313q=0.3 0.0 1.0 0 0
u1t313=7.512
u1t314q=0.3 0.0 1.0 0 0
u1t314=7.536
u1t315q=0.3 0.0 1.0 0 0
u1t315=7.56
u1t316q=0.3 0.0 1.0 0 0
u1t316=7.584
u1t317q=0.3 0.0 1.0 0 0
u1t317=7.608
u1t318q=0.3 0.0 1.0 0 0
u1t318=7.632
u1t319q=0.3 0.0 1.0 0 0
u1t319=7.656
u1t320q=0.3 0.0 1.0 0 0
u1t320=7.68
u1t321q=0.3 0.0 1.0 0 0
u1t321=7.704
u1t322q=0.3 0.0 1.0 0 0
u1t322=7.728
u1t323q=0.3 0.0 1.0 0 0
u1t323=7.752
u1t324q=0.3 0.0 1.0 0 0
u1t324=7.776
u1t325q=0.3 0.0 1.0 0 0
u1t325=7.8
u1t326q=0.3 0.0 1.0 0 0
u1t326=7.824
u1t327q=0.3 0.0 1.0 0 0
u1t327=7.848
u1t328q=0.3 0.0 1.0 0 0
u1t328=7.872
u1t329q=0.3 0.0 1.0 0 0
u1t329=7.896
u1t330q=0.3 0.0 1.0 0 0
u1t330=7.92
u1t331q=0.3 0.0 1.0 0 0
u1t331=7.944
u1t332q=0.3 0.0 1.0 0 0
u1t332=7.968
u1t333q=0.3 0.0 1.0 0 0
u1t333=7.992
u1t334q=0.3 0.0 1.0 0 0
u1t334=8.016
u1t335q=0.3 0.0 1.0 0 0
u1t335=8.04
u1t336q=0.3 0.0 1.0 0 0
u1t336=8.064
u1t337q=0.3 0.0 1.0 0 0
u1t337=8.088
u1t338q=0.3 0.0 1.0 0 0
u1t338=8.112
u1t339q=0.3 0.0 1.0 0 0
u1t339=8.136
u1t340q=0.3 0.0 1.0 0 0
u1t340=8.16
u1t341q=0.3 0.0 1.0 0 0
u1t341=8.184
u1t342q=0.3 0.0 1.0 0 0
u1t342=8.208
u1t343q=0.3 0.0 1.0 0 0
u1t343=8.232
u1t344q=0.3 0.0 1.0 0 0
u1t344=8.256
u1t345q=0.3 0.0 1.0 0 0
u1t345=8.28
u1t346q=0.3 0.0 1.0 0 0
u1t346=8.304
u1t347q=0.3 0.0 1.0 0 0
u1t347=8.328
u1t348q=0.3 0.0 1.0 0 0
u1t348=8.352
u1t349q=0.3 0.0 1.0 0 0
u1t349=8.376
u1t350q=0.3 0.0 1.0 0 0
u1t350=8.4
u1t351q=0.3 0.0 1.0 0 0
u1t351=8.424
u1t352q=0.3 0.0 1.0 0 0
u1t352=8.448
u1t353q=0.3 0.0 1.0 0 0
u1t353=8.472
u1t354q=0.3 0.0 1.0 0 0
u1t354=8.496
u1t355q=0.3 0.0 1.0 0 0
u1t355=8.52
u1t356q=0.3 0.0 1.0 0 0
u1t356=8.544
u1t357q=0.3 0.0 1.0 0 0
u1t357=8.568
u1t358q=0.3 0.0 1.0 0 0
u1t358=8.592
u1t359q=0.3 0.0 1.0 0 0
u1t359=8.616
u1t360q=0.3 0.0 1.0 0 0
u1t360=8.64
u1t361q=0.3 0.0 1.0 0 0
u1t361=8.664
u1t362q=0.3 0.0 1.0 0 0
u1t362=8.688
u1t363q=0.3 0.0 1.0 0 0
u1t363=8.712
u1t364q=0.3 0.0 1.0 0 0
u1t364=8.736
u1t365q=0.3 0.0 1.0 0 0
u1t365=8.76
u1t366q=0.3 0.0 1.0 0 0
u1t366=8.784
u1t367q=0.3 0.0 1.0 0 0
u1t367=8.808
u1t368q=0.3 0.0 1.0 0 0
u1t368=8.832
u1t369q=0.3 0.0 1.0 0 0
u1t369=8.856
u1t370q=0.3 0.0 1.0 0 0
u1t370=8.88
u1t371q=0.3 0.0 1.0 0 0
u1t371=8.904
u1t372q=0.3 0.0 1.0 0 0
u1t372=8.928
u1t373q=0.3 0.0 1.0 0 0
u1t373=8.952
u1t374q=0.3 0.0 1.0 0 0
u1t374=8.976
u1t375q=0.3 0.0 1.0 0 0
u1t375=9.0
u1t376q=0.3 0.0 1.0 0 0
u1t376=9.024
u1t377q=0.3 0.0 1.0 0 0
u1t377=9.048
u1t378q=0.3 0.0 1.0 0 0
u1t378=9.072
u1t379q=0.3 0.0 1.0 0 0
u1t379=9.096
u1t380q=0.3 0.0 1.0 0 0
u1t380=9.12
u1t381q=0.3 0.0 1.0 0 0
u1t381=9.144
u1t382q=0.3 0.0 1.0 0 0
u1t382=9.168
u1t383q=0.3 0.0 1.0 0 0
u1t383=9.192
u1t384q=0.3 0.0 1.0 0 0
u1t384=9.216
u1t385q=0.3 0.0 1.0 0 0
u1t385=9.24
u1t386q=0.3 0.0 1.0 0 0
u1t386=9.264
u1t387q=0.3 0.0 1.0 0 0
u1t387=9.288
u1t388q=0.3 0.0 1.0 0 0
u1t388=9.312
u1t389q=0.3 0.0 1.0 0 0
u1t389=9.336
u1t390q=0.3 0.0 1.0 0 0
u1t390=9.36
u1t391q=0.3 0.0 1.0 0 0
u1t391=9.384
u1t392q=0.3 0.0 1.0 0 0
u1t392=9.408
u1t393q=0.3 0.0 1.0 0 0
u1t393=9.432
u1t394q=0.3 0.0 1.0 0 0
u1t394=9.456
u1t395q=0.3 0.0 1.0 0 0
u1t395=9.48
u1t396q=0.3 0.0 1.0 0 0
u1t396=9.504
u1t397q=0.3 0.0 1.0 0 0
u1t397=9.528
u1t398q=0.3 0.0 1.0 0 0
u1t398=9.552
u1t399q=0.3 0.0 1.0 0 0
u1t399=9.576
u1t400q=0.3 0.0 1.0 0 0
u1t400=9.6
u1t401q=0.3 0.0 1.0 0 0
u1t401=9.624
u1t402q=0.3 0.0 1.0 0 0
u1t402=9.648
u1t403q=0.3 0.0 1.0 0 0
u1t403=9.672
u1t404q=0.3 0.0 1.0 0 0
u1t404=9.696
u1t405q=0.3 0.0 1.0 0 0
u1t405=9.72
u1t406q=0.3 0.0 1.0 0 0
u1t406=9.744
u1t407q=0.3 0.0 1.0 0 0
u1t407=9.768
u1t408q=0.3 0.0 1.0 0 0
u1t408=9.792
u1t409q=0.3 0.0 1.0 0 0
u1t409=9.816
u1t410q=0.3 0.0 1.0 0 0
u1t410=9.84
u1t411q=0.3 0.0 1.0 0 0
u1t411=9.864
u1t412q=0.3 0.0 1.0 0 0
u1t412=9.888
u1t413q=0.3 0.0 1.0 0 0
u1t413=9.912
u1t414q=0.3 0.0 1.0 0 0
u1t414=9.936
u1t415q=0.3 0.0 1.0 0 0
u1t415=9.96
u1t416q=0.3 0.0 1.0 0 0
u1t416=9.984
u1t417q=0.3 0.0 1.0 0 0
u1t417=10.008
u1t418q=0.3 0.0 1.0 0 0
u1t418=10.032
u1t419q=0.3 0.0 1.0 0 0
u1t419=10.056
u1t420q=0.3 0.0 1.0 0 0
u1t420=10.08
u1t421q=0.3 0.0 1.0 0 0
u1t421=10.104
u1t422q=0.3 0.0 1.0 0 0
u1t422=10.128
u1t423q=0.3 0.0 1.0 0 0
u1t423=10.152
u1t424q=0.3 0.0 1.0 0 0
u1t424=10.176
u1t425q=0.3 0.0 1.0 0 0
u1t425=10.2
u1t426q=0.3 0.0 1.0 0 0
u1t426=10.224
u1t427q=0.3 0.0 1.0 0 0
u1t427=10.248
u1t428q=0.3 0.0 1.0 0 0
u1t428=10.272
u1t429q=0.3 0.0 1.0 0 0
u1t429=10.296
u1t430q=0.3 0.0 1.0 0 0
u1t430=10.32
u1t431q=0.3 0.0 1.0 0 0
u1t431=10.344
u1t432q=0.3 0.0 1.0 0 0
u1t432=10.368
u1t433q=0.3 0.0 1.0 0 0
u1t433=10.392
u1t434q=0.3 0.0 1.0 0 0
u1t434=10.416
u1t435q=0.3 0.0 1.0 0 0
u1t435=10.44
u1t436q=0.3 0.0 1.0 0 0
u1t436=10.464
u1t437q=0.3 0.0 1.0 0 0
u1t437=10.488
u1t438q=0.3 0.0 1.0 0 0
u1t438=10.512
u1t439q=0.3 0.0 1.0 0 0
u1t439=10.536
u1t440q=0.3 0.0 1.0 0 0
u1t440=10.56
u1t441q=0.3 0.0 1.0 0 0
u1t441=10.584
u1t442q=0.3 0.0 1.0 0 0
u1t442=10.608
u1t443q=0.3 0.0 1.0 0 0
u1t443=10.632
u1t444q=0.3 0.0 1.0 0 0
u1t444=10.656
u1t445q=0.3 0.0 1.0 0 0
u1t445=10.68
u1t446q=0.3 0.0 1.0 0 0
u1t446=10.704
u1t447q=0.3 0.0 1.0 0 0
u1t447=10.728
u1t448q=0.3 0.0 1.0 0 0
u1t448=10.752
u1t449q=0.3 0.0 1.0 0 0
u1t449=10.776
u1t450q=0.3 0.0 1.0 0 0
u1t450=10.8
u1t451q=0.3 0.0 1.0 0 0
u1t451=10.824
u1t452q=0.3 0.0 1.0 0 0
u1t452=10.848
u1t453q=0.3 0.0 1.0 0 0
u1t453=10.872
u1t454q=0.3 0.0 1.0 0 0
u1t454=10.896
u1t455q=0.3 0.0 1.0 0 0
u1t455=10.92
u1t456q=0.3 0.0 1.0 0 0
u1t456=10.944
u1t457q=0.3 0.0 1.0 0 0
u1t457=10.968
u1t458q=0.3 0.0 1.0 0 0
u1t458=10.992
u1t459q=0.3 0.0 1.0 0 0
u1t459=11.016
u1t460q=0.3 0.0 1.0 0 0
u1t460=11.04
u1t461q=0.3 0.0 1.0 0 0
u1t461=11.064
u1t462q=0.3 0.0 1.0 0 0
u1t462=11.088
u1t463q=0.3 0.0 1.0 0 0
u1t463=11.112
u1t464q=0.3 0.0 1.0 0 0
u1t464=11.136
u1t465q=0.3 0.0 1.0 0 0
u1t465=11.16
u1t466q=0.3 0.0 1.0 0 0
u1t466=11.184
u1t467q=0.3 0.0 1.0 0 0
u1t467=11.208
u1t468q=0.3 0.0 1.0 0 0
u1t468=11.232
u1t469q=0.3 0.0 1.0 0 0
u1t469=11.256
u1t470q=0.3 0.0 1.0 0 0
u1t470=11.28
u1t471q=0.3 0.0 1.0 0 0
u1t471=11.304
u1t472q=0.3 0.0 1.0 0 0
u1t472=11.328
u1t473q=0.3 0.0 1.0 0 0
u1t473=11.352
u1t474q=0.3 0.0 1.0 0 0
u1t474=11.376
u1t475q=0.3 0.0 1.0 0 0
u1t475=11.4
u1t476q=0.3 0.0 1.0 0 0
u1t476=11.424
u1t477q=0.3 0.0 1.0 0 0
u1t477=11.448
u1t478q=0.3 0.0 1.0 0 0
u1t478=11.472
u1t479q=0.3 0.0 1.0 0 0
u1t479=11.496
u1t480q=0.3 0.0 1.0 0 0
u1t480=11.52
u1t481q=0.3 0.0 1.0 0 0
u1t481=11.544
u1t482q=0.3 0.0 1.0 0 0
u1t482=11.568
u1t483q=0.3 0.0 1.0 0 0
u1t483=11.592
u1t484q=0.3 0.0 1.0 0 0
u1t484=11.616
u1t485q=0.3 0.0 1.0 0 0
u1t485=11.64
u1t486q=0.3 0.0 1.0 0 0
u1t486=11.664
u1t487q=0.3 0.0 1.0 0 0
u1t487=11.688
u1t488q=0.3 0.0 1.0 0 0
u1t488=11.712
u1t489q=0.3 0.0 1.0 0 0
u1t489=11.736
u1t490q=0.3 0.0 1.0 0 0
u1t490=11.76
u1t491q=0.3 0.0 1.0 0 0
u1t491=11.784
u1t492q=0.3 0.0 1.0 0 0
u1t492=11.808
u1t493q=0.3 0.0 1.0 0 0
u1t493=11.832
u1t494q=0.3 0.0 1.0 0 0
u1t494=11.856
u1t495q=0.3 0.0 1.0 0 0
u1t495=11.88
u1t496q=0.3 0.0 1.0 0 0
u1t496=11.904
u1t497q=0.3 0.0 1.0 0 0
u1t497=11.928
u1t498q=0.3 0.0 1.0 0 0
u1t498=11.952
u1t499q=0.3 0.0 1.0 0 0
u1t499=11.976
u1t500q=0.3 0 0 0 0
u1t500=tend

[Messungen]
tAnzahl=12

t1=1
t1Anzahl=2
t1m1=mfcn1 1.0 1e-06 1
t1m2=mfcn2 1.0 1e-06 1
t1minmax=0 1e+10

t2=2.000
t2Anzahl=2
t2m1=mfcn1 1.0 1e-06 1
t2m2=mfcn2 1.0 1e-06 1
t2minmax=0 1e+10

t3=3
t3Anzahl=2
t3m1=mfcn1 1.0 1e-06 1
t3m2=mfcn2 1.0 1e-06 1
t3minmax=0 1e+10

t4=4
t4Anzahl=2
t4m1=mfcn1 1.0 1e-06 1
t4m2=mfcn2 1.0 1e-06 1
t4minmax=0 1e+10

t5=5
t5Anzahl=2
t5m1=mfcn1 1.0 1e-06 1
t5m2=mfcn2 1.0 1e-06 1
t5minmax=0 1e+10

t6=6
t6Anzahl=2
t6m1=mfcn1 1.0 1e-06 1
t6m2=mfcn2 1.0 1e-06 1
t6minmax=0 1e+10

t7=7
t7Anzahl=2
t7m1=mfcn1 1.0 1e-06 1
t7m2=mfcn2 1.0 1e-06 1
t7minmax=0 1e+10

t8=8
t8Anzahl=2
t8m1=mfcn1 1.0 1e-06 1
t8m2=mfcn2 1.0 1e-06 1
t8minmax=0 1e+10

t9=9
t9Anzahl=2
t9m1=mfcn1 1.0 1e-06 1
t9m2=mfcn2 1.0 1e-06 1
t9minmax=0 1e+10

t10=10.000
t10Anzahl=2
t10m1=mfcn1 1.0 1e-06 1
t10m2=mfcn2 1.0 1e-06 1
t10minmax=0 1e+10

t11=11.000
t11Anzahl=2
t11m1=mfcn1 1.0 1e-06 1
t11m2=mfcn2 1.0 1e-06 1
t11minmax=0 1e+10

t12=12.000
t12Anzahl=2
t12m1=mfcn1 1.0 1e-06 1
t12m2=mfcn2 1.0 1e-06 1
t12minmax=0 1e+10
[NebenbedingungenSteuergroessen]
cAnzahl=0
[Messverfahren]
mAnzahl=2
m1=mfcn1 1 0 1e+10 0
m1f1=mess3 sigma3 1
m2=mfcn2 1 0 1e+10 0
m2f1=mess4 sigma4 1
mminmaxges=0 8

[OptionenIntegration]
teps=1e-08
rtol=1e-08
atol=1e-07
stepsize=0.0001
maxorder=6
maxstepnumber=4000
minstepsize=-1
maxstepsize=-1
maxitNewton=3
realworkspace=1700000
integerworkspace=5000
printlevel=0
mcnonlinearflag=0
mcDAEflag=0
mctol=1e-07
mcmaxit=50
mclinesearch=1
mcalpha0=1
rndmethod=-1
rndeps=1e-05
rndverbose=0

</source>

ini-file for running VPLAN:
<source lang="optimica">

; ini-File fuer VPLAN

[Aktion]
;aktion=Integration
;aktion=Simulationsumgebung
;aktion=Parameterschaetzung
;aktion=Versuchsplanung
;aktion=ObjectiveTest
;aktion=DerivativeTest
;aktion={ISPS}
aktion={ISCVCS}

[Pfade]
problempath=lotka_seminar
inpath=in
outpath=simu
messpath=mess
plotpath=plot
fortranpath=fortran

[Parameter]
pAnzahl=2
p1=p2 1.0 -1e+10 1e+10 0
p2=p4 1.0 -1e+10 1e+10 0
[Versuchsplan]
expAnzahl=1
exp1=exp1.ini exp1.ini

[Guetekriterium]
Optimierungskriterium=A
AKriterium=-1
DKriterium=-1
EKriterium=-1
MKriterium=-1
covmat=covmat.m
jacmat=jacmat.m
status=undefiniert

[Residuum]
res=0

[Kosten]
costs=0 -1e+10 1e+10

[Messdatenfiles]
mess1=mess1.dat

[Outputfiles]
out1=plot2 0.05 integ.plt.1

[Residuenfiles]
rsd1=res1.txt

[ExtensionFlags]
experimenttype=0
integrator=0
dmode=0
pdeFlag=0

[OptionenAllgemein]
visflag=0
messfileflag=0
seed=-1
numberofthreads=1
robustflag=0
epsmach=0
infinity=1e+10
epsilon=1e-08
conflevel=0.95
hrobust=1e-05
computesigma=0
exitonFPE=1
iniprecision=6
clipboardflag=0
printxi=0
printconstr=0
printcolorful=-1

[OptionenParameterschaetzung]
eps=0.001
itmax=50
cond=10000
condflag=1
boundcheck=0
startflag=0
index1=1e-08
fashort=0.8
fa0=0.01
farel=0.1
famax=1.0
realworkspace=1000000
integerworkspace=1000000
printlevel=2
method=0

[OptionenVersuchsplanung]
maxit=300
opttol=1e-06
funcprec=1e-07
linfeas=1e-07
nlinfeas=0.01
maxitQP=300
maxitgesQP=10000
opttolQP=1e-06
pivottolQP=3.7e-11
steplimitLS=2
tolLS=0.9
crashtol=0.0001
elasticweight=100
superbasics=1
scaling=1
sconstraints=0
realworkspace=3000000
integerworkspace=3000000
charworkspace=500
printlevel=10
method=2

</source>

Lotka Experimental Design (VPLAN)

2016-01-19T16:28:25Z

FelixJost: /* VPLAN */

== VPLAN ==

Differential equations:

<source lang="fortran">

c RHS of the differential equations

subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), f(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

real*8 p1,p3,p5,p6, myu

c fixed parameters
p1 = 1.0
p3 = 1.0
p5 = 0.4
p6 = 0.2

c DISCRETIZE1( myu, rwh, iwh )

f(1) = p1*x(1) - p(1)*x(1)*x(2) - p5*myu*x(1)
f(2) = (-1.0)*p3*x(2) + p(2)*x(1)*x(2) - p6*myu*x(2)

end

</source>

Algebraic equations:

<source lang="fortran">

c Dummyfunction for RHS of algebraic equations

subroutine gfcn( t, x, g, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), g(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

iflag=0

end

</source>

First Measurement function:

<source lang="fortran">

c Messfunktion

subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )
implicit none

real*8 t, x(*), h, p(*), q(*), rwh(*)
integer*4 iwh(*), iflag

h = x(1)

iflag = 0

end

</source>

Second Measurement function:

<source lang="fortran">

c Messfunktion

subroutine mess4( t, x, h, p, q, rwh, iwh, iflag )
implicit none

real*8 t, x(*), h, p(*), q(*), rwh(*)
integer*4 iwh(*), iflag

h = x(2)

iflag = 0

end

</source>

Standard deviation of first measurement function

<source lang="fortran">

c Standardabweichung der Messfunktion

subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )
implicit none

real*8 rwh(*)
integer*4 iwh(*), iflag

real*8 t, x(*), p(*), q(*)
real*8 s
real*8 h

s = 1.0d+0

iflag = 0

end

</source>

Standard deviation of second measurement function:

<source lang="fortran">
c Standardabweichung der Messfunktion

subroutine sigma4( t, x, s, p, q, rwh, iwh, iflag )
implicit none

real*8 rwh(*)
integer*4 iwh(*), iflag

real*8 t, x(*), p(*), q(*)
real*8 s(*)

s(1) = 1.0

iflag = 0

end

</source>

VPLAN specific experimental setup:

<source lang="optimica">

; ini-File fuer Experiment

[Flags]
switch=1

[Kosten]
costs=0 -1e+10 1e+10

[Integrationsintervall]
t0=0
tend=12

[Modellfunktionen]
ffcn=ffcn
gfcn=gfcn

[Zustandsvariablen]
yAnzahl=2
y1=x1 0.5 -1e+10 1e+10
y2=x2 0.7 -1e+10 1e+10

zAnzahl=0

[Mehrzielknoten]
tAnzahl=12
t1=1
t2=2
t3=3
t4=4
t5=5
t6=6
t7=7
t8=8
t9=9
t10=10
t11=11
t12=12

[DynamischeNebenbedingungen]
bAnzahl=0

[GitterUeberpruefungNebenbedingungen]
tAnzahl=0

[Steuergroessen]
qAnzahl=0

[Steuerfunktionen]
uAnzahl=1
u1=myu 0 0.0 1.0
u1tAnzahl=500
u1t0=t0
u1t1q=0.3 0.0 1.0 0 0
u1t1=0.024
u1t2q=0.3 0.0 1.0 0 0
u1t2=0.048
u1t3q=0.3 0.0 1.0 0 0
u1t3=0.072
u1t4q=0.3 0.0 1.0 0 0
u1t4=0.096
u1t5q=0.3 0.0 1.0 0 0
u1t5=0.12
u1t6q=0.3 0.0 1.0 0 0
u1t6=0.144
u1t7q=0.3 0.0 1.0 0 0
u1t7=0.168
u1t8q=0.3 0.0 1.0 0 0
u1t8=0.192
u1t9q=0.3 0.0 1.0 0 0
u1t9=0.216
u1t10q=0.3 0.0 1.0 0 0
u1t10=0.24
u1t11q=0.3 0.0 1.0 0 0
u1t11=0.264
u1t12q=0.3 0.0 1.0 0 0
u1t12=0.288
u1t13q=0.3 0.0 1.0 0 0
u1t13=0.312
u1t14q=0.3 0.0 1.0 0 0
u1t14=0.336
u1t15q=0.3 0.0 1.0 0 0
u1t15=0.36
u1t16q=0.3 0.0 1.0 0 0
u1t16=0.384
u1t17q=0.3 0.0 1.0 0 0
u1t17=0.408
u1t18q=0.3 0.0 1.0 0 0
u1t18=0.432
u1t19q=0.3 0.0 1.0 0 0
u1t19=0.456
u1t20q=0.3 0.0 1.0 0 0
u1t20=0.48
u1t21q=0.3 0.0 1.0 0 0
u1t21=0.504
u1t22q=0.3 0.0 1.0 0 0
u1t22=0.528
u1t23q=0.3 0.0 1.0 0 0
u1t23=0.552
u1t24q=0.3 0.0 1.0 0 0
u1t24=0.576
u1t25q=0.3 0.0 1.0 0 0
u1t25=0.6
u1t26q=0.3 0.0 1.0 0 0
u1t26=0.624
u1t27q=0.3 0.0 1.0 0 0
u1t27=0.648
u1t28q=0.3 0.0 1.0 0 0
u1t28=0.672
u1t29q=0.3 0.0 1.0 0 0
u1t29=0.696
u1t30q=0.3 0.0 1.0 0 0
u1t30=0.72
u1t31q=0.3 0.0 1.0 0 0
u1t31=0.744
u1t32q=0.3 0.0 1.0 0 0
u1t32=0.768
u1t33q=0.3 0.0 1.0 0 0
u1t33=0.792
u1t34q=0.3 0.0 1.0 0 0
u1t34=0.816
u1t35q=0.3 0.0 1.0 0 0
u1t35=0.84
u1t36q=0.3 0.0 1.0 0 0
u1t36=0.864
u1t37q=0.3 0.0 1.0 0 0
u1t37=0.888
u1t38q=0.3 0.0 1.0 0 0
u1t38=0.912
u1t39q=0.3 0.0 1.0 0 0
u1t39=0.936
u1t40q=0.3 0.0 1.0 0 0
u1t40=0.96
u1t41q=0.3 0.0 1.0 0 0
u1t41=0.984
u1t42q=0.3 0.0 1.0 0 0
u1t42=1.008
u1t43q=0.3 0.0 1.0 0 0
u1t43=1.032
u1t44q=0.3 0.0 1.0 0 0
u1t44=1.056
u1t45q=0.3 0.0 1.0 0 0
u1t45=1.08
u1t46q=0.3 0.0 1.0 0 0
u1t46=1.104
u1t47q=0.3 0.0 1.0 0 0
u1t47=1.128
u1t48q=0.3 0.0 1.0 0 0
u1t48=1.152
u1t49q=0.3 0.0 1.0 0 0
u1t49=1.176
u1t50q=0.3 0.0 1.0 0 0
u1t50=1.2
u1t51q=0.3 0.0 1.0 0 0
u1t51=1.224
u1t52q=0.3 0.0 1.0 0 0
u1t52=1.248
u1t53q=0.3 0.0 1.0 0 0
u1t53=1.272
u1t54q=0.3 0.0 1.0 0 0
u1t54=1.296
u1t55q=0.3 0.0 1.0 0 0
u1t55=1.32
u1t56q=0.3 0.0 1.0 0 0
u1t56=1.344
u1t57q=0.3 0.0 1.0 0 0
u1t57=1.368
u1t58q=0.3 0.0 1.0 0 0
u1t58=1.392
u1t59q=0.3 0.0 1.0 0 0
u1t59=1.416
u1t60q=0.3 0.0 1.0 0 0
u1t60=1.44
u1t61q=0.3 0.0 1.0 0 0
u1t61=1.464
u1t62q=0.3 0.0 1.0 0 0
u1t62=1.488
u1t63q=0.3 0.0 1.0 0 0
u1t63=1.512
u1t64q=0.3 0.0 1.0 0 0
u1t64=1.536
u1t65q=0.3 0.0 1.0 0 0
u1t65=1.56
u1t66q=0.3 0.0 1.0 0 0
u1t66=1.584
u1t67q=0.3 0.0 1.0 0 0
u1t67=1.608
u1t68q=0.3 0.0 1.0 0 0
u1t68=1.632
u1t69q=0.3 0.0 1.0 0 0
u1t69=1.656
u1t70q=0.3 0.0 1.0 0 0
u1t70=1.68
u1t71q=0.3 0.0 1.0 0 0
u1t71=1.704
u1t72q=0.3 0.0 1.0 0 0
u1t72=1.728
u1t73q=0.3 0.0 1.0 0 0
u1t73=1.752
u1t74q=0.3 0.0 1.0 0 0
u1t74=1.776
u1t75q=0.3 0.0 1.0 0 0
u1t75=1.8
u1t76q=0.3 0.0 1.0 0 0
u1t76=1.824
u1t77q=0.3 0.0 1.0 0 0
u1t77=1.848
u1t78q=0.3 0.0 1.0 0 0
u1t78=1.872
u1t79q=0.3 0.0 1.0 0 0
u1t79=1.896
u1t80q=0.3 0.0 1.0 0 0
u1t80=1.92
u1t81q=0.3 0.0 1.0 0 0
u1t81=1.944
u1t82q=0.3 0.0 1.0 0 0
u1t82=1.968
u1t83q=0.3 0.0 1.0 0 0
u1t83=1.992
u1t84q=0.3 0.0 1.0 0 0
u1t84=2.016
u1t85q=0.3 0.0 1.0 0 0
u1t85=2.04
u1t86q=0.3 0.0 1.0 0 0
u1t86=2.064
u1t87q=0.3 0.0 1.0 0 0
u1t87=2.088
u1t88q=0.3 0.0 1.0 0 0
u1t88=2.112
u1t89q=0.3 0.0 1.0 0 0
u1t89=2.136
u1t90q=0.3 0.0 1.0 0 0
u1t90=2.16
u1t91q=0.3 0.0 1.0 0 0
u1t91=2.184
u1t92q=0.3 0.0 1.0 0 0
u1t92=2.208
u1t93q=0.3 0.0 1.0 0 0
u1t93=2.232
u1t94q=0.3 0.0 1.0 0 0
u1t94=2.256
u1t95q=0.3 0.0 1.0 0 0
u1t95=2.28
u1t96q=0.3 0.0 1.0 0 0
u1t96=2.304
u1t97q=0.3 0.0 1.0 0 0
u1t97=2.328
u1t98q=0.3 0.0 1.0 0 0
u1t98=2.352
u1t99q=0.3 0.0 1.0 0 0
u1t99=2.376
u1t100q=0.3 0.0 1.0 0 0
u1t100=2.4
u1t101q=0.3 0.0 1.0 0 0
u1t101=2.424
u1t102q=0.3 0.0 1.0 0 0
u1t102=2.448
u1t103q=0.3 0.0 1.0 0 0
u1t103=2.472
u1t104q=0.3 0.0 1.0 0 0
u1t104=2.496
u1t105q=0.3 0.0 1.0 0 0
u1t105=2.52
u1t106q=0.3 0.0 1.0 0 0
u1t106=2.544
u1t107q=0.3 0.0 1.0 0 0
u1t107=2.568
u1t108q=0.3 0.0 1.0 0 0
u1t108=2.592
u1t109q=0.3 0.0 1.0 0 0
u1t109=2.616
u1t110q=0.3 0.0 1.0 0 0
u1t110=2.64
u1t111q=0.3 0.0 1.0 0 0
u1t111=2.664
u1t112q=0.3 0.0 1.0 0 0
u1t112=2.688
u1t113q=0.3 0.0 1.0 0 0
u1t113=2.712
u1t114q=0.3 0.0 1.0 0 0
u1t114=2.736
u1t115q=0.3 0.0 1.0 0 0
u1t115=2.76
u1t116q=0.3 0.0 1.0 0 0
u1t116=2.784
u1t117q=0.3 0.0 1.0 0 0
u1t117=2.808
u1t118q=0.3 0.0 1.0 0 0
u1t118=2.832
u1t119q=0.3 0.0 1.0 0 0
u1t119=2.856
u1t120q=0.3 0.0 1.0 0 0
u1t120=2.88
u1t121q=0.3 0.0 1.0 0 0
u1t121=2.904
u1t122q=0.3 0.0 1.0 0 0
u1t122=2.928
u1t123q=0.3 0.0 1.0 0 0
u1t123=2.952
u1t124q=0.3 0.0 1.0 0 0
u1t124=2.976
u1t125q=0.3 0.0 1.0 0 0
u1t125=3.0
u1t126q=0.3 0.0 1.0 0 0
u1t126=3.024
u1t127q=0.3 0.0 1.0 0 0
u1t127=3.048
u1t128q=0.3 0.0 1.0 0 0
u1t128=3.072
u1t129q=0.3 0.0 1.0 0 0
u1t129=3.096
u1t130q=0.3 0.0 1.0 0 0
u1t130=3.12
u1t131q=0.3 0.0 1.0 0 0
u1t131=3.144
u1t132q=0.3 0.0 1.0 0 0
u1t132=3.168
u1t133q=0.3 0.0 1.0 0 0
u1t133=3.192
u1t134q=0.3 0.0 1.0 0 0
u1t134=3.216
u1t135q=0.3 0.0 1.0 0 0
u1t135=3.24
u1t136q=0.3 0.0 1.0 0 0
u1t136=3.264
u1t137q=0.3 0.0 1.0 0 0
u1t137=3.288
u1t138q=0.3 0.0 1.0 0 0
u1t138=3.312
u1t139q=0.3 0.0 1.0 0 0
u1t139=3.336
u1t140q=0.3 0.0 1.0 0 0
u1t140=3.36
u1t141q=0.3 0.0 1.0 0 0
u1t141=3.384
u1t142q=0.3 0.0 1.0 0 0
u1t142=3.408
u1t143q=0.3 0.0 1.0 0 0
u1t143=3.432
u1t144q=0.3 0.0 1.0 0 0
u1t144=3.456
u1t145q=0.3 0.0 1.0 0 0
u1t145=3.48
u1t146q=0.3 0.0 1.0 0 0
u1t146=3.504
u1t147q=0.3 0.0 1.0 0 0
u1t147=3.528
u1t148q=0.3 0.0 1.0 0 0
u1t148=3.552
u1t149q=0.3 0.0 1.0 0 0
u1t149=3.576
u1t150q=0.3 0.0 1.0 0 0
u1t150=3.6
u1t151q=0.3 0.0 1.0 0 0
u1t151=3.624
u1t152q=0.3 0.0 1.0 0 0
u1t152=3.648
u1t153q=0.3 0.0 1.0 0 0
u1t153=3.672
u1t154q=0.3 0.0 1.0 0 0
u1t154=3.696
u1t155q=0.3 0.0 1.0 0 0
u1t155=3.72
u1t156q=0.3 0.0 1.0 0 0
u1t156=3.744
u1t157q=0.3 0.0 1.0 0 0
u1t157=3.768
u1t158q=0.3 0.0 1.0 0 0
u1t158=3.792
u1t159q=0.3 0.0 1.0 0 0
u1t159=3.816
u1t160q=0.3 0.0 1.0 0 0
u1t160=3.84
u1t161q=0.3 0.0 1.0 0 0
u1t161=3.864
u1t162q=0.3 0.0 1.0 0 0
u1t162=3.888
u1t163q=0.3 0.0 1.0 0 0
u1t163=3.912
u1t164q=0.3 0.0 1.0 0 0
u1t164=3.936
u1t165q=0.3 0.0 1.0 0 0
u1t165=3.96
u1t166q=0.3 0.0 1.0 0 0
u1t166=3.984
u1t167q=0.3 0.0 1.0 0 0
u1t167=4.008
u1t168q=0.3 0.0 1.0 0 0
u1t168=4.032
u1t169q=0.3 0.0 1.0 0 0
u1t169=4.056
u1t170q=0.3 0.0 1.0 0 0
u1t170=4.08
u1t171q=0.3 0.0 1.0 0 0
u1t171=4.104
u1t172q=0.3 0.0 1.0 0 0
u1t172=4.128
u1t173q=0.3 0.0 1.0 0 0
u1t173=4.152
u1t174q=0.3 0.0 1.0 0 0
u1t174=4.176
u1t175q=0.3 0.0 1.0 0 0
u1t175=4.2
u1t176q=0.3 0.0 1.0 0 0
u1t176=4.224
u1t177q=0.3 0.0 1.0 0 0
u1t177=4.248
u1t178q=0.3 0.0 1.0 0 0
u1t178=4.272
u1t179q=0.3 0.0 1.0 0 0
u1t179=4.296
u1t180q=0.3 0.0 1.0 0 0
u1t180=4.32
u1t181q=0.3 0.0 1.0 0 0
u1t181=4.344
u1t182q=0.3 0.0 1.0 0 0
u1t182=4.368
u1t183q=0.3 0.0 1.0 0 0
u1t183=4.392
u1t184q=0.3 0.0 1.0 0 0
u1t184=4.416
u1t185q=0.3 0.0 1.0 0 0
u1t185=4.44
u1t186q=0.3 0.0 1.0 0 0
u1t186=4.464
u1t187q=0.3 0.0 1.0 0 0
u1t187=4.488
u1t188q=0.3 0.0 1.0 0 0
u1t188=4.512
u1t189q=0.3 0.0 1.0 0 0
u1t189=4.536
u1t190q=0.3 0.0 1.0 0 0
u1t190=4.56
u1t191q=0.3 0.0 1.0 0 0
u1t191=4.584
u1t192q=0.3 0.0 1.0 0 0
u1t192=4.608
u1t193q=0.3 0.0 1.0 0 0
u1t193=4.632
u1t194q=0.3 0.0 1.0 0 0
u1t194=4.656
u1t195q=0.3 0.0 1.0 0 0
u1t195=4.68
u1t196q=0.3 0.0 1.0 0 0
u1t196=4.704
u1t197q=0.3 0.0 1.0 0 0
u1t197=4.728
u1t198q=0.3 0.0 1.0 0 0
u1t198=4.752
u1t199q=0.3 0.0 1.0 0 0
u1t199=4.776
u1t200q=0.3 0.0 1.0 0 0
u1t200=4.8
u1t201q=0.3 0.0 1.0 0 0
u1t201=4.824
u1t202q=0.3 0.0 1.0 0 0
u1t202=4.848
u1t203q=0.3 0.0 1.0 0 0
u1t203=4.872
u1t204q=0.3 0.0 1.0 0 0
u1t204=4.896
u1t205q=0.3 0.0 1.0 0 0
u1t205=4.92
u1t206q=0.3 0.0 1.0 0 0
u1t206=4.944
u1t207q=0.3 0.0 1.0 0 0
u1t207=4.968
u1t208q=0.3 0.0 1.0 0 0
u1t208=4.992
u1t209q=0.3 0.0 1.0 0 0
u1t209=5.016
u1t210q=0.3 0.0 1.0 0 0
u1t210=5.04
u1t211q=0.3 0.0 1.0 0 0
u1t211=5.064
u1t212q=0.3 0.0 1.0 0 0
u1t212=5.088
u1t213q=0.3 0.0 1.0 0 0
u1t213=5.112
u1t214q=0.3 0.0 1.0 0 0
u1t214=5.136
u1t215q=0.3 0.0 1.0 0 0
u1t215=5.16
u1t216q=0.3 0.0 1.0 0 0
u1t216=5.184
u1t217q=0.3 0.0 1.0 0 0
u1t217=5.208
u1t218q=0.3 0.0 1.0 0 0
u1t218=5.232
u1t219q=0.3 0.0 1.0 0 0
u1t219=5.256
u1t220q=0.3 0.0 1.0 0 0
u1t220=5.28
u1t221q=0.3 0.0 1.0 0 0
u1t221=5.304
u1t222q=0.3 0.0 1.0 0 0
u1t222=5.328
u1t223q=0.3 0.0 1.0 0 0
u1t223=5.352
u1t224q=0.3 0.0 1.0 0 0
u1t224=5.376
u1t225q=0.3 0.0 1.0 0 0
u1t225=5.4
u1t226q=0.3 0.0 1.0 0 0
u1t226=5.424
u1t227q=0.3 0.0 1.0 0 0
u1t227=5.448
u1t228q=0.3 0.0 1.0 0 0
u1t228=5.472
u1t229q=0.3 0.0 1.0 0 0
u1t229=5.496
u1t230q=0.3 0.0 1.0 0 0
u1t230=5.52
u1t231q=0.3 0.0 1.0 0 0
u1t231=5.544
u1t232q=0.3 0.0 1.0 0 0
u1t232=5.568
u1t233q=0.3 0.0 1.0 0 0
u1t233=5.592
u1t234q=0.3 0.0 1.0 0 0
u1t234=5.616
u1t235q=0.3 0.0 1.0 0 0
u1t235=5.64
u1t236q=0.3 0.0 1.0 0 0
u1t236=5.664
u1t237q=0.3 0.0 1.0 0 0
u1t237=5.688
u1t238q=0.3 0.0 1.0 0 0
u1t238=5.712
u1t239q=0.3 0.0 1.0 0 0
u1t239=5.736
u1t240q=0.3 0.0 1.0 0 0
u1t240=5.76
u1t241q=0.3 0.0 1.0 0 0
u1t241=5.784
u1t242q=0.3 0.0 1.0 0 0
u1t242=5.808
u1t243q=0.3 0.0 1.0 0 0
u1t243=5.832
u1t244q=0.3 0.0 1.0 0 0
u1t244=5.856
u1t245q=0.3 0.0 1.0 0 0
u1t245=5.88
u1t246q=0.3 0.0 1.0 0 0
u1t246=5.904
u1t247q=0.3 0.0 1.0 0 0
u1t247=5.928
u1t248q=0.3 0.0 1.0 0 0
u1t248=5.952
u1t249q=0.3 0.0 1.0 0 0
u1t249=5.976
u1t250q=0.3 0.0 1.0 0 0
u1t250=6.0
u1t251q=0.3 0.0 1.0 0 0
u1t251=6.024
u1t252q=0.3 0.0 1.0 0 0
u1t252=6.048
u1t253q=0.3 0.0 1.0 0 0
u1t253=6.072
u1t254q=0.3 0.0 1.0 0 0
u1t254=6.096
u1t255q=0.3 0.0 1.0 0 0
u1t255=6.12
u1t256q=0.3 0.0 1.0 0 0
u1t256=6.144
u1t257q=0.3 0.0 1.0 0 0
u1t257=6.168
u1t258q=0.3 0.0 1.0 0 0
u1t258=6.192
u1t259q=0.3 0.0 1.0 0 0
u1t259=6.216
u1t260q=0.3 0.0 1.0 0 0
u1t260=6.24
u1t261q=0.3 0.0 1.0 0 0
u1t261=6.264
u1t262q=0.3 0.0 1.0 0 0
u1t262=6.288
u1t263q=0.3 0.0 1.0 0 0
u1t263=6.312
u1t264q=0.3 0.0 1.0 0 0
u1t264=6.336
u1t265q=0.3 0.0 1.0 0 0
u1t265=6.36
u1t266q=0.3 0.0 1.0 0 0
u1t266=6.384
u1t267q=0.3 0.0 1.0 0 0
u1t267=6.408
u1t268q=0.3 0.0 1.0 0 0
u1t268=6.432
u1t269q=0.3 0.0 1.0 0 0
u1t269=6.456
u1t270q=0.3 0.0 1.0 0 0
u1t270=6.48
u1t271q=0.3 0.0 1.0 0 0
u1t271=6.504
u1t272q=0.3 0.0 1.0 0 0
u1t272=6.528
u1t273q=0.3 0.0 1.0 0 0
u1t273=6.552
u1t274q=0.3 0.0 1.0 0 0
u1t274=6.576
u1t275q=0.3 0.0 1.0 0 0
u1t275=6.6
u1t276q=0.3 0.0 1.0 0 0
u1t276=6.624
u1t277q=0.3 0.0 1.0 0 0
u1t277=6.648
u1t278q=0.3 0.0 1.0 0 0
u1t278=6.672
u1t279q=0.3 0.0 1.0 0 0
u1t279=6.696
u1t280q=0.3 0.0 1.0 0 0
u1t280=6.72
u1t281q=0.3 0.0 1.0 0 0
u1t281=6.744
u1t282q=0.3 0.0 1.0 0 0
u1t282=6.768
u1t283q=0.3 0.0 1.0 0 0
u1t283=6.792
u1t284q=0.3 0.0 1.0 0 0
u1t284=6.816
u1t285q=0.3 0.0 1.0 0 0
u1t285=6.84
u1t286q=0.3 0.0 1.0 0 0
u1t286=6.864
u1t287q=0.3 0.0 1.0 0 0
u1t287=6.888
u1t288q=0.3 0.0 1.0 0 0
u1t288=6.912
u1t289q=0.3 0.0 1.0 0 0
u1t289=6.936
u1t290q=0.3 0.0 1.0 0 0
u1t290=6.96
u1t291q=0.3 0.0 1.0 0 0
u1t291=6.984
u1t292q=0.3 0.0 1.0 0 0
u1t292=7.008
u1t293q=0.3 0.0 1.0 0 0
u1t293=7.032
u1t294q=0.3 0.0 1.0 0 0
u1t294=7.056
u1t295q=0.3 0.0 1.0 0 0
u1t295=7.08
u1t296q=0.3 0.0 1.0 0 0
u1t296=7.104
u1t297q=0.3 0.0 1.0 0 0
u1t297=7.128
u1t298q=0.3 0.0 1.0 0 0
u1t298=7.152
u1t299q=0.3 0.0 1.0 0 0
u1t299=7.176
u1t300q=0.3 0.0 1.0 0 0
u1t300=7.2
u1t301q=0.3 0.0 1.0 0 0
u1t301=7.224
u1t302q=0.3 0.0 1.0 0 0
u1t302=7.248
u1t303q=0.3 0.0 1.0 0 0
u1t303=7.272
u1t304q=0.3 0.0 1.0 0 0
u1t304=7.296
u1t305q=0.3 0.0 1.0 0 0
u1t305=7.32
u1t306q=0.3 0.0 1.0 0 0
u1t306=7.344
u1t307q=0.3 0.0 1.0 0 0
u1t307=7.368
u1t308q=0.3 0.0 1.0 0 0
u1t308=7.392
u1t309q=0.3 0.0 1.0 0 0
u1t309=7.416
u1t310q=0.3 0.0 1.0 0 0
u1t310=7.44
u1t311q=0.3 0.0 1.0 0 0
u1t311=7.464
u1t312q=0.3 0.0 1.0 0 0
u1t312=7.488
u1t313q=0.3 0.0 1.0 0 0
u1t313=7.512
u1t314q=0.3 0.0 1.0 0 0
u1t314=7.536
u1t315q=0.3 0.0 1.0 0 0
u1t315=7.56
u1t316q=0.3 0.0 1.0 0 0
u1t316=7.584
u1t317q=0.3 0.0 1.0 0 0
u1t317=7.608
u1t318q=0.3 0.0 1.0 0 0
u1t318=7.632
u1t319q=0.3 0.0 1.0 0 0
u1t319=7.656
u1t320q=0.3 0.0 1.0 0 0
u1t320=7.68
u1t321q=0.3 0.0 1.0 0 0
u1t321=7.704
u1t322q=0.3 0.0 1.0 0 0
u1t322=7.728
u1t323q=0.3 0.0 1.0 0 0
u1t323=7.752
u1t324q=0.3 0.0 1.0 0 0
u1t324=7.776
u1t325q=0.3 0.0 1.0 0 0
u1t325=7.8
u1t326q=0.3 0.0 1.0 0 0
u1t326=7.824
u1t327q=0.3 0.0 1.0 0 0
u1t327=7.848
u1t328q=0.3 0.0 1.0 0 0
u1t328=7.872
u1t329q=0.3 0.0 1.0 0 0
u1t329=7.896
u1t330q=0.3 0.0 1.0 0 0
u1t330=7.92
u1t331q=0.3 0.0 1.0 0 0
u1t331=7.944
u1t332q=0.3 0.0 1.0 0 0
u1t332=7.968
u1t333q=0.3 0.0 1.0 0 0
u1t333=7.992
u1t334q=0.3 0.0 1.0 0 0
u1t334=8.016
u1t335q=0.3 0.0 1.0 0 0
u1t335=8.04
u1t336q=0.3 0.0 1.0 0 0
u1t336=8.064
u1t337q=0.3 0.0 1.0 0 0
u1t337=8.088
u1t338q=0.3 0.0 1.0 0 0
u1t338=8.112
u1t339q=0.3 0.0 1.0 0 0
u1t339=8.136
u1t340q=0.3 0.0 1.0 0 0
u1t340=8.16
u1t341q=0.3 0.0 1.0 0 0
u1t341=8.184
u1t342q=0.3 0.0 1.0 0 0
u1t342=8.208
u1t343q=0.3 0.0 1.0 0 0
u1t343=8.232
u1t344q=0.3 0.0 1.0 0 0
u1t344=8.256
u1t345q=0.3 0.0 1.0 0 0
u1t345=8.28
u1t346q=0.3 0.0 1.0 0 0
u1t346=8.304
u1t347q=0.3 0.0 1.0 0 0
u1t347=8.328
u1t348q=0.3 0.0 1.0 0 0
u1t348=8.352
u1t349q=0.3 0.0 1.0 0 0
u1t349=8.376
u1t350q=0.3 0.0 1.0 0 0
u1t350=8.4
u1t351q=0.3 0.0 1.0 0 0
u1t351=8.424
u1t352q=0.3 0.0 1.0 0 0
u1t352=8.448
u1t353q=0.3 0.0 1.0 0 0
u1t353=8.472
u1t354q=0.3 0.0 1.0 0 0
u1t354=8.496
u1t355q=0.3 0.0 1.0 0 0
u1t355=8.52
u1t356q=0.3 0.0 1.0 0 0
u1t356=8.544
u1t357q=0.3 0.0 1.0 0 0
u1t357=8.568
u1t358q=0.3 0.0 1.0 0 0
u1t358=8.592
u1t359q=0.3 0.0 1.0 0 0
u1t359=8.616
u1t360q=0.3 0.0 1.0 0 0
u1t360=8.64
u1t361q=0.3 0.0 1.0 0 0
u1t361=8.664
u1t362q=0.3 0.0 1.0 0 0
u1t362=8.688
u1t363q=0.3 0.0 1.0 0 0
u1t363=8.712
u1t364q=0.3 0.0 1.0 0 0
u1t364=8.736
u1t365q=0.3 0.0 1.0 0 0
u1t365=8.76
u1t366q=0.3 0.0 1.0 0 0
u1t366=8.784
u1t367q=0.3 0.0 1.0 0 0
u1t367=8.808
u1t368q=0.3 0.0 1.0 0 0
u1t368=8.832
u1t369q=0.3 0.0 1.0 0 0
u1t369=8.856
u1t370q=0.3 0.0 1.0 0 0
u1t370=8.88
u1t371q=0.3 0.0 1.0 0 0
u1t371=8.904
u1t372q=0.3 0.0 1.0 0 0
u1t372=8.928
u1t373q=0.3 0.0 1.0 0 0
u1t373=8.952
u1t374q=0.3 0.0 1.0 0 0
u1t374=8.976
u1t375q=0.3 0.0 1.0 0 0
u1t375=9.0
u1t376q=0.3 0.0 1.0 0 0
u1t376=9.024
u1t377q=0.3 0.0 1.0 0 0
u1t377=9.048
u1t378q=0.3 0.0 1.0 0 0
u1t378=9.072
u1t379q=0.3 0.0 1.0 0 0
u1t379=9.096
u1t380q=0.3 0.0 1.0 0 0
u1t380=9.12
u1t381q=0.3 0.0 1.0 0 0
u1t381=9.144
u1t382q=0.3 0.0 1.0 0 0
u1t382=9.168
u1t383q=0.3 0.0 1.0 0 0
u1t383=9.192
u1t384q=0.3 0.0 1.0 0 0
u1t384=9.216
u1t385q=0.3 0.0 1.0 0 0
u1t385=9.24
u1t386q=0.3 0.0 1.0 0 0
u1t386=9.264
u1t387q=0.3 0.0 1.0 0 0
u1t387=9.288
u1t388q=0.3 0.0 1.0 0 0
u1t388=9.312
u1t389q=0.3 0.0 1.0 0 0
u1t389=9.336
u1t390q=0.3 0.0 1.0 0 0
u1t390=9.36
u1t391q=0.3 0.0 1.0 0 0
u1t391=9.384
u1t392q=0.3 0.0 1.0 0 0
u1t392=9.408
u1t393q=0.3 0.0 1.0 0 0
u1t393=9.432
u1t394q=0.3 0.0 1.0 0 0
u1t394=9.456
u1t395q=0.3 0.0 1.0 0 0
u1t395=9.48
u1t396q=0.3 0.0 1.0 0 0
u1t396=9.504
u1t397q=0.3 0.0 1.0 0 0
u1t397=9.528
u1t398q=0.3 0.0 1.0 0 0
u1t398=9.552
u1t399q=0.3 0.0 1.0 0 0
u1t399=9.576
u1t400q=0.3 0.0 1.0 0 0
u1t400=9.6
u1t401q=0.3 0.0 1.0 0 0
u1t401=9.624
u1t402q=0.3 0.0 1.0 0 0
u1t402=9.648
u1t403q=0.3 0.0 1.0 0 0
u1t403=9.672
u1t404q=0.3 0.0 1.0 0 0
u1t404=9.696
u1t405q=0.3 0.0 1.0 0 0
u1t405=9.72
u1t406q=0.3 0.0 1.0 0 0
u1t406=9.744
u1t407q=0.3 0.0 1.0 0 0
u1t407=9.768
u1t408q=0.3 0.0 1.0 0 0
u1t408=9.792
u1t409q=0.3 0.0 1.0 0 0
u1t409=9.816
u1t410q=0.3 0.0 1.0 0 0
u1t410=9.84
u1t411q=0.3 0.0 1.0 0 0
u1t411=9.864
u1t412q=0.3 0.0 1.0 0 0
u1t412=9.888
u1t413q=0.3 0.0 1.0 0 0
u1t413=9.912
u1t414q=0.3 0.0 1.0 0 0
u1t414=9.936
u1t415q=0.3 0.0 1.0 0 0
u1t415=9.96
u1t416q=0.3 0.0 1.0 0 0
u1t416=9.984
u1t417q=0.3 0.0 1.0 0 0
u1t417=10.008
u1t418q=0.3 0.0 1.0 0 0
u1t418=10.032
u1t419q=0.3 0.0 1.0 0 0
u1t419=10.056
u1t420q=0.3 0.0 1.0 0 0
u1t420=10.08
u1t421q=0.3 0.0 1.0 0 0
u1t421=10.104
u1t422q=0.3 0.0 1.0 0 0
u1t422=10.128
u1t423q=0.3 0.0 1.0 0 0
u1t423=10.152
u1t424q=0.3 0.0 1.0 0 0
u1t424=10.176
u1t425q=0.3 0.0 1.0 0 0
u1t425=10.2
u1t426q=0.3 0.0 1.0 0 0
u1t426=10.224
u1t427q=0.3 0.0 1.0 0 0
u1t427=10.248
u1t428q=0.3 0.0 1.0 0 0
u1t428=10.272
u1t429q=0.3 0.0 1.0 0 0
u1t429=10.296
u1t430q=0.3 0.0 1.0 0 0
u1t430=10.32
u1t431q=0.3 0.0 1.0 0 0
u1t431=10.344
u1t432q=0.3 0.0 1.0 0 0
u1t432=10.368
u1t433q=0.3 0.0 1.0 0 0
u1t433=10.392
u1t434q=0.3 0.0 1.0 0 0
u1t434=10.416
u1t435q=0.3 0.0 1.0 0 0
u1t435=10.44
u1t436q=0.3 0.0 1.0 0 0
u1t436=10.464
u1t437q=0.3 0.0 1.0 0 0
u1t437=10.488
u1t438q=0.3 0.0 1.0 0 0
u1t438=10.512
u1t439q=0.3 0.0 1.0 0 0
u1t439=10.536
u1t440q=0.3 0.0 1.0 0 0
u1t440=10.56
u1t441q=0.3 0.0 1.0 0 0
u1t441=10.584
u1t442q=0.3 0.0 1.0 0 0
u1t442=10.608
u1t443q=0.3 0.0 1.0 0 0
u1t443=10.632
u1t444q=0.3 0.0 1.0 0 0
u1t444=10.656
u1t445q=0.3 0.0 1.0 0 0
u1t445=10.68
u1t446q=0.3 0.0 1.0 0 0
u1t446=10.704
u1t447q=0.3 0.0 1.0 0 0
u1t447=10.728
u1t448q=0.3 0.0 1.0 0 0
u1t448=10.752
u1t449q=0.3 0.0 1.0 0 0
u1t449=10.776
u1t450q=0.3 0.0 1.0 0 0
u1t450=10.8
u1t451q=0.3 0.0 1.0 0 0
u1t451=10.824
u1t452q=0.3 0.0 1.0 0 0
u1t452=10.848
u1t453q=0.3 0.0 1.0 0 0
u1t453=10.872
u1t454q=0.3 0.0 1.0 0 0
u1t454=10.896
u1t455q=0.3 0.0 1.0 0 0
u1t455=10.92
u1t456q=0.3 0.0 1.0 0 0
u1t456=10.944
u1t457q=0.3 0.0 1.0 0 0
u1t457=10.968
u1t458q=0.3 0.0 1.0 0 0
u1t458=10.992
u1t459q=0.3 0.0 1.0 0 0
u1t459=11.016
u1t460q=0.3 0.0 1.0 0 0
u1t460=11.04
u1t461q=0.3 0.0 1.0 0 0
u1t461=11.064
u1t462q=0.3 0.0 1.0 0 0
u1t462=11.088
u1t463q=0.3 0.0 1.0 0 0
u1t463=11.112
u1t464q=0.3 0.0 1.0 0 0
u1t464=11.136
u1t465q=0.3 0.0 1.0 0 0
u1t465=11.16
u1t466q=0.3 0.0 1.0 0 0
u1t466=11.184
u1t467q=0.3 0.0 1.0 0 0
u1t467=11.208
u1t468q=0.3 0.0 1.0 0 0
u1t468=11.232
u1t469q=0.3 0.0 1.0 0 0
u1t469=11.256
u1t470q=0.3 0.0 1.0 0 0
u1t470=11.28
u1t471q=0.3 0.0 1.0 0 0
u1t471=11.304
u1t472q=0.3 0.0 1.0 0 0
u1t472=11.328
u1t473q=0.3 0.0 1.0 0 0
u1t473=11.352
u1t474q=0.3 0.0 1.0 0 0
u1t474=11.376
u1t475q=0.3 0.0 1.0 0 0
u1t475=11.4
u1t476q=0.3 0.0 1.0 0 0
u1t476=11.424
u1t477q=0.3 0.0 1.0 0 0
u1t477=11.448
u1t478q=0.3 0.0 1.0 0 0
u1t478=11.472
u1t479q=0.3 0.0 1.0 0 0
u1t479=11.496
u1t480q=0.3 0.0 1.0 0 0
u1t480=11.52
u1t481q=0.3 0.0 1.0 0 0
u1t481=11.544
u1t482q=0.3 0.0 1.0 0 0
u1t482=11.568
u1t483q=0.3 0.0 1.0 0 0
u1t483=11.592
u1t484q=0.3 0.0 1.0 0 0
u1t484=11.616
u1t485q=0.3 0.0 1.0 0 0
u1t485=11.64
u1t486q=0.3 0.0 1.0 0 0
u1t486=11.664
u1t487q=0.3 0.0 1.0 0 0
u1t487=11.688
u1t488q=0.3 0.0 1.0 0 0
u1t488=11.712
u1t489q=0.3 0.0 1.0 0 0
u1t489=11.736
u1t490q=0.3 0.0 1.0 0 0
u1t490=11.76
u1t491q=0.3 0.0 1.0 0 0
u1t491=11.784
u1t492q=0.3 0.0 1.0 0 0
u1t492=11.808
u1t493q=0.3 0.0 1.0 0 0
u1t493=11.832
u1t494q=0.3 0.0 1.0 0 0
u1t494=11.856
u1t495q=0.3 0.0 1.0 0 0
u1t495=11.88
u1t496q=0.3 0.0 1.0 0 0
u1t496=11.904
u1t497q=0.3 0.0 1.0 0 0
u1t497=11.928
u1t498q=0.3 0.0 1.0 0 0
u1t498=11.952
u1t499q=0.3 0.0 1.0 0 0
u1t499=11.976
u1t500q=0.3 0 0 0 0
u1t500=tend

[Messungen]
tAnzahl=12

t1=1
t1Anzahl=2
t1m1=mfcn1 1.0 1e-06 1
t1m2=mfcn2 1.0 1e-06 1
t1minmax=0 1e+10

t2=2.000
t2Anzahl=2
t2m1=mfcn1 1.0 1e-06 1
t2m2=mfcn2 1.0 1e-06 1
t2minmax=0 1e+10

t3=3
t3Anzahl=2
t3m1=mfcn1 1.0 1e-06 1
t3m2=mfcn2 1.0 1e-06 1
t3minmax=0 1e+10

t4=4
t4Anzahl=2
t4m1=mfcn1 1.0 1e-06 1
t4m2=mfcn2 1.0 1e-06 1
t4minmax=0 1e+10

t5=5
t5Anzahl=2
t5m1=mfcn1 1.0 1e-06 1
t5m2=mfcn2 1.0 1e-06 1
t5minmax=0 1e+10

t6=6
t6Anzahl=2
t6m1=mfcn1 1.0 1e-06 1
t6m2=mfcn2 1.0 1e-06 1
t6minmax=0 1e+10

t7=7
t7Anzahl=2
t7m1=mfcn1 1.0 1e-06 1
t7m2=mfcn2 1.0 1e-06 1
t7minmax=0 1e+10

t8=8
t8Anzahl=2
t8m1=mfcn1 1.0 1e-06 1
t8m2=mfcn2 1.0 1e-06 1
t8minmax=0 1e+10

t9=9
t9Anzahl=2
t9m1=mfcn1 1.0 1e-06 1
t9m2=mfcn2 1.0 1e-06 1
t9minmax=0 1e+10

t10=10.000
t10Anzahl=2
t10m1=mfcn1 1.0 1e-06 1
t10m2=mfcn2 1.0 1e-06 1
t10minmax=0 1e+10

t11=11.000
t11Anzahl=2
t11m1=mfcn1 1.0 1e-06 1
t11m2=mfcn2 1.0 1e-06 1
t11minmax=0 1e+10

t12=12.000
t12Anzahl=2
t12m1=mfcn1 1.0 1e-06 1
t12m2=mfcn2 1.0 1e-06 1
t12minmax=0 1e+10
[NebenbedingungenSteuergroessen]
cAnzahl=0
[Messverfahren]
mAnzahl=2
m1=mfcn1 1 0 1e+10 0
m1f1=mess3 sigma3 1
m2=mfcn2 1 0 1e+10 0
m2f1=mess4 sigma4 1
mminmaxges=0 8

[OptionenIntegration]
teps=1e-08
rtol=1e-08
atol=1e-07
stepsize=0.0001
maxorder=6
maxstepnumber=4000
minstepsize=-1
maxstepsize=-1
maxitNewton=3
realworkspace=1700000
integerworkspace=5000
printlevel=0
mcnonlinearflag=0
mcDAEflag=0
mctol=1e-07
mcmaxit=50
mclinesearch=1
mcalpha0=1
rndmethod=-1
rndeps=1e-05
rndverbose=0

</source>

ini-file for running VPLAN:

<source lang="optimica">

; ini-File fuer VPLAN

[Aktion]
;aktion=Integration
;aktion=Simulationsumgebung
;aktion=Parameterschaetzung
;aktion=Versuchsplanung
;aktion=ObjectiveTest
;aktion=DerivativeTest
;aktion={ISPS}
aktion={ISCVCS}

[Pfade]
problempath=lotka_seminar
inpath=in
outpath=simu
messpath=mess
plotpath=plot
fortranpath=fortran

[Parameter]
pAnzahl=2
p1=p2 1.0 -1e+10 1e+10 0
p2=p4 1.0 -1e+10 1e+10 0
[Versuchsplan]
expAnzahl=1
exp1=exp1.ini exp1.ini

[Guetekriterium]
Optimierungskriterium=A
AKriterium=-1
DKriterium=-1
EKriterium=-1
MKriterium=-1
covmat=covmat.m
jacmat=jacmat.m
status=undefiniert

[Residuum]
res=0

[Kosten]
costs=0 -1e+10 1e+10

[Messdatenfiles]
mess1=mess1.dat

[Outputfiles]
out1=plot2 0.05 integ.plt.1

[Residuenfiles]
rsd1=res1.txt

[ExtensionFlags]
experimenttype=0
integrator=0
dmode=0
pdeFlag=0

[OptionenAllgemein]
visflag=0
messfileflag=0
seed=-1
numberofthreads=1
robustflag=0
epsmach=0
infinity=1e+10
epsilon=1e-08
conflevel=0.95
hrobust=1e-05
computesigma=0
exitonFPE=1
iniprecision=6
clipboardflag=0
printxi=0
printconstr=0
printcolorful=-1

[OptionenParameterschaetzung]
eps=0.001
itmax=50
cond=10000
condflag=1
boundcheck=0
startflag=0
index1=1e-08
fashort=0.8
fa0=0.01
farel=0.1
famax=1.0
realworkspace=1000000
integerworkspace=1000000
printlevel=2
method=0

[OptionenVersuchsplanung]
maxit=300
opttol=1e-06
funcprec=1e-07
linfeas=1e-07
nlinfeas=0.01
maxitQP=300
maxitgesQP=10000
opttolQP=1e-06
pivottolQP=3.7e-11
steplimitLS=2
tolLS=0.9
crashtol=0.0001
elasticweight=100
superbasics=1
scaling=1
sconstraints=0
realworkspace=3000000
integerworkspace=3000000
charworkspace=500
printlevel=10
method=2

</source>

Lotka Experimental Design (VPLAN)

2016-01-19T16:26:17Z

FelixJost: /* VPLAN */

== VPLAN ==

Differential equations:

<source lang="fortran">

c RHS of the differential equations

subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), f(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

real*8 p1,p3,p5,p6, myu

c fixed parameters
p1 = 1.0
p3 = 1.0
p5 = 0.4
p6 = 0.2

c DISCRETIZE1( myu, rwh, iwh )

f(1) = p1*x(1) - p(1)*x(1)*x(2) - p5*myu*x(1)
f(2) = (-1.0)*p3*x(2) + p(2)*x(1)*x(2) - p6*myu*x(2)

end

</source>

Algebraic equations:

<source lang="fortran">

c Dummyfunction for RHS of algebraic equations

subroutine gfcn( t, x, g, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), g(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

iflag=0

end

</source>

First Measurement function:

<source lang="fortran">

c Messfunktion

subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )
implicit none

real*8 t, x(*), h, p(*), q(*), rwh(*)
integer*4 iwh(*), iflag

h = x(1)

iflag = 0

end

</source>

Second Measurement function:

<source lang="fortran">

c Messfunktion

subroutine mess4( t, x, h, p, q, rwh, iwh, iflag )
implicit none

real*8 t, x(*), h, p(*), q(*), rwh(*)
integer*4 iwh(*), iflag

h = x(2)

iflag = 0

end

</source>

Standard deviation of first measurement function

<source lang="fortran">

c Standardabweichung der Messfunktion

subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )
implicit none

real*8 rwh(*)
integer*4 iwh(*), iflag

real*8 t, x(*), p(*), q(*)
real*8 s
real*8 h

s = 1.0d+0

iflag = 0

end

</source>

Standard deviation of second measurement function:

<source lang="fortran">
c Standardabweichung der Messfunktion

subroutine sigma4( t, x, s, p, q, rwh, iwh, iflag )
implicit none

real*8 rwh(*)
integer*4 iwh(*), iflag

real*8 t, x(*), p(*), q(*)
real*8 s(*)

s(1) = 1.0

iflag = 0

end

</source>

VPLAN specific experimental setup:

<source lang="optimica">

; ini-File fuer Experiment

[Flags]
switch=1

[Kosten]
costs=0 -1e+10 1e+10

[Integrationsintervall]
t0=0
tend=12

[Modellfunktionen]
ffcn=ffcn
gfcn=gfcn

[Zustandsvariablen]
yAnzahl=2
y1=x1 0.5 -1e+10 1e+10
y2=x2 0.7 -1e+10 1e+10

zAnzahl=0

[Mehrzielknoten]
tAnzahl=12
t1=1
t2=2
t3=3
t4=4
t5=5
t6=6
t7=7
t8=8
t9=9
t10=10
t11=11
t12=12

[DynamischeNebenbedingungen]
bAnzahl=0

[GitterUeberpruefungNebenbedingungen]
tAnzahl=0

[Steuergroessen]
qAnzahl=0

[Steuerfunktionen]
uAnzahl=1
u1=myu 0 0.0 1.0
u1tAnzahl=500
u1t0=t0
u1t1q=0.3 0.0 1.0 0 0
u1t1=0.024
u1t2q=0.3 0.0 1.0 0 0
u1t2=0.048
u1t3q=0.3 0.0 1.0 0 0
u1t3=0.072
u1t4q=0.3 0.0 1.0 0 0
u1t4=0.096
u1t5q=0.3 0.0 1.0 0 0
u1t5=0.12
u1t6q=0.3 0.0 1.0 0 0
u1t6=0.144
u1t7q=0.3 0.0 1.0 0 0
u1t7=0.168
u1t8q=0.3 0.0 1.0 0 0
u1t8=0.192
u1t9q=0.3 0.0 1.0 0 0
u1t9=0.216
u1t10q=0.3 0.0 1.0 0 0
u1t10=0.24
u1t11q=0.3 0.0 1.0 0 0
u1t11=0.264
u1t12q=0.3 0.0 1.0 0 0
u1t12=0.288
u1t13q=0.3 0.0 1.0 0 0
u1t13=0.312
u1t14q=0.3 0.0 1.0 0 0
u1t14=0.336
u1t15q=0.3 0.0 1.0 0 0
u1t15=0.36
u1t16q=0.3 0.0 1.0 0 0
u1t16=0.384
u1t17q=0.3 0.0 1.0 0 0
u1t17=0.408
u1t18q=0.3 0.0 1.0 0 0
u1t18=0.432
u1t19q=0.3 0.0 1.0 0 0
u1t19=0.456
u1t20q=0.3 0.0 1.0 0 0
u1t20=0.48
u1t21q=0.3 0.0 1.0 0 0
u1t21=0.504
u1t22q=0.3 0.0 1.0 0 0
u1t22=0.528
u1t23q=0.3 0.0 1.0 0 0
u1t23=0.552
u1t24q=0.3 0.0 1.0 0 0
u1t24=0.576
u1t25q=0.3 0.0 1.0 0 0
u1t25=0.6
u1t26q=0.3 0.0 1.0 0 0
u1t26=0.624
u1t27q=0.3 0.0 1.0 0 0
u1t27=0.648
u1t28q=0.3 0.0 1.0 0 0
u1t28=0.672
u1t29q=0.3 0.0 1.0 0 0
u1t29=0.696
u1t30q=0.3 0.0 1.0 0 0
u1t30=0.72
u1t31q=0.3 0.0 1.0 0 0
u1t31=0.744
u1t32q=0.3 0.0 1.0 0 0
u1t32=0.768
u1t33q=0.3 0.0 1.0 0 0
u1t33=0.792
u1t34q=0.3 0.0 1.0 0 0
u1t34=0.816
u1t35q=0.3 0.0 1.0 0 0
u1t35=0.84
u1t36q=0.3 0.0 1.0 0 0
u1t36=0.864
u1t37q=0.3 0.0 1.0 0 0
u1t37=0.888
u1t38q=0.3 0.0 1.0 0 0
u1t38=0.912
u1t39q=0.3 0.0 1.0 0 0
u1t39=0.936
u1t40q=0.3 0.0 1.0 0 0
u1t40=0.96
u1t41q=0.3 0.0 1.0 0 0
u1t41=0.984
u1t42q=0.3 0.0 1.0 0 0
u1t42=1.008
u1t43q=0.3 0.0 1.0 0 0
u1t43=1.032
u1t44q=0.3 0.0 1.0 0 0
u1t44=1.056
u1t45q=0.3 0.0 1.0 0 0
u1t45=1.08
u1t46q=0.3 0.0 1.0 0 0
u1t46=1.104
u1t47q=0.3 0.0 1.0 0 0
u1t47=1.128
u1t48q=0.3 0.0 1.0 0 0
u1t48=1.152
u1t49q=0.3 0.0 1.0 0 0
u1t49=1.176
u1t50q=0.3 0.0 1.0 0 0
u1t50=1.2
u1t51q=0.3 0.0 1.0 0 0
u1t51=1.224
u1t52q=0.3 0.0 1.0 0 0
u1t52=1.248
u1t53q=0.3 0.0 1.0 0 0
u1t53=1.272
u1t54q=0.3 0.0 1.0 0 0
u1t54=1.296
u1t55q=0.3 0.0 1.0 0 0
u1t55=1.32
u1t56q=0.3 0.0 1.0 0 0
u1t56=1.344
u1t57q=0.3 0.0 1.0 0 0
u1t57=1.368
u1t58q=0.3 0.0 1.0 0 0
u1t58=1.392
u1t59q=0.3 0.0 1.0 0 0
u1t59=1.416
u1t60q=0.3 0.0 1.0 0 0
u1t60=1.44
u1t61q=0.3 0.0 1.0 0 0
u1t61=1.464
u1t62q=0.3 0.0 1.0 0 0
u1t62=1.488
u1t63q=0.3 0.0 1.0 0 0
u1t63=1.512
u1t64q=0.3 0.0 1.0 0 0
u1t64=1.536
u1t65q=0.3 0.0 1.0 0 0
u1t65=1.56
u1t66q=0.3 0.0 1.0 0 0
u1t66=1.584
u1t67q=0.3 0.0 1.0 0 0
u1t67=1.608
u1t68q=0.3 0.0 1.0 0 0
u1t68=1.632
u1t69q=0.3 0.0 1.0 0 0
u1t69=1.656
u1t70q=0.3 0.0 1.0 0 0
u1t70=1.68
u1t71q=0.3 0.0 1.0 0 0
u1t71=1.704
u1t72q=0.3 0.0 1.0 0 0
u1t72=1.728
u1t73q=0.3 0.0 1.0 0 0
u1t73=1.752
u1t74q=0.3 0.0 1.0 0 0
u1t74=1.776
u1t75q=0.3 0.0 1.0 0 0
u1t75=1.8
u1t76q=0.3 0.0 1.0 0 0
u1t76=1.824
u1t77q=0.3 0.0 1.0 0 0
u1t77=1.848
u1t78q=0.3 0.0 1.0 0 0
u1t78=1.872
u1t79q=0.3 0.0 1.0 0 0
u1t79=1.896
u1t80q=0.3 0.0 1.0 0 0
u1t80=1.92
u1t81q=0.3 0.0 1.0 0 0
u1t81=1.944
u1t82q=0.3 0.0 1.0 0 0
u1t82=1.968
u1t83q=0.3 0.0 1.0 0 0
u1t83=1.992
u1t84q=0.3 0.0 1.0 0 0
u1t84=2.016
u1t85q=0.3 0.0 1.0 0 0
u1t85=2.04
u1t86q=0.3 0.0 1.0 0 0
u1t86=2.064
u1t87q=0.3 0.0 1.0 0 0
u1t87=2.088
u1t88q=0.3 0.0 1.0 0 0
u1t88=2.112
u1t89q=0.3 0.0 1.0 0 0
u1t89=2.136
u1t90q=0.3 0.0 1.0 0 0
u1t90=2.16
u1t91q=0.3 0.0 1.0 0 0
u1t91=2.184
u1t92q=0.3 0.0 1.0 0 0
u1t92=2.208
u1t93q=0.3 0.0 1.0 0 0
u1t93=2.232
u1t94q=0.3 0.0 1.0 0 0
u1t94=2.256
u1t95q=0.3 0.0 1.0 0 0
u1t95=2.28
u1t96q=0.3 0.0 1.0 0 0
u1t96=2.304
u1t97q=0.3 0.0 1.0 0 0
u1t97=2.328
u1t98q=0.3 0.0 1.0 0 0
u1t98=2.352
u1t99q=0.3 0.0 1.0 0 0
u1t99=2.376
u1t100q=0.3 0.0 1.0 0 0
u1t100=2.4
u1t101q=0.3 0.0 1.0 0 0
u1t101=2.424
u1t102q=0.3 0.0 1.0 0 0
u1t102=2.448
u1t103q=0.3 0.0 1.0 0 0
u1t103=2.472
u1t104q=0.3 0.0 1.0 0 0
u1t104=2.496
u1t105q=0.3 0.0 1.0 0 0
u1t105=2.52
u1t106q=0.3 0.0 1.0 0 0
u1t106=2.544
u1t107q=0.3 0.0 1.0 0 0
u1t107=2.568
u1t108q=0.3 0.0 1.0 0 0
u1t108=2.592
u1t109q=0.3 0.0 1.0 0 0
u1t109=2.616
u1t110q=0.3 0.0 1.0 0 0
u1t110=2.64
u1t111q=0.3 0.0 1.0 0 0
u1t111=2.664
u1t112q=0.3 0.0 1.0 0 0
u1t112=2.688
u1t113q=0.3 0.0 1.0 0 0
u1t113=2.712
u1t114q=0.3 0.0 1.0 0 0
u1t114=2.736
u1t115q=0.3 0.0 1.0 0 0
u1t115=2.76
u1t116q=0.3 0.0 1.0 0 0
u1t116=2.784
u1t117q=0.3 0.0 1.0 0 0
u1t117=2.808
u1t118q=0.3 0.0 1.0 0 0
u1t118=2.832
u1t119q=0.3 0.0 1.0 0 0
u1t119=2.856
u1t120q=0.3 0.0 1.0 0 0
u1t120=2.88
u1t121q=0.3 0.0 1.0 0 0
u1t121=2.904
u1t122q=0.3 0.0 1.0 0 0
u1t122=2.928
u1t123q=0.3 0.0 1.0 0 0
u1t123=2.952
u1t124q=0.3 0.0 1.0 0 0
u1t124=2.976
u1t125q=0.3 0.0 1.0 0 0
u1t125=3.0
u1t126q=0.3 0.0 1.0 0 0
u1t126=3.024
u1t127q=0.3 0.0 1.0 0 0
u1t127=3.048
u1t128q=0.3 0.0 1.0 0 0
u1t128=3.072
u1t129q=0.3 0.0 1.0 0 0
u1t129=3.096
u1t130q=0.3 0.0 1.0 0 0
u1t130=3.12
u1t131q=0.3 0.0 1.0 0 0
u1t131=3.144
u1t132q=0.3 0.0 1.0 0 0
u1t132=3.168
u1t133q=0.3 0.0 1.0 0 0
u1t133=3.192
u1t134q=0.3 0.0 1.0 0 0
u1t134=3.216
u1t135q=0.3 0.0 1.0 0 0
u1t135=3.24
u1t136q=0.3 0.0 1.0 0 0
u1t136=3.264
u1t137q=0.3 0.0 1.0 0 0
u1t137=3.288
u1t138q=0.3 0.0 1.0 0 0
u1t138=3.312
u1t139q=0.3 0.0 1.0 0 0
u1t139=3.336
u1t140q=0.3 0.0 1.0 0 0
u1t140=3.36
u1t141q=0.3 0.0 1.0 0 0
u1t141=3.384
u1t142q=0.3 0.0 1.0 0 0
u1t142=3.408
u1t143q=0.3 0.0 1.0 0 0
u1t143=3.432
u1t144q=0.3 0.0 1.0 0 0
u1t144=3.456
u1t145q=0.3 0.0 1.0 0 0
u1t145=3.48
u1t146q=0.3 0.0 1.0 0 0
u1t146=3.504
u1t147q=0.3 0.0 1.0 0 0
u1t147=3.528
u1t148q=0.3 0.0 1.0 0 0
u1t148=3.552
u1t149q=0.3 0.0 1.0 0 0
u1t149=3.576
u1t150q=0.3 0.0 1.0 0 0
u1t150=3.6
u1t151q=0.3 0.0 1.0 0 0
u1t151=3.624
u1t152q=0.3 0.0 1.0 0 0
u1t152=3.648
u1t153q=0.3 0.0 1.0 0 0
u1t153=3.672
u1t154q=0.3 0.0 1.0 0 0
u1t154=3.696
u1t155q=0.3 0.0 1.0 0 0
u1t155=3.72
u1t156q=0.3 0.0 1.0 0 0
u1t156=3.744
u1t157q=0.3 0.0 1.0 0 0
u1t157=3.768
u1t158q=0.3 0.0 1.0 0 0
u1t158=3.792
u1t159q=0.3 0.0 1.0 0 0
u1t159=3.816
u1t160q=0.3 0.0 1.0 0 0
u1t160=3.84
u1t161q=0.3 0.0 1.0 0 0
u1t161=3.864
u1t162q=0.3 0.0 1.0 0 0
u1t162=3.888
u1t163q=0.3 0.0 1.0 0 0
u1t163=3.912
u1t164q=0.3 0.0 1.0 0 0
u1t164=3.936
u1t165q=0.3 0.0 1.0 0 0
u1t165=3.96
u1t166q=0.3 0.0 1.0 0 0
u1t166=3.984
u1t167q=0.3 0.0 1.0 0 0
u1t167=4.008
u1t168q=0.3 0.0 1.0 0 0
u1t168=4.032
u1t169q=0.3 0.0 1.0 0 0
u1t169=4.056
u1t170q=0.3 0.0 1.0 0 0
u1t170=4.08
u1t171q=0.3 0.0 1.0 0 0
u1t171=4.104
u1t172q=0.3 0.0 1.0 0 0
u1t172=4.128
u1t173q=0.3 0.0 1.0 0 0
u1t173=4.152
u1t174q=0.3 0.0 1.0 0 0
u1t174=4.176
u1t175q=0.3 0.0 1.0 0 0
u1t175=4.2
u1t176q=0.3 0.0 1.0 0 0
u1t176=4.224
u1t177q=0.3 0.0 1.0 0 0
u1t177=4.248
u1t178q=0.3 0.0 1.0 0 0
u1t178=4.272
u1t179q=0.3 0.0 1.0 0 0
u1t179=4.296
u1t180q=0.3 0.0 1.0 0 0
u1t180=4.32
u1t181q=0.3 0.0 1.0 0 0
u1t181=4.344
u1t182q=0.3 0.0 1.0 0 0
u1t182=4.368
u1t183q=0.3 0.0 1.0 0 0
u1t183=4.392
u1t184q=0.3 0.0 1.0 0 0
u1t184=4.416
u1t185q=0.3 0.0 1.0 0 0
u1t185=4.44
u1t186q=0.3 0.0 1.0 0 0
u1t186=4.464
u1t187q=0.3 0.0 1.0 0 0
u1t187=4.488
u1t188q=0.3 0.0 1.0 0 0
u1t188=4.512
u1t189q=0.3 0.0 1.0 0 0
u1t189=4.536
u1t190q=0.3 0.0 1.0 0 0
u1t190=4.56
u1t191q=0.3 0.0 1.0 0 0
u1t191=4.584
u1t192q=0.3 0.0 1.0 0 0
u1t192=4.608
u1t193q=0.3 0.0 1.0 0 0
u1t193=4.632
u1t194q=0.3 0.0 1.0 0 0
u1t194=4.656
u1t195q=0.3 0.0 1.0 0 0
u1t195=4.68
u1t196q=0.3 0.0 1.0 0 0
u1t196=4.704
u1t197q=0.3 0.0 1.0 0 0
u1t197=4.728
u1t198q=0.3 0.0 1.0 0 0
u1t198=4.752
u1t199q=0.3 0.0 1.0 0 0
u1t199=4.776
u1t200q=0.3 0.0 1.0 0 0
u1t200=4.8
u1t201q=0.3 0.0 1.0 0 0
u1t201=4.824
u1t202q=0.3 0.0 1.0 0 0
u1t202=4.848
u1t203q=0.3 0.0 1.0 0 0
u1t203=4.872
u1t204q=0.3 0.0 1.0 0 0
u1t204=4.896
u1t205q=0.3 0.0 1.0 0 0
u1t205=4.92
u1t206q=0.3 0.0 1.0 0 0
u1t206=4.944
u1t207q=0.3 0.0 1.0 0 0
u1t207=4.968
u1t208q=0.3 0.0 1.0 0 0
u1t208=4.992
u1t209q=0.3 0.0 1.0 0 0
u1t209=5.016
u1t210q=0.3 0.0 1.0 0 0
u1t210=5.04
u1t211q=0.3 0.0 1.0 0 0
u1t211=5.064
u1t212q=0.3 0.0 1.0 0 0
u1t212=5.088
u1t213q=0.3 0.0 1.0 0 0
u1t213=5.112
u1t214q=0.3 0.0 1.0 0 0
u1t214=5.136
u1t215q=0.3 0.0 1.0 0 0
u1t215=5.16
u1t216q=0.3 0.0 1.0 0 0
u1t216=5.184
u1t217q=0.3 0.0 1.0 0 0
u1t217=5.208
u1t218q=0.3 0.0 1.0 0 0
u1t218=5.232
u1t219q=0.3 0.0 1.0 0 0
u1t219=5.256
u1t220q=0.3 0.0 1.0 0 0
u1t220=5.28
u1t221q=0.3 0.0 1.0 0 0
u1t221=5.304
u1t222q=0.3 0.0 1.0 0 0
u1t222=5.328
u1t223q=0.3 0.0 1.0 0 0
u1t223=5.352
u1t224q=0.3 0.0 1.0 0 0
u1t224=5.376
u1t225q=0.3 0.0 1.0 0 0
u1t225=5.4
u1t226q=0.3 0.0 1.0 0 0
u1t226=5.424
u1t227q=0.3 0.0 1.0 0 0
u1t227=5.448
u1t228q=0.3 0.0 1.0 0 0
u1t228=5.472
u1t229q=0.3 0.0 1.0 0 0
u1t229=5.496
u1t230q=0.3 0.0 1.0 0 0
u1t230=5.52
u1t231q=0.3 0.0 1.0 0 0
u1t231=5.544
u1t232q=0.3 0.0 1.0 0 0
u1t232=5.568
u1t233q=0.3 0.0 1.0 0 0
u1t233=5.592
u1t234q=0.3 0.0 1.0 0 0
u1t234=5.616
u1t235q=0.3 0.0 1.0 0 0
u1t235=5.64
u1t236q=0.3 0.0 1.0 0 0
u1t236=5.664
u1t237q=0.3 0.0 1.0 0 0
u1t237=5.688
u1t238q=0.3 0.0 1.0 0 0
u1t238=5.712
u1t239q=0.3 0.0 1.0 0 0
u1t239=5.736
u1t240q=0.3 0.0 1.0 0 0
u1t240=5.76
u1t241q=0.3 0.0 1.0 0 0
u1t241=5.784
u1t242q=0.3 0.0 1.0 0 0
u1t242=5.808
u1t243q=0.3 0.0 1.0 0 0
u1t243=5.832
u1t244q=0.3 0.0 1.0 0 0
u1t244=5.856
u1t245q=0.3 0.0 1.0 0 0
u1t245=5.88
u1t246q=0.3 0.0 1.0 0 0
u1t246=5.904
u1t247q=0.3 0.0 1.0 0 0
u1t247=5.928
u1t248q=0.3 0.0 1.0 0 0
u1t248=5.952
u1t249q=0.3 0.0 1.0 0 0
u1t249=5.976
u1t250q=0.3 0.0 1.0 0 0
u1t250=6.0
u1t251q=0.3 0.0 1.0 0 0
u1t251=6.024
u1t252q=0.3 0.0 1.0 0 0
u1t252=6.048
u1t253q=0.3 0.0 1.0 0 0
u1t253=6.072
u1t254q=0.3 0.0 1.0 0 0
u1t254=6.096
u1t255q=0.3 0.0 1.0 0 0
u1t255=6.12
u1t256q=0.3 0.0 1.0 0 0
u1t256=6.144
u1t257q=0.3 0.0 1.0 0 0
u1t257=6.168
u1t258q=0.3 0.0 1.0 0 0
u1t258=6.192
u1t259q=0.3 0.0 1.0 0 0
u1t259=6.216
u1t260q=0.3 0.0 1.0 0 0
u1t260=6.24
u1t261q=0.3 0.0 1.0 0 0
u1t261=6.264
u1t262q=0.3 0.0 1.0 0 0
u1t262=6.288
u1t263q=0.3 0.0 1.0 0 0
u1t263=6.312
u1t264q=0.3 0.0 1.0 0 0
u1t264=6.336
u1t265q=0.3 0.0 1.0 0 0
u1t265=6.36
u1t266q=0.3 0.0 1.0 0 0
u1t266=6.384
u1t267q=0.3 0.0 1.0 0 0
u1t267=6.408
u1t268q=0.3 0.0 1.0 0 0
u1t268=6.432
u1t269q=0.3 0.0 1.0 0 0
u1t269=6.456
u1t270q=0.3 0.0 1.0 0 0
u1t270=6.48
u1t271q=0.3 0.0 1.0 0 0
u1t271=6.504
u1t272q=0.3 0.0 1.0 0 0
u1t272=6.528
u1t273q=0.3 0.0 1.0 0 0
u1t273=6.552
u1t274q=0.3 0.0 1.0 0 0
u1t274=6.576
u1t275q=0.3 0.0 1.0 0 0
u1t275=6.6
u1t276q=0.3 0.0 1.0 0 0
u1t276=6.624
u1t277q=0.3 0.0 1.0 0 0
u1t277=6.648
u1t278q=0.3 0.0 1.0 0 0
u1t278=6.672
u1t279q=0.3 0.0 1.0 0 0
u1t279=6.696
u1t280q=0.3 0.0 1.0 0 0
u1t280=6.72
u1t281q=0.3 0.0 1.0 0 0
u1t281=6.744
u1t282q=0.3 0.0 1.0 0 0
u1t282=6.768
u1t283q=0.3 0.0 1.0 0 0
u1t283=6.792
u1t284q=0.3 0.0 1.0 0 0
u1t284=6.816
u1t285q=0.3 0.0 1.0 0 0
u1t285=6.84
u1t286q=0.3 0.0 1.0 0 0
u1t286=6.864
u1t287q=0.3 0.0 1.0 0 0
u1t287=6.888
u1t288q=0.3 0.0 1.0 0 0
u1t288=6.912
u1t289q=0.3 0.0 1.0 0 0
u1t289=6.936
u1t290q=0.3 0.0 1.0 0 0
u1t290=6.96
u1t291q=0.3 0.0 1.0 0 0
u1t291=6.984
u1t292q=0.3 0.0 1.0 0 0
u1t292=7.008
u1t293q=0.3 0.0 1.0 0 0
u1t293=7.032
u1t294q=0.3 0.0 1.0 0 0
u1t294=7.056
u1t295q=0.3 0.0 1.0 0 0
u1t295=7.08
u1t296q=0.3 0.0 1.0 0 0
u1t296=7.104
u1t297q=0.3 0.0 1.0 0 0
u1t297=7.128
u1t298q=0.3 0.0 1.0 0 0
u1t298=7.152
u1t299q=0.3 0.0 1.0 0 0
u1t299=7.176
u1t300q=0.3 0.0 1.0 0 0
u1t300=7.2
u1t301q=0.3 0.0 1.0 0 0
u1t301=7.224
u1t302q=0.3 0.0 1.0 0 0
u1t302=7.248
u1t303q=0.3 0.0 1.0 0 0
u1t303=7.272
u1t304q=0.3 0.0 1.0 0 0
u1t304=7.296
u1t305q=0.3 0.0 1.0 0 0
u1t305=7.32
u1t306q=0.3 0.0 1.0 0 0
u1t306=7.344
u1t307q=0.3 0.0 1.0 0 0
u1t307=7.368
u1t308q=0.3 0.0 1.0 0 0
u1t308=7.392
u1t309q=0.3 0.0 1.0 0 0
u1t309=7.416
u1t310q=0.3 0.0 1.0 0 0
u1t310=7.44
u1t311q=0.3 0.0 1.0 0 0
u1t311=7.464
u1t312q=0.3 0.0 1.0 0 0
u1t312=7.488
u1t313q=0.3 0.0 1.0 0 0
u1t313=7.512
u1t314q=0.3 0.0 1.0 0 0
u1t314=7.536
u1t315q=0.3 0.0 1.0 0 0
u1t315=7.56
u1t316q=0.3 0.0 1.0 0 0
u1t316=7.584
u1t317q=0.3 0.0 1.0 0 0
u1t317=7.608
u1t318q=0.3 0.0 1.0 0 0
u1t318=7.632
u1t319q=0.3 0.0 1.0 0 0
u1t319=7.656
u1t320q=0.3 0.0 1.0 0 0
u1t320=7.68
u1t321q=0.3 0.0 1.0 0 0
u1t321=7.704
u1t322q=0.3 0.0 1.0 0 0
u1t322=7.728
u1t323q=0.3 0.0 1.0 0 0
u1t323=7.752
u1t324q=0.3 0.0 1.0 0 0
u1t324=7.776
u1t325q=0.3 0.0 1.0 0 0
u1t325=7.8
u1t326q=0.3 0.0 1.0 0 0
u1t326=7.824
u1t327q=0.3 0.0 1.0 0 0
u1t327=7.848
u1t328q=0.3 0.0 1.0 0 0
u1t328=7.872
u1t329q=0.3 0.0 1.0 0 0
u1t329=7.896
u1t330q=0.3 0.0 1.0 0 0
u1t330=7.92
u1t331q=0.3 0.0 1.0 0 0
u1t331=7.944
u1t332q=0.3 0.0 1.0 0 0
u1t332=7.968
u1t333q=0.3 0.0 1.0 0 0
u1t333=7.992
u1t334q=0.3 0.0 1.0 0 0
u1t334=8.016
u1t335q=0.3 0.0 1.0 0 0
u1t335=8.04
u1t336q=0.3 0.0 1.0 0 0
u1t336=8.064
u1t337q=0.3 0.0 1.0 0 0
u1t337=8.088
u1t338q=0.3 0.0 1.0 0 0
u1t338=8.112
u1t339q=0.3 0.0 1.0 0 0
u1t339=8.136
u1t340q=0.3 0.0 1.0 0 0
u1t340=8.16
u1t341q=0.3 0.0 1.0 0 0
u1t341=8.184
u1t342q=0.3 0.0 1.0 0 0
u1t342=8.208
u1t343q=0.3 0.0 1.0 0 0
u1t343=8.232
u1t344q=0.3 0.0 1.0 0 0
u1t344=8.256
u1t345q=0.3 0.0 1.0 0 0
u1t345=8.28
u1t346q=0.3 0.0 1.0 0 0
u1t346=8.304
u1t347q=0.3 0.0 1.0 0 0
u1t347=8.328
u1t348q=0.3 0.0 1.0 0 0
u1t348=8.352
u1t349q=0.3 0.0 1.0 0 0
u1t349=8.376
u1t350q=0.3 0.0 1.0 0 0
u1t350=8.4
u1t351q=0.3 0.0 1.0 0 0
u1t351=8.424
u1t352q=0.3 0.0 1.0 0 0
u1t352=8.448
u1t353q=0.3 0.0 1.0 0 0
u1t353=8.472
u1t354q=0.3 0.0 1.0 0 0
u1t354=8.496
u1t355q=0.3 0.0 1.0 0 0
u1t355=8.52
u1t356q=0.3 0.0 1.0 0 0
u1t356=8.544
u1t357q=0.3 0.0 1.0 0 0
u1t357=8.568
u1t358q=0.3 0.0 1.0 0 0
u1t358=8.592
u1t359q=0.3 0.0 1.0 0 0
u1t359=8.616
u1t360q=0.3 0.0 1.0 0 0
u1t360=8.64
u1t361q=0.3 0.0 1.0 0 0
u1t361=8.664
u1t362q=0.3 0.0 1.0 0 0
u1t362=8.688
u1t363q=0.3 0.0 1.0 0 0
u1t363=8.712
u1t364q=0.3 0.0 1.0 0 0
u1t364=8.736
u1t365q=0.3 0.0 1.0 0 0
u1t365=8.76
u1t366q=0.3 0.0 1.0 0 0
u1t366=8.784
u1t367q=0.3 0.0 1.0 0 0
u1t367=8.808
u1t368q=0.3 0.0 1.0 0 0
u1t368=8.832
u1t369q=0.3 0.0 1.0 0 0
u1t369=8.856
u1t370q=0.3 0.0 1.0 0 0
u1t370=8.88
u1t371q=0.3 0.0 1.0 0 0
u1t371=8.904
u1t372q=0.3 0.0 1.0 0 0
u1t372=8.928
u1t373q=0.3 0.0 1.0 0 0
u1t373=8.952
u1t374q=0.3 0.0 1.0 0 0
u1t374=8.976
u1t375q=0.3 0.0 1.0 0 0
u1t375=9.0
u1t376q=0.3 0.0 1.0 0 0
u1t376=9.024
u1t377q=0.3 0.0 1.0 0 0
u1t377=9.048
u1t378q=0.3 0.0 1.0 0 0
u1t378=9.072
u1t379q=0.3 0.0 1.0 0 0
u1t379=9.096
u1t380q=0.3 0.0 1.0 0 0
u1t380=9.12
u1t381q=0.3 0.0 1.0 0 0
u1t381=9.144
u1t382q=0.3 0.0 1.0 0 0
u1t382=9.168
u1t383q=0.3 0.0 1.0 0 0
u1t383=9.192
u1t384q=0.3 0.0 1.0 0 0
u1t384=9.216
u1t385q=0.3 0.0 1.0 0 0
u1t385=9.24
u1t386q=0.3 0.0 1.0 0 0
u1t386=9.264
u1t387q=0.3 0.0 1.0 0 0
u1t387=9.288
u1t388q=0.3 0.0 1.0 0 0
u1t388=9.312
u1t389q=0.3 0.0 1.0 0 0
u1t389=9.336
u1t390q=0.3 0.0 1.0 0 0
u1t390=9.36
u1t391q=0.3 0.0 1.0 0 0
u1t391=9.384
u1t392q=0.3 0.0 1.0 0 0
u1t392=9.408
u1t393q=0.3 0.0 1.0 0 0
u1t393=9.432
u1t394q=0.3 0.0 1.0 0 0
u1t394=9.456
u1t395q=0.3 0.0 1.0 0 0
u1t395=9.48
u1t396q=0.3 0.0 1.0 0 0
u1t396=9.504
u1t397q=0.3 0.0 1.0 0 0
u1t397=9.528
u1t398q=0.3 0.0 1.0 0 0
u1t398=9.552
u1t399q=0.3 0.0 1.0 0 0
u1t399=9.576
u1t400q=0.3 0.0 1.0 0 0
u1t400=9.6
u1t401q=0.3 0.0 1.0 0 0
u1t401=9.624
u1t402q=0.3 0.0 1.0 0 0
u1t402=9.648
u1t403q=0.3 0.0 1.0 0 0
u1t403=9.672
u1t404q=0.3 0.0 1.0 0 0
u1t404=9.696
u1t405q=0.3 0.0 1.0 0 0
u1t405=9.72
u1t406q=0.3 0.0 1.0 0 0
u1t406=9.744
u1t407q=0.3 0.0 1.0 0 0
u1t407=9.768
u1t408q=0.3 0.0 1.0 0 0
u1t408=9.792
u1t409q=0.3 0.0 1.0 0 0
u1t409=9.816
u1t410q=0.3 0.0 1.0 0 0
u1t410=9.84
u1t411q=0.3 0.0 1.0 0 0
u1t411=9.864
u1t412q=0.3 0.0 1.0 0 0
u1t412=9.888
u1t413q=0.3 0.0 1.0 0 0
u1t413=9.912
u1t414q=0.3 0.0 1.0 0 0
u1t414=9.936
u1t415q=0.3 0.0 1.0 0 0
u1t415=9.96
u1t416q=0.3 0.0 1.0 0 0
u1t416=9.984
u1t417q=0.3 0.0 1.0 0 0
u1t417=10.008
u1t418q=0.3 0.0 1.0 0 0
u1t418=10.032
u1t419q=0.3 0.0 1.0 0 0
u1t419=10.056
u1t420q=0.3 0.0 1.0 0 0
u1t420=10.08
u1t421q=0.3 0.0 1.0 0 0
u1t421=10.104
u1t422q=0.3 0.0 1.0 0 0
u1t422=10.128
u1t423q=0.3 0.0 1.0 0 0
u1t423=10.152
u1t424q=0.3 0.0 1.0 0 0
u1t424=10.176
u1t425q=0.3 0.0 1.0 0 0
u1t425=10.2
u1t426q=0.3 0.0 1.0 0 0
u1t426=10.224
u1t427q=0.3 0.0 1.0 0 0
u1t427=10.248
u1t428q=0.3 0.0 1.0 0 0
u1t428=10.272
u1t429q=0.3 0.0 1.0 0 0
u1t429=10.296
u1t430q=0.3 0.0 1.0 0 0
u1t430=10.32
u1t431q=0.3 0.0 1.0 0 0
u1t431=10.344
u1t432q=0.3 0.0 1.0 0 0
u1t432=10.368
u1t433q=0.3 0.0 1.0 0 0
u1t433=10.392
u1t434q=0.3 0.0 1.0 0 0
u1t434=10.416
u1t435q=0.3 0.0 1.0 0 0
u1t435=10.44
u1t436q=0.3 0.0 1.0 0 0
u1t436=10.464
u1t437q=0.3 0.0 1.0 0 0
u1t437=10.488
u1t438q=0.3 0.0 1.0 0 0
u1t438=10.512
u1t439q=0.3 0.0 1.0 0 0
u1t439=10.536
u1t440q=0.3 0.0 1.0 0 0
u1t440=10.56
u1t441q=0.3 0.0 1.0 0 0
u1t441=10.584
u1t442q=0.3 0.0 1.0 0 0
u1t442=10.608
u1t443q=0.3 0.0 1.0 0 0
u1t443=10.632
u1t444q=0.3 0.0 1.0 0 0
u1t444=10.656
u1t445q=0.3 0.0 1.0 0 0
u1t445=10.68
u1t446q=0.3 0.0 1.0 0 0
u1t446=10.704
u1t447q=0.3 0.0 1.0 0 0
u1t447=10.728
u1t448q=0.3 0.0 1.0 0 0
u1t448=10.752
u1t449q=0.3 0.0 1.0 0 0
u1t449=10.776
u1t450q=0.3 0.0 1.0 0 0
u1t450=10.8
u1t451q=0.3 0.0 1.0 0 0
u1t451=10.824
u1t452q=0.3 0.0 1.0 0 0
u1t452=10.848
u1t453q=0.3 0.0 1.0 0 0
u1t453=10.872
u1t454q=0.3 0.0 1.0 0 0
u1t454=10.896
u1t455q=0.3 0.0 1.0 0 0
u1t455=10.92
u1t456q=0.3 0.0 1.0 0 0
u1t456=10.944
u1t457q=0.3 0.0 1.0 0 0
u1t457=10.968
u1t458q=0.3 0.0 1.0 0 0
u1t458=10.992
u1t459q=0.3 0.0 1.0 0 0
u1t459=11.016
u1t460q=0.3 0.0 1.0 0 0
u1t460=11.04
u1t461q=0.3 0.0 1.0 0 0
u1t461=11.064
u1t462q=0.3 0.0 1.0 0 0
u1t462=11.088
u1t463q=0.3 0.0 1.0 0 0
u1t463=11.112
u1t464q=0.3 0.0 1.0 0 0
u1t464=11.136
u1t465q=0.3 0.0 1.0 0 0
u1t465=11.16
u1t466q=0.3 0.0 1.0 0 0
u1t466=11.184
u1t467q=0.3 0.0 1.0 0 0
u1t467=11.208
u1t468q=0.3 0.0 1.0 0 0
u1t468=11.232
u1t469q=0.3 0.0 1.0 0 0
u1t469=11.256
u1t470q=0.3 0.0 1.0 0 0
u1t470=11.28
u1t471q=0.3 0.0 1.0 0 0
u1t471=11.304
u1t472q=0.3 0.0 1.0 0 0
u1t472=11.328
u1t473q=0.3 0.0 1.0 0 0
u1t473=11.352
u1t474q=0.3 0.0 1.0 0 0
u1t474=11.376
u1t475q=0.3 0.0 1.0 0 0
u1t475=11.4
u1t476q=0.3 0.0 1.0 0 0
u1t476=11.424
u1t477q=0.3 0.0 1.0 0 0
u1t477=11.448
u1t478q=0.3 0.0 1.0 0 0
u1t478=11.472
u1t479q=0.3 0.0 1.0 0 0
u1t479=11.496
u1t480q=0.3 0.0 1.0 0 0
u1t480=11.52
u1t481q=0.3 0.0 1.0 0 0
u1t481=11.544
u1t482q=0.3 0.0 1.0 0 0
u1t482=11.568
u1t483q=0.3 0.0 1.0 0 0
u1t483=11.592
u1t484q=0.3 0.0 1.0 0 0
u1t484=11.616
u1t485q=0.3 0.0 1.0 0 0
u1t485=11.64
u1t486q=0.3 0.0 1.0 0 0
u1t486=11.664
u1t487q=0.3 0.0 1.0 0 0
u1t487=11.688
u1t488q=0.3 0.0 1.0 0 0
u1t488=11.712
u1t489q=0.3 0.0 1.0 0 0
u1t489=11.736
u1t490q=0.3 0.0 1.0 0 0
u1t490=11.76
u1t491q=0.3 0.0 1.0 0 0
u1t491=11.784
u1t492q=0.3 0.0 1.0 0 0
u1t492=11.808
u1t493q=0.3 0.0 1.0 0 0
u1t493=11.832
u1t494q=0.3 0.0 1.0 0 0
u1t494=11.856
u1t495q=0.3 0.0 1.0 0 0
u1t495=11.88
u1t496q=0.3 0.0 1.0 0 0
u1t496=11.904
u1t497q=0.3 0.0 1.0 0 0
u1t497=11.928
u1t498q=0.3 0.0 1.0 0 0
u1t498=11.952
u1t499q=0.3 0.0 1.0 0 0
u1t499=11.976
u1t500q=0.3 0 0 0 0
u1t500=tend

[Messungen]
tAnzahl=12

t1=1
t1Anzahl=2
t1m1=mfcn1 1.0 1e-06 1
t1m2=mfcn2 1.0 1e-06 1
t1minmax=0 1e+10

t2=2.000
t2Anzahl=2
t2m1=mfcn1 1.0 1e-06 1
t2m2=mfcn2 1.0 1e-06 1
t2minmax=0 1e+10

t3=3
t3Anzahl=2
t3m1=mfcn1 1.0 1e-06 1
t3m2=mfcn2 1.0 1e-06 1
t3minmax=0 1e+10

t4=4
t4Anzahl=2
t4m1=mfcn1 1.0 1e-06 1
t4m2=mfcn2 1.0 1e-06 1
t4minmax=0 1e+10

t5=5
t5Anzahl=2
t5m1=mfcn1 1.0 1e-06 1
t5m2=mfcn2 1.0 1e-06 1
t5minmax=0 1e+10

t6=6
t6Anzahl=2
t6m1=mfcn1 1.0 1e-06 1
t6m2=mfcn2 1.0 1e-06 1
t6minmax=0 1e+10

t7=7
t7Anzahl=2
t7m1=mfcn1 1.0 1e-06 1
t7m2=mfcn2 1.0 1e-06 1
t7minmax=0 1e+10

t8=8
t8Anzahl=2
t8m1=mfcn1 1.0 1e-06 1
t8m2=mfcn2 1.0 1e-06 1
t8minmax=0 1e+10

t9=9
t9Anzahl=2
t9m1=mfcn1 1.0 1e-06 1
t9m2=mfcn2 1.0 1e-06 1
t9minmax=0 1e+10

t10=10.000
t10Anzahl=2
t10m1=mfcn1 1.0 1e-06 1
t10m2=mfcn2 1.0 1e-06 1
t10minmax=0 1e+10

t11=11.000
t11Anzahl=2
t11m1=mfcn1 1.0 1e-06 1
t11m2=mfcn2 1.0 1e-06 1
t11minmax=0 1e+10

t12=12.000
t12Anzahl=2
t12m1=mfcn1 1.0 1e-06 1
t12m2=mfcn2 1.0 1e-06 1
t12minmax=0 1e+10
[NebenbedingungenSteuergroessen]
cAnzahl=0
[Messverfahren]
mAnzahl=2
m1=mfcn1 1 0 1e+10 0
m1f1=mess3 sigma3 1
m2=mfcn2 1 0 1e+10 0
m2f1=mess4 sigma4 1
mminmaxges=0 8

[OptionenIntegration]
teps=1e-08
rtol=1e-08
atol=1e-07
stepsize=0.0001
maxorder=6
maxstepnumber=4000
minstepsize=-1
maxstepsize=-1
maxitNewton=3
realworkspace=1700000
integerworkspace=5000
printlevel=0
mcnonlinearflag=0
mcDAEflag=0
mctol=1e-07
mcmaxit=50
mclinesearch=1
mcalpha0=1
rndmethod=-1
rndeps=1e-05
rndverbose=0

</source>

Lotka Experimental Design (VPLAN)

2016-01-19T16:25:27Z

FelixJost: /* VPLAN */

== VPLAN ==

Differential equations:

<source lang="fortran">

c RHS of the differential equations

subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), f(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

real*8 p1,p3,p5,p6, myu

c fixed parameters
p1 = 1.0
p3 = 1.0
p5 = 0.4
p6 = 0.2

c DISCRETIZE1( myu, rwh, iwh )

f(1) = p1*x(1) - p(1)*x(1)*x(2) - p5*myu*x(1)
f(2) = (-1.0)*p3*x(2) + p(2)*x(1)*x(2) - p6*myu*x(2)

end

</source>

Algebraic equations:

<source lang="fortran">

c Dummyfunction for RHS of algebraic equations

subroutine gfcn( t, x, g, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), g(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

iflag=0

end

</source>

First Measurement function:

<source lang="fortran">

c Messfunktion

subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )
implicit none

real*8 t, x(*), h, p(*), q(*), rwh(*)
integer*4 iwh(*), iflag

h = x(1)

iflag = 0

end

</source>

Second Measurement function:

<source lang="fortran">

c Messfunktion

subroutine mess4( t, x, h, p, q, rwh, iwh, iflag )
implicit none

real*8 t, x(*), h, p(*), q(*), rwh(*)
integer*4 iwh(*), iflag

h = x(2)

iflag = 0

end

</source>

Standard deviation of first measurement function

<source lang="fortran">

c Standardabweichung der Messfunktion

subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )
implicit none

real*8 rwh(*)
integer*4 iwh(*), iflag

real*8 t, x(*), p(*), q(*)
real*8 s
real*8 h

s = 1.0d+0

iflag = 0

end

</source>

Standard deviation of second measurement function:

<source lang="optimica">
c Standardabweichung der Messfunktion

subroutine sigma4( t, x, s, p, q, rwh, iwh, iflag )
implicit none

real*8 rwh(*)
integer*4 iwh(*), iflag

real*8 t, x(*), p(*), q(*)
real*8 s(*)

s(1) = 1.0

iflag = 0

end

</source>

VPLAN specific experimental setup:

<source lang="vplan">

; ini-File fuer Experiment

[Flags]
switch=1

[Kosten]
costs=0 -1e+10 1e+10

[Integrationsintervall]
t0=0
tend=12

[Modellfunktionen]
ffcn=ffcn
gfcn=gfcn

[Zustandsvariablen]
yAnzahl=2
y1=x1 0.5 -1e+10 1e+10
y2=x2 0.7 -1e+10 1e+10

zAnzahl=0

[Mehrzielknoten]
tAnzahl=12
t1=1
t2=2
t3=3
t4=4
t5=5
t6=6
t7=7
t8=8
t9=9
t10=10
t11=11
t12=12

[DynamischeNebenbedingungen]
bAnzahl=0

[GitterUeberpruefungNebenbedingungen]
tAnzahl=0

[Steuergroessen]
qAnzahl=0

[Steuerfunktionen]
uAnzahl=1
u1=myu 0 0.0 1.0
u1tAnzahl=500
u1t0=t0
u1t1q=0.3 0.0 1.0 0 0
u1t1=0.024
u1t2q=0.3 0.0 1.0 0 0
u1t2=0.048
u1t3q=0.3 0.0 1.0 0 0
u1t3=0.072
u1t4q=0.3 0.0 1.0 0 0
u1t4=0.096
u1t5q=0.3 0.0 1.0 0 0
u1t5=0.12
u1t6q=0.3 0.0 1.0 0 0
u1t6=0.144
u1t7q=0.3 0.0 1.0 0 0
u1t7=0.168
u1t8q=0.3 0.0 1.0 0 0
u1t8=0.192
u1t9q=0.3 0.0 1.0 0 0
u1t9=0.216
u1t10q=0.3 0.0 1.0 0 0
u1t10=0.24
u1t11q=0.3 0.0 1.0 0 0
u1t11=0.264
u1t12q=0.3 0.0 1.0 0 0
u1t12=0.288
u1t13q=0.3 0.0 1.0 0 0
u1t13=0.312
u1t14q=0.3 0.0 1.0 0 0
u1t14=0.336
u1t15q=0.3 0.0 1.0 0 0
u1t15=0.36
u1t16q=0.3 0.0 1.0 0 0
u1t16=0.384
u1t17q=0.3 0.0 1.0 0 0
u1t17=0.408
u1t18q=0.3 0.0 1.0 0 0
u1t18=0.432
u1t19q=0.3 0.0 1.0 0 0
u1t19=0.456
u1t20q=0.3 0.0 1.0 0 0
u1t20=0.48
u1t21q=0.3 0.0 1.0 0 0
u1t21=0.504
u1t22q=0.3 0.0 1.0 0 0
u1t22=0.528
u1t23q=0.3 0.0 1.0 0 0
u1t23=0.552
u1t24q=0.3 0.0 1.0 0 0
u1t24=0.576
u1t25q=0.3 0.0 1.0 0 0
u1t25=0.6
u1t26q=0.3 0.0 1.0 0 0
u1t26=0.624
u1t27q=0.3 0.0 1.0 0 0
u1t27=0.648
u1t28q=0.3 0.0 1.0 0 0
u1t28=0.672
u1t29q=0.3 0.0 1.0 0 0
u1t29=0.696
u1t30q=0.3 0.0 1.0 0 0
u1t30=0.72
u1t31q=0.3 0.0 1.0 0 0
u1t31=0.744
u1t32q=0.3 0.0 1.0 0 0
u1t32=0.768
u1t33q=0.3 0.0 1.0 0 0
u1t33=0.792
u1t34q=0.3 0.0 1.0 0 0
u1t34=0.816
u1t35q=0.3 0.0 1.0 0 0
u1t35=0.84
u1t36q=0.3 0.0 1.0 0 0
u1t36=0.864
u1t37q=0.3 0.0 1.0 0 0
u1t37=0.888
u1t38q=0.3 0.0 1.0 0 0
u1t38=0.912
u1t39q=0.3 0.0 1.0 0 0
u1t39=0.936
u1t40q=0.3 0.0 1.0 0 0
u1t40=0.96
u1t41q=0.3 0.0 1.0 0 0
u1t41=0.984
u1t42q=0.3 0.0 1.0 0 0
u1t42=1.008
u1t43q=0.3 0.0 1.0 0 0
u1t43=1.032
u1t44q=0.3 0.0 1.0 0 0
u1t44=1.056
u1t45q=0.3 0.0 1.0 0 0
u1t45=1.08
u1t46q=0.3 0.0 1.0 0 0
u1t46=1.104
u1t47q=0.3 0.0 1.0 0 0
u1t47=1.128
u1t48q=0.3 0.0 1.0 0 0
u1t48=1.152
u1t49q=0.3 0.0 1.0 0 0
u1t49=1.176
u1t50q=0.3 0.0 1.0 0 0
u1t50=1.2
u1t51q=0.3 0.0 1.0 0 0
u1t51=1.224
u1t52q=0.3 0.0 1.0 0 0
u1t52=1.248
u1t53q=0.3 0.0 1.0 0 0
u1t53=1.272
u1t54q=0.3 0.0 1.0 0 0
u1t54=1.296
u1t55q=0.3 0.0 1.0 0 0
u1t55=1.32
u1t56q=0.3 0.0 1.0 0 0
u1t56=1.344
u1t57q=0.3 0.0 1.0 0 0
u1t57=1.368
u1t58q=0.3 0.0 1.0 0 0
u1t58=1.392
u1t59q=0.3 0.0 1.0 0 0
u1t59=1.416
u1t60q=0.3 0.0 1.0 0 0
u1t60=1.44
u1t61q=0.3 0.0 1.0 0 0
u1t61=1.464
u1t62q=0.3 0.0 1.0 0 0
u1t62=1.488
u1t63q=0.3 0.0 1.0 0 0
u1t63=1.512
u1t64q=0.3 0.0 1.0 0 0
u1t64=1.536
u1t65q=0.3 0.0 1.0 0 0
u1t65=1.56
u1t66q=0.3 0.0 1.0 0 0
u1t66=1.584
u1t67q=0.3 0.0 1.0 0 0
u1t67=1.608
u1t68q=0.3 0.0 1.0 0 0
u1t68=1.632
u1t69q=0.3 0.0 1.0 0 0
u1t69=1.656
u1t70q=0.3 0.0 1.0 0 0
u1t70=1.68
u1t71q=0.3 0.0 1.0 0 0
u1t71=1.704
u1t72q=0.3 0.0 1.0 0 0
u1t72=1.728
u1t73q=0.3 0.0 1.0 0 0
u1t73=1.752
u1t74q=0.3 0.0 1.0 0 0
u1t74=1.776
u1t75q=0.3 0.0 1.0 0 0
u1t75=1.8
u1t76q=0.3 0.0 1.0 0 0
u1t76=1.824
u1t77q=0.3 0.0 1.0 0 0
u1t77=1.848
u1t78q=0.3 0.0 1.0 0 0
u1t78=1.872
u1t79q=0.3 0.0 1.0 0 0
u1t79=1.896
u1t80q=0.3 0.0 1.0 0 0
u1t80=1.92
u1t81q=0.3 0.0 1.0 0 0
u1t81=1.944
u1t82q=0.3 0.0 1.0 0 0
u1t82=1.968
u1t83q=0.3 0.0 1.0 0 0
u1t83=1.992
u1t84q=0.3 0.0 1.0 0 0
u1t84=2.016
u1t85q=0.3 0.0 1.0 0 0
u1t85=2.04
u1t86q=0.3 0.0 1.0 0 0
u1t86=2.064
u1t87q=0.3 0.0 1.0 0 0
u1t87=2.088
u1t88q=0.3 0.0 1.0 0 0
u1t88=2.112
u1t89q=0.3 0.0 1.0 0 0
u1t89=2.136
u1t90q=0.3 0.0 1.0 0 0
u1t90=2.16
u1t91q=0.3 0.0 1.0 0 0
u1t91=2.184
u1t92q=0.3 0.0 1.0 0 0
u1t92=2.208
u1t93q=0.3 0.0 1.0 0 0
u1t93=2.232
u1t94q=0.3 0.0 1.0 0 0
u1t94=2.256
u1t95q=0.3 0.0 1.0 0 0
u1t95=2.28
u1t96q=0.3 0.0 1.0 0 0
u1t96=2.304
u1t97q=0.3 0.0 1.0 0 0
u1t97=2.328
u1t98q=0.3 0.0 1.0 0 0
u1t98=2.352
u1t99q=0.3 0.0 1.0 0 0
u1t99=2.376
u1t100q=0.3 0.0 1.0 0 0
u1t100=2.4
u1t101q=0.3 0.0 1.0 0 0
u1t101=2.424
u1t102q=0.3 0.0 1.0 0 0
u1t102=2.448
u1t103q=0.3 0.0 1.0 0 0
u1t103=2.472
u1t104q=0.3 0.0 1.0 0 0
u1t104=2.496
u1t105q=0.3 0.0 1.0 0 0
u1t105=2.52
u1t106q=0.3 0.0 1.0 0 0
u1t106=2.544
u1t107q=0.3 0.0 1.0 0 0
u1t107=2.568
u1t108q=0.3 0.0 1.0 0 0
u1t108=2.592
u1t109q=0.3 0.0 1.0 0 0
u1t109=2.616
u1t110q=0.3 0.0 1.0 0 0
u1t110=2.64
u1t111q=0.3 0.0 1.0 0 0
u1t111=2.664
u1t112q=0.3 0.0 1.0 0 0
u1t112=2.688
u1t113q=0.3 0.0 1.0 0 0
u1t113=2.712
u1t114q=0.3 0.0 1.0 0 0
u1t114=2.736
u1t115q=0.3 0.0 1.0 0 0
u1t115=2.76
u1t116q=0.3 0.0 1.0 0 0
u1t116=2.784
u1t117q=0.3 0.0 1.0 0 0
u1t117=2.808
u1t118q=0.3 0.0 1.0 0 0
u1t118=2.832
u1t119q=0.3 0.0 1.0 0 0
u1t119=2.856
u1t120q=0.3 0.0 1.0 0 0
u1t120=2.88
u1t121q=0.3 0.0 1.0 0 0
u1t121=2.904
u1t122q=0.3 0.0 1.0 0 0
u1t122=2.928
u1t123q=0.3 0.0 1.0 0 0
u1t123=2.952
u1t124q=0.3 0.0 1.0 0 0
u1t124=2.976
u1t125q=0.3 0.0 1.0 0 0
u1t125=3.0
u1t126q=0.3 0.0 1.0 0 0
u1t126=3.024
u1t127q=0.3 0.0 1.0 0 0
u1t127=3.048
u1t128q=0.3 0.0 1.0 0 0
u1t128=3.072
u1t129q=0.3 0.0 1.0 0 0
u1t129=3.096
u1t130q=0.3 0.0 1.0 0 0
u1t130=3.12
u1t131q=0.3 0.0 1.0 0 0
u1t131=3.144
u1t132q=0.3 0.0 1.0 0 0
u1t132=3.168
u1t133q=0.3 0.0 1.0 0 0
u1t133=3.192
u1t134q=0.3 0.0 1.0 0 0
u1t134=3.216
u1t135q=0.3 0.0 1.0 0 0
u1t135=3.24
u1t136q=0.3 0.0 1.0 0 0
u1t136=3.264
u1t137q=0.3 0.0 1.0 0 0
u1t137=3.288
u1t138q=0.3 0.0 1.0 0 0
u1t138=3.312
u1t139q=0.3 0.0 1.0 0 0
u1t139=3.336
u1t140q=0.3 0.0 1.0 0 0
u1t140=3.36
u1t141q=0.3 0.0 1.0 0 0
u1t141=3.384
u1t142q=0.3 0.0 1.0 0 0
u1t142=3.408
u1t143q=0.3 0.0 1.0 0 0
u1t143=3.432
u1t144q=0.3 0.0 1.0 0 0
u1t144=3.456
u1t145q=0.3 0.0 1.0 0 0
u1t145=3.48
u1t146q=0.3 0.0 1.0 0 0
u1t146=3.504
u1t147q=0.3 0.0 1.0 0 0
u1t147=3.528
u1t148q=0.3 0.0 1.0 0 0
u1t148=3.552
u1t149q=0.3 0.0 1.0 0 0
u1t149=3.576
u1t150q=0.3 0.0 1.0 0 0
u1t150=3.6
u1t151q=0.3 0.0 1.0 0 0
u1t151=3.624
u1t152q=0.3 0.0 1.0 0 0
u1t152=3.648
u1t153q=0.3 0.0 1.0 0 0
u1t153=3.672
u1t154q=0.3 0.0 1.0 0 0
u1t154=3.696
u1t155q=0.3 0.0 1.0 0 0
u1t155=3.72
u1t156q=0.3 0.0 1.0 0 0
u1t156=3.744
u1t157q=0.3 0.0 1.0 0 0
u1t157=3.768
u1t158q=0.3 0.0 1.0 0 0
u1t158=3.792
u1t159q=0.3 0.0 1.0 0 0
u1t159=3.816
u1t160q=0.3 0.0 1.0 0 0
u1t160=3.84
u1t161q=0.3 0.0 1.0 0 0
u1t161=3.864
u1t162q=0.3 0.0 1.0 0 0
u1t162=3.888
u1t163q=0.3 0.0 1.0 0 0
u1t163=3.912
u1t164q=0.3 0.0 1.0 0 0
u1t164=3.936
u1t165q=0.3 0.0 1.0 0 0
u1t165=3.96
u1t166q=0.3 0.0 1.0 0 0
u1t166=3.984
u1t167q=0.3 0.0 1.0 0 0
u1t167=4.008
u1t168q=0.3 0.0 1.0 0 0
u1t168=4.032
u1t169q=0.3 0.0 1.0 0 0
u1t169=4.056
u1t170q=0.3 0.0 1.0 0 0
u1t170=4.08
u1t171q=0.3 0.0 1.0 0 0
u1t171=4.104
u1t172q=0.3 0.0 1.0 0 0
u1t172=4.128
u1t173q=0.3 0.0 1.0 0 0
u1t173=4.152
u1t174q=0.3 0.0 1.0 0 0
u1t174=4.176
u1t175q=0.3 0.0 1.0 0 0
u1t175=4.2
u1t176q=0.3 0.0 1.0 0 0
u1t176=4.224
u1t177q=0.3 0.0 1.0 0 0
u1t177=4.248
u1t178q=0.3 0.0 1.0 0 0
u1t178=4.272
u1t179q=0.3 0.0 1.0 0 0
u1t179=4.296
u1t180q=0.3 0.0 1.0 0 0
u1t180=4.32
u1t181q=0.3 0.0 1.0 0 0
u1t181=4.344
u1t182q=0.3 0.0 1.0 0 0
u1t182=4.368
u1t183q=0.3 0.0 1.0 0 0
u1t183=4.392
u1t184q=0.3 0.0 1.0 0 0
u1t184=4.416
u1t185q=0.3 0.0 1.0 0 0
u1t185=4.44
u1t186q=0.3 0.0 1.0 0 0
u1t186=4.464
u1t187q=0.3 0.0 1.0 0 0
u1t187=4.488
u1t188q=0.3 0.0 1.0 0 0
u1t188=4.512
u1t189q=0.3 0.0 1.0 0 0
u1t189=4.536
u1t190q=0.3 0.0 1.0 0 0
u1t190=4.56
u1t191q=0.3 0.0 1.0 0 0
u1t191=4.584
u1t192q=0.3 0.0 1.0 0 0
u1t192=4.608
u1t193q=0.3 0.0 1.0 0 0
u1t193=4.632
u1t194q=0.3 0.0 1.0 0 0
u1t194=4.656
u1t195q=0.3 0.0 1.0 0 0
u1t195=4.68
u1t196q=0.3 0.0 1.0 0 0
u1t196=4.704
u1t197q=0.3 0.0 1.0 0 0
u1t197=4.728
u1t198q=0.3 0.0 1.0 0 0
u1t198=4.752
u1t199q=0.3 0.0 1.0 0 0
u1t199=4.776
u1t200q=0.3 0.0 1.0 0 0
u1t200=4.8
u1t201q=0.3 0.0 1.0 0 0
u1t201=4.824
u1t202q=0.3 0.0 1.0 0 0
u1t202=4.848
u1t203q=0.3 0.0 1.0 0 0
u1t203=4.872
u1t204q=0.3 0.0 1.0 0 0
u1t204=4.896
u1t205q=0.3 0.0 1.0 0 0
u1t205=4.92
u1t206q=0.3 0.0 1.0 0 0
u1t206=4.944
u1t207q=0.3 0.0 1.0 0 0
u1t207=4.968
u1t208q=0.3 0.0 1.0 0 0
u1t208=4.992
u1t209q=0.3 0.0 1.0 0 0
u1t209=5.016
u1t210q=0.3 0.0 1.0 0 0
u1t210=5.04
u1t211q=0.3 0.0 1.0 0 0
u1t211=5.064
u1t212q=0.3 0.0 1.0 0 0
u1t212=5.088
u1t213q=0.3 0.0 1.0 0 0
u1t213=5.112
u1t214q=0.3 0.0 1.0 0 0
u1t214=5.136
u1t215q=0.3 0.0 1.0 0 0
u1t215=5.16
u1t216q=0.3 0.0 1.0 0 0
u1t216=5.184
u1t217q=0.3 0.0 1.0 0 0
u1t217=5.208
u1t218q=0.3 0.0 1.0 0 0
u1t218=5.232
u1t219q=0.3 0.0 1.0 0 0
u1t219=5.256
u1t220q=0.3 0.0 1.0 0 0
u1t220=5.28
u1t221q=0.3 0.0 1.0 0 0
u1t221=5.304
u1t222q=0.3 0.0 1.0 0 0
u1t222=5.328
u1t223q=0.3 0.0 1.0 0 0
u1t223=5.352
u1t224q=0.3 0.0 1.0 0 0
u1t224=5.376
u1t225q=0.3 0.0 1.0 0 0
u1t225=5.4
u1t226q=0.3 0.0 1.0 0 0
u1t226=5.424
u1t227q=0.3 0.0 1.0 0 0
u1t227=5.448
u1t228q=0.3 0.0 1.0 0 0
u1t228=5.472
u1t229q=0.3 0.0 1.0 0 0
u1t229=5.496
u1t230q=0.3 0.0 1.0 0 0
u1t230=5.52
u1t231q=0.3 0.0 1.0 0 0
u1t231=5.544
u1t232q=0.3 0.0 1.0 0 0
u1t232=5.568
u1t233q=0.3 0.0 1.0 0 0
u1t233=5.592
u1t234q=0.3 0.0 1.0 0 0
u1t234=5.616
u1t235q=0.3 0.0 1.0 0 0
u1t235=5.64
u1t236q=0.3 0.0 1.0 0 0
u1t236=5.664
u1t237q=0.3 0.0 1.0 0 0
u1t237=5.688
u1t238q=0.3 0.0 1.0 0 0
u1t238=5.712
u1t239q=0.3 0.0 1.0 0 0
u1t239=5.736
u1t240q=0.3 0.0 1.0 0 0
u1t240=5.76
u1t241q=0.3 0.0 1.0 0 0
u1t241=5.784
u1t242q=0.3 0.0 1.0 0 0
u1t242=5.808
u1t243q=0.3 0.0 1.0 0 0
u1t243=5.832
u1t244q=0.3 0.0 1.0 0 0
u1t244=5.856
u1t245q=0.3 0.0 1.0 0 0
u1t245=5.88
u1t246q=0.3 0.0 1.0 0 0
u1t246=5.904
u1t247q=0.3 0.0 1.0 0 0
u1t247=5.928
u1t248q=0.3 0.0 1.0 0 0
u1t248=5.952
u1t249q=0.3 0.0 1.0 0 0
u1t249=5.976
u1t250q=0.3 0.0 1.0 0 0
u1t250=6.0
u1t251q=0.3 0.0 1.0 0 0
u1t251=6.024
u1t252q=0.3 0.0 1.0 0 0
u1t252=6.048
u1t253q=0.3 0.0 1.0 0 0
u1t253=6.072
u1t254q=0.3 0.0 1.0 0 0
u1t254=6.096
u1t255q=0.3 0.0 1.0 0 0
u1t255=6.12
u1t256q=0.3 0.0 1.0 0 0
u1t256=6.144
u1t257q=0.3 0.0 1.0 0 0
u1t257=6.168
u1t258q=0.3 0.0 1.0 0 0
u1t258=6.192
u1t259q=0.3 0.0 1.0 0 0
u1t259=6.216
u1t260q=0.3 0.0 1.0 0 0
u1t260=6.24
u1t261q=0.3 0.0 1.0 0 0
u1t261=6.264
u1t262q=0.3 0.0 1.0 0 0
u1t262=6.288
u1t263q=0.3 0.0 1.0 0 0
u1t263=6.312
u1t264q=0.3 0.0 1.0 0 0
u1t264=6.336
u1t265q=0.3 0.0 1.0 0 0
u1t265=6.36
u1t266q=0.3 0.0 1.0 0 0
u1t266=6.384
u1t267q=0.3 0.0 1.0 0 0
u1t267=6.408
u1t268q=0.3 0.0 1.0 0 0
u1t268=6.432
u1t269q=0.3 0.0 1.0 0 0
u1t269=6.456
u1t270q=0.3 0.0 1.0 0 0
u1t270=6.48
u1t271q=0.3 0.0 1.0 0 0
u1t271=6.504
u1t272q=0.3 0.0 1.0 0 0
u1t272=6.528
u1t273q=0.3 0.0 1.0 0 0
u1t273=6.552
u1t274q=0.3 0.0 1.0 0 0
u1t274=6.576
u1t275q=0.3 0.0 1.0 0 0
u1t275=6.6
u1t276q=0.3 0.0 1.0 0 0
u1t276=6.624
u1t277q=0.3 0.0 1.0 0 0
u1t277=6.648
u1t278q=0.3 0.0 1.0 0 0
u1t278=6.672
u1t279q=0.3 0.0 1.0 0 0
u1t279=6.696
u1t280q=0.3 0.0 1.0 0 0
u1t280=6.72
u1t281q=0.3 0.0 1.0 0 0
u1t281=6.744
u1t282q=0.3 0.0 1.0 0 0
u1t282=6.768
u1t283q=0.3 0.0 1.0 0 0
u1t283=6.792
u1t284q=0.3 0.0 1.0 0 0
u1t284=6.816
u1t285q=0.3 0.0 1.0 0 0
u1t285=6.84
u1t286q=0.3 0.0 1.0 0 0
u1t286=6.864
u1t287q=0.3 0.0 1.0 0 0
u1t287=6.888
u1t288q=0.3 0.0 1.0 0 0
u1t288=6.912
u1t289q=0.3 0.0 1.0 0 0
u1t289=6.936
u1t290q=0.3 0.0 1.0 0 0
u1t290=6.96
u1t291q=0.3 0.0 1.0 0 0
u1t291=6.984
u1t292q=0.3 0.0 1.0 0 0
u1t292=7.008
u1t293q=0.3 0.0 1.0 0 0
u1t293=7.032
u1t294q=0.3 0.0 1.0 0 0
u1t294=7.056
u1t295q=0.3 0.0 1.0 0 0
u1t295=7.08
u1t296q=0.3 0.0 1.0 0 0
u1t296=7.104
u1t297q=0.3 0.0 1.0 0 0
u1t297=7.128
u1t298q=0.3 0.0 1.0 0 0
u1t298=7.152
u1t299q=0.3 0.0 1.0 0 0
u1t299=7.176
u1t300q=0.3 0.0 1.0 0 0
u1t300=7.2
u1t301q=0.3 0.0 1.0 0 0
u1t301=7.224
u1t302q=0.3 0.0 1.0 0 0
u1t302=7.248
u1t303q=0.3 0.0 1.0 0 0
u1t303=7.272
u1t304q=0.3 0.0 1.0 0 0
u1t304=7.296
u1t305q=0.3 0.0 1.0 0 0
u1t305=7.32
u1t306q=0.3 0.0 1.0 0 0
u1t306=7.344
u1t307q=0.3 0.0 1.0 0 0
u1t307=7.368
u1t308q=0.3 0.0 1.0 0 0
u1t308=7.392
u1t309q=0.3 0.0 1.0 0 0
u1t309=7.416
u1t310q=0.3 0.0 1.0 0 0
u1t310=7.44
u1t311q=0.3 0.0 1.0 0 0
u1t311=7.464
u1t312q=0.3 0.0 1.0 0 0
u1t312=7.488
u1t313q=0.3 0.0 1.0 0 0
u1t313=7.512
u1t314q=0.3 0.0 1.0 0 0
u1t314=7.536
u1t315q=0.3 0.0 1.0 0 0
u1t315=7.56
u1t316q=0.3 0.0 1.0 0 0
u1t316=7.584
u1t317q=0.3 0.0 1.0 0 0
u1t317=7.608
u1t318q=0.3 0.0 1.0 0 0
u1t318=7.632
u1t319q=0.3 0.0 1.0 0 0
u1t319=7.656
u1t320q=0.3 0.0 1.0 0 0
u1t320=7.68
u1t321q=0.3 0.0 1.0 0 0
u1t321=7.704
u1t322q=0.3 0.0 1.0 0 0
u1t322=7.728
u1t323q=0.3 0.0 1.0 0 0
u1t323=7.752
u1t324q=0.3 0.0 1.0 0 0
u1t324=7.776
u1t325q=0.3 0.0 1.0 0 0
u1t325=7.8
u1t326q=0.3 0.0 1.0 0 0
u1t326=7.824
u1t327q=0.3 0.0 1.0 0 0
u1t327=7.848
u1t328q=0.3 0.0 1.0 0 0
u1t328=7.872
u1t329q=0.3 0.0 1.0 0 0
u1t329=7.896
u1t330q=0.3 0.0 1.0 0 0
u1t330=7.92
u1t331q=0.3 0.0 1.0 0 0
u1t331=7.944
u1t332q=0.3 0.0 1.0 0 0
u1t332=7.968
u1t333q=0.3 0.0 1.0 0 0
u1t333=7.992
u1t334q=0.3 0.0 1.0 0 0
u1t334=8.016
u1t335q=0.3 0.0 1.0 0 0
u1t335=8.04
u1t336q=0.3 0.0 1.0 0 0
u1t336=8.064
u1t337q=0.3 0.0 1.0 0 0
u1t337=8.088
u1t338q=0.3 0.0 1.0 0 0
u1t338=8.112
u1t339q=0.3 0.0 1.0 0 0
u1t339=8.136
u1t340q=0.3 0.0 1.0 0 0
u1t340=8.16
u1t341q=0.3 0.0 1.0 0 0
u1t341=8.184
u1t342q=0.3 0.0 1.0 0 0
u1t342=8.208
u1t343q=0.3 0.0 1.0 0 0
u1t343=8.232
u1t344q=0.3 0.0 1.0 0 0
u1t344=8.256
u1t345q=0.3 0.0 1.0 0 0
u1t345=8.28
u1t346q=0.3 0.0 1.0 0 0
u1t346=8.304
u1t347q=0.3 0.0 1.0 0 0
u1t347=8.328
u1t348q=0.3 0.0 1.0 0 0
u1t348=8.352
u1t349q=0.3 0.0 1.0 0 0
u1t349=8.376
u1t350q=0.3 0.0 1.0 0 0
u1t350=8.4
u1t351q=0.3 0.0 1.0 0 0
u1t351=8.424
u1t352q=0.3 0.0 1.0 0 0
u1t352=8.448
u1t353q=0.3 0.0 1.0 0 0
u1t353=8.472
u1t354q=0.3 0.0 1.0 0 0
u1t354=8.496
u1t355q=0.3 0.0 1.0 0 0
u1t355=8.52
u1t356q=0.3 0.0 1.0 0 0
u1t356=8.544
u1t357q=0.3 0.0 1.0 0 0
u1t357=8.568
u1t358q=0.3 0.0 1.0 0 0
u1t358=8.592
u1t359q=0.3 0.0 1.0 0 0
u1t359=8.616
u1t360q=0.3 0.0 1.0 0 0
u1t360=8.64
u1t361q=0.3 0.0 1.0 0 0
u1t361=8.664
u1t362q=0.3 0.0 1.0 0 0
u1t362=8.688
u1t363q=0.3 0.0 1.0 0 0
u1t363=8.712
u1t364q=0.3 0.0 1.0 0 0
u1t364=8.736
u1t365q=0.3 0.0 1.0 0 0
u1t365=8.76
u1t366q=0.3 0.0 1.0 0 0
u1t366=8.784
u1t367q=0.3 0.0 1.0 0 0
u1t367=8.808
u1t368q=0.3 0.0 1.0 0 0
u1t368=8.832
u1t369q=0.3 0.0 1.0 0 0
u1t369=8.856
u1t370q=0.3 0.0 1.0 0 0
u1t370=8.88
u1t371q=0.3 0.0 1.0 0 0
u1t371=8.904
u1t372q=0.3 0.0 1.0 0 0
u1t372=8.928
u1t373q=0.3 0.0 1.0 0 0
u1t373=8.952
u1t374q=0.3 0.0 1.0 0 0
u1t374=8.976
u1t375q=0.3 0.0 1.0 0 0
u1t375=9.0
u1t376q=0.3 0.0 1.0 0 0
u1t376=9.024
u1t377q=0.3 0.0 1.0 0 0
u1t377=9.048
u1t378q=0.3 0.0 1.0 0 0
u1t378=9.072
u1t379q=0.3 0.0 1.0 0 0
u1t379=9.096
u1t380q=0.3 0.0 1.0 0 0
u1t380=9.12
u1t381q=0.3 0.0 1.0 0 0
u1t381=9.144
u1t382q=0.3 0.0 1.0 0 0
u1t382=9.168
u1t383q=0.3 0.0 1.0 0 0
u1t383=9.192
u1t384q=0.3 0.0 1.0 0 0
u1t384=9.216
u1t385q=0.3 0.0 1.0 0 0
u1t385=9.24
u1t386q=0.3 0.0 1.0 0 0
u1t386=9.264
u1t387q=0.3 0.0 1.0 0 0
u1t387=9.288
u1t388q=0.3 0.0 1.0 0 0
u1t388=9.312
u1t389q=0.3 0.0 1.0 0 0
u1t389=9.336
u1t390q=0.3 0.0 1.0 0 0
u1t390=9.36
u1t391q=0.3 0.0 1.0 0 0
u1t391=9.384
u1t392q=0.3 0.0 1.0 0 0
u1t392=9.408
u1t393q=0.3 0.0 1.0 0 0
u1t393=9.432
u1t394q=0.3 0.0 1.0 0 0
u1t394=9.456
u1t395q=0.3 0.0 1.0 0 0
u1t395=9.48
u1t396q=0.3 0.0 1.0 0 0
u1t396=9.504
u1t397q=0.3 0.0 1.0 0 0
u1t397=9.528
u1t398q=0.3 0.0 1.0 0 0
u1t398=9.552
u1t399q=0.3 0.0 1.0 0 0
u1t399=9.576
u1t400q=0.3 0.0 1.0 0 0
u1t400=9.6
u1t401q=0.3 0.0 1.0 0 0
u1t401=9.624
u1t402q=0.3 0.0 1.0 0 0
u1t402=9.648
u1t403q=0.3 0.0 1.0 0 0
u1t403=9.672
u1t404q=0.3 0.0 1.0 0 0
u1t404=9.696
u1t405q=0.3 0.0 1.0 0 0
u1t405=9.72
u1t406q=0.3 0.0 1.0 0 0
u1t406=9.744
u1t407q=0.3 0.0 1.0 0 0
u1t407=9.768
u1t408q=0.3 0.0 1.0 0 0
u1t408=9.792
u1t409q=0.3 0.0 1.0 0 0
u1t409=9.816
u1t410q=0.3 0.0 1.0 0 0
u1t410=9.84
u1t411q=0.3 0.0 1.0 0 0
u1t411=9.864
u1t412q=0.3 0.0 1.0 0 0
u1t412=9.888
u1t413q=0.3 0.0 1.0 0 0
u1t413=9.912
u1t414q=0.3 0.0 1.0 0 0
u1t414=9.936
u1t415q=0.3 0.0 1.0 0 0
u1t415=9.96
u1t416q=0.3 0.0 1.0 0 0
u1t416=9.984
u1t417q=0.3 0.0 1.0 0 0
u1t417=10.008
u1t418q=0.3 0.0 1.0 0 0
u1t418=10.032
u1t419q=0.3 0.0 1.0 0 0
u1t419=10.056
u1t420q=0.3 0.0 1.0 0 0
u1t420=10.08
u1t421q=0.3 0.0 1.0 0 0
u1t421=10.104
u1t422q=0.3 0.0 1.0 0 0
u1t422=10.128
u1t423q=0.3 0.0 1.0 0 0
u1t423=10.152
u1t424q=0.3 0.0 1.0 0 0
u1t424=10.176
u1t425q=0.3 0.0 1.0 0 0
u1t425=10.2
u1t426q=0.3 0.0 1.0 0 0
u1t426=10.224
u1t427q=0.3 0.0 1.0 0 0
u1t427=10.248
u1t428q=0.3 0.0 1.0 0 0
u1t428=10.272
u1t429q=0.3 0.0 1.0 0 0
u1t429=10.296
u1t430q=0.3 0.0 1.0 0 0
u1t430=10.32
u1t431q=0.3 0.0 1.0 0 0
u1t431=10.344
u1t432q=0.3 0.0 1.0 0 0
u1t432=10.368
u1t433q=0.3 0.0 1.0 0 0
u1t433=10.392
u1t434q=0.3 0.0 1.0 0 0
u1t434=10.416
u1t435q=0.3 0.0 1.0 0 0
u1t435=10.44
u1t436q=0.3 0.0 1.0 0 0
u1t436=10.464
u1t437q=0.3 0.0 1.0 0 0
u1t437=10.488
u1t438q=0.3 0.0 1.0 0 0
u1t438=10.512
u1t439q=0.3 0.0 1.0 0 0
u1t439=10.536
u1t440q=0.3 0.0 1.0 0 0
u1t440=10.56
u1t441q=0.3 0.0 1.0 0 0
u1t441=10.584
u1t442q=0.3 0.0 1.0 0 0
u1t442=10.608
u1t443q=0.3 0.0 1.0 0 0
u1t443=10.632
u1t444q=0.3 0.0 1.0 0 0
u1t444=10.656
u1t445q=0.3 0.0 1.0 0 0
u1t445=10.68
u1t446q=0.3 0.0 1.0 0 0
u1t446=10.704
u1t447q=0.3 0.0 1.0 0 0
u1t447=10.728
u1t448q=0.3 0.0 1.0 0 0
u1t448=10.752
u1t449q=0.3 0.0 1.0 0 0
u1t449=10.776
u1t450q=0.3 0.0 1.0 0 0
u1t450=10.8
u1t451q=0.3 0.0 1.0 0 0
u1t451=10.824
u1t452q=0.3 0.0 1.0 0 0
u1t452=10.848
u1t453q=0.3 0.0 1.0 0 0
u1t453=10.872
u1t454q=0.3 0.0 1.0 0 0
u1t454=10.896
u1t455q=0.3 0.0 1.0 0 0
u1t455=10.92
u1t456q=0.3 0.0 1.0 0 0
u1t456=10.944
u1t457q=0.3 0.0 1.0 0 0
u1t457=10.968
u1t458q=0.3 0.0 1.0 0 0
u1t458=10.992
u1t459q=0.3 0.0 1.0 0 0
u1t459=11.016
u1t460q=0.3 0.0 1.0 0 0
u1t460=11.04
u1t461q=0.3 0.0 1.0 0 0
u1t461=11.064
u1t462q=0.3 0.0 1.0 0 0
u1t462=11.088
u1t463q=0.3 0.0 1.0 0 0
u1t463=11.112
u1t464q=0.3 0.0 1.0 0 0
u1t464=11.136
u1t465q=0.3 0.0 1.0 0 0
u1t465=11.16
u1t466q=0.3 0.0 1.0 0 0
u1t466=11.184
u1t467q=0.3 0.0 1.0 0 0
u1t467=11.208
u1t468q=0.3 0.0 1.0 0 0
u1t468=11.232
u1t469q=0.3 0.0 1.0 0 0
u1t469=11.256
u1t470q=0.3 0.0 1.0 0 0
u1t470=11.28
u1t471q=0.3 0.0 1.0 0 0
u1t471=11.304
u1t472q=0.3 0.0 1.0 0 0
u1t472=11.328
u1t473q=0.3 0.0 1.0 0 0
u1t473=11.352
u1t474q=0.3 0.0 1.0 0 0
u1t474=11.376
u1t475q=0.3 0.0 1.0 0 0
u1t475=11.4
u1t476q=0.3 0.0 1.0 0 0
u1t476=11.424
u1t477q=0.3 0.0 1.0 0 0
u1t477=11.448
u1t478q=0.3 0.0 1.0 0 0
u1t478=11.472
u1t479q=0.3 0.0 1.0 0 0
u1t479=11.496
u1t480q=0.3 0.0 1.0 0 0
u1t480=11.52
u1t481q=0.3 0.0 1.0 0 0
u1t481=11.544
u1t482q=0.3 0.0 1.0 0 0
u1t482=11.568
u1t483q=0.3 0.0 1.0 0 0
u1t483=11.592
u1t484q=0.3 0.0 1.0 0 0
u1t484=11.616
u1t485q=0.3 0.0 1.0 0 0
u1t485=11.64
u1t486q=0.3 0.0 1.0 0 0
u1t486=11.664
u1t487q=0.3 0.0 1.0 0 0
u1t487=11.688
u1t488q=0.3 0.0 1.0 0 0
u1t488=11.712
u1t489q=0.3 0.0 1.0 0 0
u1t489=11.736
u1t490q=0.3 0.0 1.0 0 0
u1t490=11.76
u1t491q=0.3 0.0 1.0 0 0
u1t491=11.784
u1t492q=0.3 0.0 1.0 0 0
u1t492=11.808
u1t493q=0.3 0.0 1.0 0 0
u1t493=11.832
u1t494q=0.3 0.0 1.0 0 0
u1t494=11.856
u1t495q=0.3 0.0 1.0 0 0
u1t495=11.88
u1t496q=0.3 0.0 1.0 0 0
u1t496=11.904
u1t497q=0.3 0.0 1.0 0 0
u1t497=11.928
u1t498q=0.3 0.0 1.0 0 0
u1t498=11.952
u1t499q=0.3 0.0 1.0 0 0
u1t499=11.976
u1t500q=0.3 0 0 0 0
u1t500=tend

[Messungen]
tAnzahl=12

t1=1
t1Anzahl=2
t1m1=mfcn1 1.0 1e-06 1
t1m2=mfcn2 1.0 1e-06 1
t1minmax=0 1e+10

t2=2.000
t2Anzahl=2
t2m1=mfcn1 1.0 1e-06 1
t2m2=mfcn2 1.0 1e-06 1
t2minmax=0 1e+10

t3=3
t3Anzahl=2
t3m1=mfcn1 1.0 1e-06 1
t3m2=mfcn2 1.0 1e-06 1
t3minmax=0 1e+10

t4=4
t4Anzahl=2
t4m1=mfcn1 1.0 1e-06 1
t4m2=mfcn2 1.0 1e-06 1
t4minmax=0 1e+10

t5=5
t5Anzahl=2
t5m1=mfcn1 1.0 1e-06 1
t5m2=mfcn2 1.0 1e-06 1
t5minmax=0 1e+10

t6=6
t6Anzahl=2
t6m1=mfcn1 1.0 1e-06 1
t6m2=mfcn2 1.0 1e-06 1
t6minmax=0 1e+10

t7=7
t7Anzahl=2
t7m1=mfcn1 1.0 1e-06 1
t7m2=mfcn2 1.0 1e-06 1
t7minmax=0 1e+10

t8=8
t8Anzahl=2
t8m1=mfcn1 1.0 1e-06 1
t8m2=mfcn2 1.0 1e-06 1
t8minmax=0 1e+10

t9=9
t9Anzahl=2
t9m1=mfcn1 1.0 1e-06 1
t9m2=mfcn2 1.0 1e-06 1
t9minmax=0 1e+10

t10=10.000
t10Anzahl=2
t10m1=mfcn1 1.0 1e-06 1
t10m2=mfcn2 1.0 1e-06 1
t10minmax=0 1e+10

t11=11.000
t11Anzahl=2
t11m1=mfcn1 1.0 1e-06 1
t11m2=mfcn2 1.0 1e-06 1
t11minmax=0 1e+10

t12=12.000
t12Anzahl=2
t12m1=mfcn1 1.0 1e-06 1
t12m2=mfcn2 1.0 1e-06 1
t12minmax=0 1e+10
[NebenbedingungenSteuergroessen]
cAnzahl=0
[Messverfahren]
mAnzahl=2
m1=mfcn1 1 0 1e+10 0
m1f1=mess3 sigma3 1
m2=mfcn2 1 0 1e+10 0
m2f1=mess4 sigma4 1
mminmaxges=0 8

[OptionenIntegration]
teps=1e-08
rtol=1e-08
atol=1e-07
stepsize=0.0001
maxorder=6
maxstepnumber=4000
minstepsize=-1
maxstepsize=-1
maxitNewton=3
realworkspace=1700000
integerworkspace=5000
printlevel=0
mcnonlinearflag=0
mcDAEflag=0
mctol=1e-07
mcmaxit=50
mclinesearch=1
mcalpha0=1
rndmethod=-1
rndeps=1e-05
rndverbose=0

</source>

Lotka Experimental Design (VPLAN)

2016-01-19T16:23:19Z

FelixJost: /* VPLAN */

== VPLAN ==

Differential equations:

<source lang="fortran">

c RHS of the differential equations

subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), f(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

real*8 p1,p3,p5,p6, myu

c fixed parameters
p1 = 1.0
p3 = 1.0
p5 = 0.4
p6 = 0.2

c DISCRETIZE1( myu, rwh, iwh )

f(1) = p1*x(1) - p(1)*x(1)*x(2) - p5*myu*x(1)
f(2) = (-1.0)*p3*x(2) + p(2)*x(1)*x(2) - p6*myu*x(2)

end

</source>

Algebraic equations:

<source lang="fortran">

c Dummyfunction for RHS of algebraic equations

subroutine gfcn( t, x, g, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), g(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

iflag=0

end

</source>

First Measurement function:

<source lang="fortran">

c Messfunktion

subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )
implicit none

real*8 t, x(*), h, p(*), q(*), rwh(*)
integer*4 iwh(*), iflag

h = x(1)

iflag = 0

end

</source>

Second Measurement function:

<source lang="fortran">

c Messfunktion

subroutine mess4( t, x, h, p, q, rwh, iwh, iflag )
implicit none

real*8 t, x(*), h, p(*), q(*), rwh(*)
integer*4 iwh(*), iflag

h = x(2)

iflag = 0

end

</source>

Standard deviation of first measurement function

<source lang="fortran">

c Standardabweichung der Messfunktion

subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )
implicit none

real*8 rwh(*)
integer*4 iwh(*), iflag

real*8 t, x(*), p(*), q(*)
real*8 s
real*8 h

s = 1.0d+0

iflag = 0

end

</source>

Standard deviation of second measurement function:

<source lang="fortran">
c Standardabweichung der Messfunktion

subroutine sigma4( t, x, s, p, q, rwh, iwh, iflag )
implicit none

real*8 rwh(*)
integer*4 iwh(*), iflag

real*8 t, x(*), p(*), q(*)
real*8 s(*)

s(1) = 1.0

iflag = 0

end

</source>

VPLAN specific experimental setup:

<source lang="vplan">

; ini-File fuer Experiment

[Flags]
switch=1

[Kosten]
costs=0 -1e+10 1e+10

[Integrationsintervall]
t0=0
tend=12

[Modellfunktionen]
ffcn=ffcn
gfcn=gfcn

[Zustandsvariablen]
yAnzahl=2
y1=x1 0.5 -1e+10 1e+10
y2=x2 0.7 -1e+10 1e+10

zAnzahl=0

[Mehrzielknoten]
tAnzahl=12
t1=1
t2=2
t3=3
t4=4
t5=5
t6=6
t7=7
t8=8
t9=9
t10=10
t11=11
t12=12

[DynamischeNebenbedingungen]
bAnzahl=0

[GitterUeberpruefungNebenbedingungen]
tAnzahl=0

[Steuergroessen]
qAnzahl=0

[Steuerfunktionen]
uAnzahl=1
u1=myu 0 0.0 1.0
u1tAnzahl=500
u1t0=t0
u1t1q=0.3 0.0 1.0 0 0
u1t1=0.024
u1t2q=0.3 0.0 1.0 0 0
u1t2=0.048
u1t3q=0.3 0.0 1.0 0 0
u1t3=0.072
u1t4q=0.3 0.0 1.0 0 0
u1t4=0.096
u1t5q=0.3 0.0 1.0 0 0
u1t5=0.12
u1t6q=0.3 0.0 1.0 0 0
u1t6=0.144
u1t7q=0.3 0.0 1.0 0 0
u1t7=0.168
u1t8q=0.3 0.0 1.0 0 0
u1t8=0.192
u1t9q=0.3 0.0 1.0 0 0
u1t9=0.216
u1t10q=0.3 0.0 1.0 0 0
u1t10=0.24
u1t11q=0.3 0.0 1.0 0 0
u1t11=0.264
u1t12q=0.3 0.0 1.0 0 0
u1t12=0.288
u1t13q=0.3 0.0 1.0 0 0
u1t13=0.312
u1t14q=0.3 0.0 1.0 0 0
u1t14=0.336
u1t15q=0.3 0.0 1.0 0 0
u1t15=0.36
u1t16q=0.3 0.0 1.0 0 0
u1t16=0.384
u1t17q=0.3 0.0 1.0 0 0
u1t17=0.408
u1t18q=0.3 0.0 1.0 0 0
u1t18=0.432
u1t19q=0.3 0.0 1.0 0 0
u1t19=0.456
u1t20q=0.3 0.0 1.0 0 0
u1t20=0.48
u1t21q=0.3 0.0 1.0 0 0
u1t21=0.504
u1t22q=0.3 0.0 1.0 0 0
u1t22=0.528
u1t23q=0.3 0.0 1.0 0 0
u1t23=0.552
u1t24q=0.3 0.0 1.0 0 0
u1t24=0.576
u1t25q=0.3 0.0 1.0 0 0
u1t25=0.6
u1t26q=0.3 0.0 1.0 0 0
u1t26=0.624
u1t27q=0.3 0.0 1.0 0 0
u1t27=0.648
u1t28q=0.3 0.0 1.0 0 0
u1t28=0.672
u1t29q=0.3 0.0 1.0 0 0
u1t29=0.696
u1t30q=0.3 0.0 1.0 0 0
u1t30=0.72
u1t31q=0.3 0.0 1.0 0 0
u1t31=0.744
u1t32q=0.3 0.0 1.0 0 0
u1t32=0.768
u1t33q=0.3 0.0 1.0 0 0
u1t33=0.792
u1t34q=0.3 0.0 1.0 0 0
u1t34=0.816
u1t35q=0.3 0.0 1.0 0 0
u1t35=0.84
u1t36q=0.3 0.0 1.0 0 0
u1t36=0.864
u1t37q=0.3 0.0 1.0 0 0
u1t37=0.888
u1t38q=0.3 0.0 1.0 0 0
u1t38=0.912
u1t39q=0.3 0.0 1.0 0 0
u1t39=0.936
u1t40q=0.3 0.0 1.0 0 0
u1t40=0.96
u1t41q=0.3 0.0 1.0 0 0
u1t41=0.984
u1t42q=0.3 0.0 1.0 0 0
u1t42=1.008
u1t43q=0.3 0.0 1.0 0 0
u1t43=1.032
u1t44q=0.3 0.0 1.0 0 0
u1t44=1.056
u1t45q=0.3 0.0 1.0 0 0
u1t45=1.08
u1t46q=0.3 0.0 1.0 0 0
u1t46=1.104
u1t47q=0.3 0.0 1.0 0 0
u1t47=1.128
u1t48q=0.3 0.0 1.0 0 0
u1t48=1.152
u1t49q=0.3 0.0 1.0 0 0
u1t49=1.176
u1t50q=0.3 0.0 1.0 0 0
u1t50=1.2
u1t51q=0.3 0.0 1.0 0 0
u1t51=1.224
u1t52q=0.3 0.0 1.0 0 0
u1t52=1.248
u1t53q=0.3 0.0 1.0 0 0
u1t53=1.272
u1t54q=0.3 0.0 1.0 0 0
u1t54=1.296
u1t55q=0.3 0.0 1.0 0 0
u1t55=1.32
u1t56q=0.3 0.0 1.0 0 0
u1t56=1.344
u1t57q=0.3 0.0 1.0 0 0
u1t57=1.368
u1t58q=0.3 0.0 1.0 0 0
u1t58=1.392
u1t59q=0.3 0.0 1.0 0 0
u1t59=1.416
u1t60q=0.3 0.0 1.0 0 0
u1t60=1.44
u1t61q=0.3 0.0 1.0 0 0
u1t61=1.464
u1t62q=0.3 0.0 1.0 0 0
u1t62=1.488
u1t63q=0.3 0.0 1.0 0 0
u1t63=1.512
u1t64q=0.3 0.0 1.0 0 0
u1t64=1.536
u1t65q=0.3 0.0 1.0 0 0
u1t65=1.56
u1t66q=0.3 0.0 1.0 0 0
u1t66=1.584
u1t67q=0.3 0.0 1.0 0 0
u1t67=1.608
u1t68q=0.3 0.0 1.0 0 0
u1t68=1.632
u1t69q=0.3 0.0 1.0 0 0
u1t69=1.656
u1t70q=0.3 0.0 1.0 0 0
u1t70=1.68
u1t71q=0.3 0.0 1.0 0 0
u1t71=1.704
u1t72q=0.3 0.0 1.0 0 0
u1t72=1.728
u1t73q=0.3 0.0 1.0 0 0
u1t73=1.752
u1t74q=0.3 0.0 1.0 0 0
u1t74=1.776
u1t75q=0.3 0.0 1.0 0 0
u1t75=1.8
u1t76q=0.3 0.0 1.0 0 0
u1t76=1.824
u1t77q=0.3 0.0 1.0 0 0
u1t77=1.848
u1t78q=0.3 0.0 1.0 0 0
u1t78=1.872
u1t79q=0.3 0.0 1.0 0 0
u1t79=1.896
u1t80q=0.3 0.0 1.0 0 0
u1t80=1.92
u1t81q=0.3 0.0 1.0 0 0
u1t81=1.944
u1t82q=0.3 0.0 1.0 0 0
u1t82=1.968
u1t83q=0.3 0.0 1.0 0 0
u1t83=1.992
u1t84q=0.3 0.0 1.0 0 0
u1t84=2.016
u1t85q=0.3 0.0 1.0 0 0
u1t85=2.04
u1t86q=0.3 0.0 1.0 0 0
u1t86=2.064
u1t87q=0.3 0.0 1.0 0 0
u1t87=2.088
u1t88q=0.3 0.0 1.0 0 0
u1t88=2.112
u1t89q=0.3 0.0 1.0 0 0
u1t89=2.136
u1t90q=0.3 0.0 1.0 0 0
u1t90=2.16
u1t91q=0.3 0.0 1.0 0 0
u1t91=2.184
u1t92q=0.3 0.0 1.0 0 0
u1t92=2.208
u1t93q=0.3 0.0 1.0 0 0
u1t93=2.232
u1t94q=0.3 0.0 1.0 0 0
u1t94=2.256
u1t95q=0.3 0.0 1.0 0 0
u1t95=2.28
u1t96q=0.3 0.0 1.0 0 0
u1t96=2.304
u1t97q=0.3 0.0 1.0 0 0
u1t97=2.328
u1t98q=0.3 0.0 1.0 0 0
u1t98=2.352
u1t99q=0.3 0.0 1.0 0 0
u1t99=2.376
u1t100q=0.3 0.0 1.0 0 0
u1t100=2.4
u1t101q=0.3 0.0 1.0 0 0
u1t101=2.424
u1t102q=0.3 0.0 1.0 0 0
u1t102=2.448
u1t103q=0.3 0.0 1.0 0 0
u1t103=2.472
u1t104q=0.3 0.0 1.0 0 0
u1t104=2.496
u1t105q=0.3 0.0 1.0 0 0
u1t105=2.52
u1t106q=0.3 0.0 1.0 0 0
u1t106=2.544
u1t107q=0.3 0.0 1.0 0 0
u1t107=2.568
u1t108q=0.3 0.0 1.0 0 0
u1t108=2.592
u1t109q=0.3 0.0 1.0 0 0
u1t109=2.616
u1t110q=0.3 0.0 1.0 0 0
u1t110=2.64
u1t111q=0.3 0.0 1.0 0 0
u1t111=2.664
u1t112q=0.3 0.0 1.0 0 0
u1t112=2.688
u1t113q=0.3 0.0 1.0 0 0
u1t113=2.712
u1t114q=0.3 0.0 1.0 0 0
u1t114=2.736
u1t115q=0.3 0.0 1.0 0 0
u1t115=2.76
u1t116q=0.3 0.0 1.0 0 0
u1t116=2.784
u1t117q=0.3 0.0 1.0 0 0
u1t117=2.808
u1t118q=0.3 0.0 1.0 0 0
u1t118=2.832
u1t119q=0.3 0.0 1.0 0 0
u1t119=2.856
u1t120q=0.3 0.0 1.0 0 0
u1t120=2.88
u1t121q=0.3 0.0 1.0 0 0
u1t121=2.904
u1t122q=0.3 0.0 1.0 0 0
u1t122=2.928
u1t123q=0.3 0.0 1.0 0 0
u1t123=2.952
u1t124q=0.3 0.0 1.0 0 0
u1t124=2.976
u1t125q=0.3 0.0 1.0 0 0
u1t125=3.0
u1t126q=0.3 0.0 1.0 0 0
u1t126=3.024
u1t127q=0.3 0.0 1.0 0 0
u1t127=3.048
u1t128q=0.3 0.0 1.0 0 0
u1t128=3.072
u1t129q=0.3 0.0 1.0 0 0
u1t129=3.096
u1t130q=0.3 0.0 1.0 0 0
u1t130=3.12
u1t131q=0.3 0.0 1.0 0 0
u1t131=3.144
u1t132q=0.3 0.0 1.0 0 0
u1t132=3.168
u1t133q=0.3 0.0 1.0 0 0
u1t133=3.192
u1t134q=0.3 0.0 1.0 0 0
u1t134=3.216
u1t135q=0.3 0.0 1.0 0 0
u1t135=3.24
u1t136q=0.3 0.0 1.0 0 0
u1t136=3.264
u1t137q=0.3 0.0 1.0 0 0
u1t137=3.288
u1t138q=0.3 0.0 1.0 0 0
u1t138=3.312
u1t139q=0.3 0.0 1.0 0 0
u1t139=3.336
u1t140q=0.3 0.0 1.0 0 0
u1t140=3.36
u1t141q=0.3 0.0 1.0 0 0
u1t141=3.384
u1t142q=0.3 0.0 1.0 0 0
u1t142=3.408
u1t143q=0.3 0.0 1.0 0 0
u1t143=3.432
u1t144q=0.3 0.0 1.0 0 0
u1t144=3.456
u1t145q=0.3 0.0 1.0 0 0
u1t145=3.48
u1t146q=0.3 0.0 1.0 0 0
u1t146=3.504
u1t147q=0.3 0.0 1.0 0 0
u1t147=3.528
u1t148q=0.3 0.0 1.0 0 0
u1t148=3.552
u1t149q=0.3 0.0 1.0 0 0
u1t149=3.576
u1t150q=0.3 0.0 1.0 0 0
u1t150=3.6
u1t151q=0.3 0.0 1.0 0 0
u1t151=3.624
u1t152q=0.3 0.0 1.0 0 0
u1t152=3.648
u1t153q=0.3 0.0 1.0 0 0
u1t153=3.672
u1t154q=0.3 0.0 1.0 0 0
u1t154=3.696
u1t155q=0.3 0.0 1.0 0 0
u1t155=3.72
u1t156q=0.3 0.0 1.0 0 0
u1t156=3.744
u1t157q=0.3 0.0 1.0 0 0
u1t157=3.768
u1t158q=0.3 0.0 1.0 0 0
u1t158=3.792
u1t159q=0.3 0.0 1.0 0 0
u1t159=3.816
u1t160q=0.3 0.0 1.0 0 0
u1t160=3.84
u1t161q=0.3 0.0 1.0 0 0
u1t161=3.864
u1t162q=0.3 0.0 1.0 0 0
u1t162=3.888
u1t163q=0.3 0.0 1.0 0 0
u1t163=3.912
u1t164q=0.3 0.0 1.0 0 0
u1t164=3.936
u1t165q=0.3 0.0 1.0 0 0
u1t165=3.96
u1t166q=0.3 0.0 1.0 0 0
u1t166=3.984
u1t167q=0.3 0.0 1.0 0 0
u1t167=4.008
u1t168q=0.3 0.0 1.0 0 0
u1t168=4.032
u1t169q=0.3 0.0 1.0 0 0
u1t169=4.056
u1t170q=0.3 0.0 1.0 0 0
u1t170=4.08
u1t171q=0.3 0.0 1.0 0 0
u1t171=4.104
u1t172q=0.3 0.0 1.0 0 0
u1t172=4.128
u1t173q=0.3 0.0 1.0 0 0
u1t173=4.152
u1t174q=0.3 0.0 1.0 0 0
u1t174=4.176
u1t175q=0.3 0.0 1.0 0 0
u1t175=4.2
u1t176q=0.3 0.0 1.0 0 0
u1t176=4.224
u1t177q=0.3 0.0 1.0 0 0
u1t177=4.248
u1t178q=0.3 0.0 1.0 0 0
u1t178=4.272
u1t179q=0.3 0.0 1.0 0 0
u1t179=4.296
u1t180q=0.3 0.0 1.0 0 0
u1t180=4.32
u1t181q=0.3 0.0 1.0 0 0
u1t181=4.344
u1t182q=0.3 0.0 1.0 0 0
u1t182=4.368
u1t183q=0.3 0.0 1.0 0 0
u1t183=4.392
u1t184q=0.3 0.0 1.0 0 0
u1t184=4.416
u1t185q=0.3 0.0 1.0 0 0
u1t185=4.44
u1t186q=0.3 0.0 1.0 0 0
u1t186=4.464
u1t187q=0.3 0.0 1.0 0 0
u1t187=4.488
u1t188q=0.3 0.0 1.0 0 0
u1t188=4.512
u1t189q=0.3 0.0 1.0 0 0
u1t189=4.536
u1t190q=0.3 0.0 1.0 0 0
u1t190=4.56
u1t191q=0.3 0.0 1.0 0 0
u1t191=4.584
u1t192q=0.3 0.0 1.0 0 0
u1t192=4.608
u1t193q=0.3 0.0 1.0 0 0
u1t193=4.632
u1t194q=0.3 0.0 1.0 0 0
u1t194=4.656
u1t195q=0.3 0.0 1.0 0 0
u1t195=4.68
u1t196q=0.3 0.0 1.0 0 0
u1t196=4.704
u1t197q=0.3 0.0 1.0 0 0
u1t197=4.728
u1t198q=0.3 0.0 1.0 0 0
u1t198=4.752
u1t199q=0.3 0.0 1.0 0 0
u1t199=4.776
u1t200q=0.3 0.0 1.0 0 0
u1t200=4.8
u1t201q=0.3 0.0 1.0 0 0
u1t201=4.824
u1t202q=0.3 0.0 1.0 0 0
u1t202=4.848
u1t203q=0.3 0.0 1.0 0 0
u1t203=4.872
u1t204q=0.3 0.0 1.0 0 0
u1t204=4.896
u1t205q=0.3 0.0 1.0 0 0
u1t205=4.92
u1t206q=0.3 0.0 1.0 0 0
u1t206=4.944
u1t207q=0.3 0.0 1.0 0 0
u1t207=4.968
u1t208q=0.3 0.0 1.0 0 0
u1t208=4.992
u1t209q=0.3 0.0 1.0 0 0
u1t209=5.016
u1t210q=0.3 0.0 1.0 0 0
u1t210=5.04
u1t211q=0.3 0.0 1.0 0 0
u1t211=5.064
u1t212q=0.3 0.0 1.0 0 0
u1t212=5.088
u1t213q=0.3 0.0 1.0 0 0
u1t213=5.112
u1t214q=0.3 0.0 1.0 0 0
u1t214=5.136
u1t215q=0.3 0.0 1.0 0 0
u1t215=5.16
u1t216q=0.3 0.0 1.0 0 0
u1t216=5.184
u1t217q=0.3 0.0 1.0 0 0
u1t217=5.208
u1t218q=0.3 0.0 1.0 0 0
u1t218=5.232
u1t219q=0.3 0.0 1.0 0 0
u1t219=5.256
u1t220q=0.3 0.0 1.0 0 0
u1t220=5.28
u1t221q=0.3 0.0 1.0 0 0
u1t221=5.304
u1t222q=0.3 0.0 1.0 0 0
u1t222=5.328
u1t223q=0.3 0.0 1.0 0 0
u1t223=5.352
u1t224q=0.3 0.0 1.0 0 0
u1t224=5.376
u1t225q=0.3 0.0 1.0 0 0
u1t225=5.4
u1t226q=0.3 0.0 1.0 0 0
u1t226=5.424
u1t227q=0.3 0.0 1.0 0 0
u1t227=5.448
u1t228q=0.3 0.0 1.0 0 0
u1t228=5.472
u1t229q=0.3 0.0 1.0 0 0
u1t229=5.496
u1t230q=0.3 0.0 1.0 0 0
u1t230=5.52
u1t231q=0.3 0.0 1.0 0 0
u1t231=5.544
u1t232q=0.3 0.0 1.0 0 0
u1t232=5.568
u1t233q=0.3 0.0 1.0 0 0
u1t233=5.592
u1t234q=0.3 0.0 1.0 0 0
u1t234=5.616
u1t235q=0.3 0.0 1.0 0 0
u1t235=5.64
u1t236q=0.3 0.0 1.0 0 0
u1t236=5.664
u1t237q=0.3 0.0 1.0 0 0
u1t237=5.688
u1t238q=0.3 0.0 1.0 0 0
u1t238=5.712
u1t239q=0.3 0.0 1.0 0 0
u1t239=5.736
u1t240q=0.3 0.0 1.0 0 0
u1t240=5.76
u1t241q=0.3 0.0 1.0 0 0
u1t241=5.784
u1t242q=0.3 0.0 1.0 0 0
u1t242=5.808
u1t243q=0.3 0.0 1.0 0 0
u1t243=5.832
u1t244q=0.3 0.0 1.0 0 0
u1t244=5.856
u1t245q=0.3 0.0 1.0 0 0
u1t245=5.88
u1t246q=0.3 0.0 1.0 0 0
u1t246=5.904
u1t247q=0.3 0.0 1.0 0 0
u1t247=5.928
u1t248q=0.3 0.0 1.0 0 0
u1t248=5.952
u1t249q=0.3 0.0 1.0 0 0
u1t249=5.976
u1t250q=0.3 0.0 1.0 0 0
u1t250=6.0
u1t251q=0.3 0.0 1.0 0 0
u1t251=6.024
u1t252q=0.3 0.0 1.0 0 0
u1t252=6.048
u1t253q=0.3 0.0 1.0 0 0
u1t253=6.072
u1t254q=0.3 0.0 1.0 0 0
u1t254=6.096
u1t255q=0.3 0.0 1.0 0 0
u1t255=6.12
u1t256q=0.3 0.0 1.0 0 0
u1t256=6.144
u1t257q=0.3 0.0 1.0 0 0
u1t257=6.168
u1t258q=0.3 0.0 1.0 0 0
u1t258=6.192
u1t259q=0.3 0.0 1.0 0 0
u1t259=6.216
u1t260q=0.3 0.0 1.0 0 0
u1t260=6.24
u1t261q=0.3 0.0 1.0 0 0
u1t261=6.264
u1t262q=0.3 0.0 1.0 0 0
u1t262=6.288
u1t263q=0.3 0.0 1.0 0 0
u1t263=6.312
u1t264q=0.3 0.0 1.0 0 0
u1t264=6.336
u1t265q=0.3 0.0 1.0 0 0
u1t265=6.36
u1t266q=0.3 0.0 1.0 0 0
u1t266=6.384
u1t267q=0.3 0.0 1.0 0 0
u1t267=6.408
u1t268q=0.3 0.0 1.0 0 0
u1t268=6.432
u1t269q=0.3 0.0 1.0 0 0
u1t269=6.456
u1t270q=0.3 0.0 1.0 0 0
u1t270=6.48
u1t271q=0.3 0.0 1.0 0 0
u1t271=6.504
u1t272q=0.3 0.0 1.0 0 0
u1t272=6.528
u1t273q=0.3 0.0 1.0 0 0
u1t273=6.552
u1t274q=0.3 0.0 1.0 0 0
u1t274=6.576
u1t275q=0.3 0.0 1.0 0 0
u1t275=6.6
u1t276q=0.3 0.0 1.0 0 0
u1t276=6.624
u1t277q=0.3 0.0 1.0 0 0
u1t277=6.648
u1t278q=0.3 0.0 1.0 0 0
u1t278=6.672
u1t279q=0.3 0.0 1.0 0 0
u1t279=6.696
u1t280q=0.3 0.0 1.0 0 0
u1t280=6.72
u1t281q=0.3 0.0 1.0 0 0
u1t281=6.744
u1t282q=0.3 0.0 1.0 0 0
u1t282=6.768
u1t283q=0.3 0.0 1.0 0 0
u1t283=6.792
u1t284q=0.3 0.0 1.0 0 0
u1t284=6.816
u1t285q=0.3 0.0 1.0 0 0
u1t285=6.84
u1t286q=0.3 0.0 1.0 0 0
u1t286=6.864
u1t287q=0.3 0.0 1.0 0 0
u1t287=6.888
u1t288q=0.3 0.0 1.0 0 0
u1t288=6.912
u1t289q=0.3 0.0 1.0 0 0
u1t289=6.936
u1t290q=0.3 0.0 1.0 0 0
u1t290=6.96
u1t291q=0.3 0.0 1.0 0 0
u1t291=6.984
u1t292q=0.3 0.0 1.0 0 0
u1t292=7.008
u1t293q=0.3 0.0 1.0 0 0
u1t293=7.032
u1t294q=0.3 0.0 1.0 0 0
u1t294=7.056
u1t295q=0.3 0.0 1.0 0 0
u1t295=7.08
u1t296q=0.3 0.0 1.0 0 0
u1t296=7.104
u1t297q=0.3 0.0 1.0 0 0
u1t297=7.128
u1t298q=0.3 0.0 1.0 0 0
u1t298=7.152
u1t299q=0.3 0.0 1.0 0 0
u1t299=7.176
u1t300q=0.3 0.0 1.0 0 0
u1t300=7.2
u1t301q=0.3 0.0 1.0 0 0
u1t301=7.224
u1t302q=0.3 0.0 1.0 0 0
u1t302=7.248
u1t303q=0.3 0.0 1.0 0 0
u1t303=7.272
u1t304q=0.3 0.0 1.0 0 0
u1t304=7.296
u1t305q=0.3 0.0 1.0 0 0
u1t305=7.32
u1t306q=0.3 0.0 1.0 0 0
u1t306=7.344
u1t307q=0.3 0.0 1.0 0 0
u1t307=7.368
u1t308q=0.3 0.0 1.0 0 0
u1t308=7.392
u1t309q=0.3 0.0 1.0 0 0
u1t309=7.416
u1t310q=0.3 0.0 1.0 0 0
u1t310=7.44
u1t311q=0.3 0.0 1.0 0 0
u1t311=7.464
u1t312q=0.3 0.0 1.0 0 0
u1t312=7.488
u1t313q=0.3 0.0 1.0 0 0
u1t313=7.512
u1t314q=0.3 0.0 1.0 0 0
u1t314=7.536
u1t315q=0.3 0.0 1.0 0 0
u1t315=7.56
u1t316q=0.3 0.0 1.0 0 0
u1t316=7.584
u1t317q=0.3 0.0 1.0 0 0
u1t317=7.608
u1t318q=0.3 0.0 1.0 0 0
u1t318=7.632
u1t319q=0.3 0.0 1.0 0 0
u1t319=7.656
u1t320q=0.3 0.0 1.0 0 0
u1t320=7.68
u1t321q=0.3 0.0 1.0 0 0
u1t321=7.704
u1t322q=0.3 0.0 1.0 0 0
u1t322=7.728
u1t323q=0.3 0.0 1.0 0 0
u1t323=7.752
u1t324q=0.3 0.0 1.0 0 0
u1t324=7.776
u1t325q=0.3 0.0 1.0 0 0
u1t325=7.8
u1t326q=0.3 0.0 1.0 0 0
u1t326=7.824
u1t327q=0.3 0.0 1.0 0 0
u1t327=7.848
u1t328q=0.3 0.0 1.0 0 0
u1t328=7.872
u1t329q=0.3 0.0 1.0 0 0
u1t329=7.896
u1t330q=0.3 0.0 1.0 0 0
u1t330=7.92
u1t331q=0.3 0.0 1.0 0 0
u1t331=7.944
u1t332q=0.3 0.0 1.0 0 0
u1t332=7.968
u1t333q=0.3 0.0 1.0 0 0
u1t333=7.992
u1t334q=0.3 0.0 1.0 0 0
u1t334=8.016
u1t335q=0.3 0.0 1.0 0 0
u1t335=8.04
u1t336q=0.3 0.0 1.0 0 0
u1t336=8.064
u1t337q=0.3 0.0 1.0 0 0
u1t337=8.088
u1t338q=0.3 0.0 1.0 0 0
u1t338=8.112
u1t339q=0.3 0.0 1.0 0 0
u1t339=8.136
u1t340q=0.3 0.0 1.0 0 0
u1t340=8.16
u1t341q=0.3 0.0 1.0 0 0
u1t341=8.184
u1t342q=0.3 0.0 1.0 0 0
u1t342=8.208
u1t343q=0.3 0.0 1.0 0 0
u1t343=8.232
u1t344q=0.3 0.0 1.0 0 0
u1t344=8.256
u1t345q=0.3 0.0 1.0 0 0
u1t345=8.28
u1t346q=0.3 0.0 1.0 0 0
u1t346=8.304
u1t347q=0.3 0.0 1.0 0 0
u1t347=8.328
u1t348q=0.3 0.0 1.0 0 0
u1t348=8.352
u1t349q=0.3 0.0 1.0 0 0
u1t349=8.376
u1t350q=0.3 0.0 1.0 0 0
u1t350=8.4
u1t351q=0.3 0.0 1.0 0 0
u1t351=8.424
u1t352q=0.3 0.0 1.0 0 0
u1t352=8.448
u1t353q=0.3 0.0 1.0 0 0
u1t353=8.472
u1t354q=0.3 0.0 1.0 0 0
u1t354=8.496
u1t355q=0.3 0.0 1.0 0 0
u1t355=8.52
u1t356q=0.3 0.0 1.0 0 0
u1t356=8.544
u1t357q=0.3 0.0 1.0 0 0
u1t357=8.568
u1t358q=0.3 0.0 1.0 0 0
u1t358=8.592
u1t359q=0.3 0.0 1.0 0 0
u1t359=8.616
u1t360q=0.3 0.0 1.0 0 0
u1t360=8.64
u1t361q=0.3 0.0 1.0 0 0
u1t361=8.664
u1t362q=0.3 0.0 1.0 0 0
u1t362=8.688
u1t363q=0.3 0.0 1.0 0 0
u1t363=8.712
u1t364q=0.3 0.0 1.0 0 0
u1t364=8.736
u1t365q=0.3 0.0 1.0 0 0
u1t365=8.76
u1t366q=0.3 0.0 1.0 0 0
u1t366=8.784
u1t367q=0.3 0.0 1.0 0 0
u1t367=8.808
u1t368q=0.3 0.0 1.0 0 0
u1t368=8.832
u1t369q=0.3 0.0 1.0 0 0
u1t369=8.856
u1t370q=0.3 0.0 1.0 0 0
u1t370=8.88
u1t371q=0.3 0.0 1.0 0 0
u1t371=8.904
u1t372q=0.3 0.0 1.0 0 0
u1t372=8.928
u1t373q=0.3 0.0 1.0 0 0
u1t373=8.952
u1t374q=0.3 0.0 1.0 0 0
u1t374=8.976
u1t375q=0.3 0.0 1.0 0 0
u1t375=9.0
u1t376q=0.3 0.0 1.0 0 0
u1t376=9.024
u1t377q=0.3 0.0 1.0 0 0
u1t377=9.048
u1t378q=0.3 0.0 1.0 0 0
u1t378=9.072
u1t379q=0.3 0.0 1.0 0 0
u1t379=9.096
u1t380q=0.3 0.0 1.0 0 0
u1t380=9.12
u1t381q=0.3 0.0 1.0 0 0
u1t381=9.144
u1t382q=0.3 0.0 1.0 0 0
u1t382=9.168
u1t383q=0.3 0.0 1.0 0 0
u1t383=9.192
u1t384q=0.3 0.0 1.0 0 0
u1t384=9.216
u1t385q=0.3 0.0 1.0 0 0
u1t385=9.24
u1t386q=0.3 0.0 1.0 0 0
u1t386=9.264
u1t387q=0.3 0.0 1.0 0 0
u1t387=9.288
u1t388q=0.3 0.0 1.0 0 0
u1t388=9.312
u1t389q=0.3 0.0 1.0 0 0
u1t389=9.336
u1t390q=0.3 0.0 1.0 0 0
u1t390=9.36
u1t391q=0.3 0.0 1.0 0 0
u1t391=9.384
u1t392q=0.3 0.0 1.0 0 0
u1t392=9.408
u1t393q=0.3 0.0 1.0 0 0
u1t393=9.432
u1t394q=0.3 0.0 1.0 0 0
u1t394=9.456
u1t395q=0.3 0.0 1.0 0 0
u1t395=9.48
u1t396q=0.3 0.0 1.0 0 0
u1t396=9.504
u1t397q=0.3 0.0 1.0 0 0
u1t397=9.528
u1t398q=0.3 0.0 1.0 0 0
u1t398=9.552
u1t399q=0.3 0.0 1.0 0 0
u1t399=9.576
u1t400q=0.3 0.0 1.0 0 0
u1t400=9.6
u1t401q=0.3 0.0 1.0 0 0
u1t401=9.624
u1t402q=0.3 0.0 1.0 0 0
u1t402=9.648
u1t403q=0.3 0.0 1.0 0 0
u1t403=9.672
u1t404q=0.3 0.0 1.0 0 0
u1t404=9.696
u1t405q=0.3 0.0 1.0 0 0
u1t405=9.72
u1t406q=0.3 0.0 1.0 0 0
u1t406=9.744
u1t407q=0.3 0.0 1.0 0 0
u1t407=9.768
u1t408q=0.3 0.0 1.0 0 0
u1t408=9.792
u1t409q=0.3 0.0 1.0 0 0
u1t409=9.816
u1t410q=0.3 0.0 1.0 0 0
u1t410=9.84
u1t411q=0.3 0.0 1.0 0 0
u1t411=9.864
u1t412q=0.3 0.0 1.0 0 0
u1t412=9.888
u1t413q=0.3 0.0 1.0 0 0
u1t413=9.912
u1t414q=0.3 0.0 1.0 0 0
u1t414=9.936
u1t415q=0.3 0.0 1.0 0 0
u1t415=9.96
u1t416q=0.3 0.0 1.0 0 0
u1t416=9.984
u1t417q=0.3 0.0 1.0 0 0
u1t417=10.008
u1t418q=0.3 0.0 1.0 0 0
u1t418=10.032
u1t419q=0.3 0.0 1.0 0 0
u1t419=10.056
u1t420q=0.3 0.0 1.0 0 0
u1t420=10.08
u1t421q=0.3 0.0 1.0 0 0
u1t421=10.104
u1t422q=0.3 0.0 1.0 0 0
u1t422=10.128
u1t423q=0.3 0.0 1.0 0 0
u1t423=10.152
u1t424q=0.3 0.0 1.0 0 0
u1t424=10.176
u1t425q=0.3 0.0 1.0 0 0
u1t425=10.2
u1t426q=0.3 0.0 1.0 0 0
u1t426=10.224
u1t427q=0.3 0.0 1.0 0 0
u1t427=10.248
u1t428q=0.3 0.0 1.0 0 0
u1t428=10.272
u1t429q=0.3 0.0 1.0 0 0
u1t429=10.296
u1t430q=0.3 0.0 1.0 0 0
u1t430=10.32
u1t431q=0.3 0.0 1.0 0 0
u1t431=10.344
u1t432q=0.3 0.0 1.0 0 0
u1t432=10.368
u1t433q=0.3 0.0 1.0 0 0
u1t433=10.392
u1t434q=0.3 0.0 1.0 0 0
u1t434=10.416
u1t435q=0.3 0.0 1.0 0 0
u1t435=10.44
u1t436q=0.3 0.0 1.0 0 0
u1t436=10.464
u1t437q=0.3 0.0 1.0 0 0
u1t437=10.488
u1t438q=0.3 0.0 1.0 0 0
u1t438=10.512
u1t439q=0.3 0.0 1.0 0 0
u1t439=10.536
u1t440q=0.3 0.0 1.0 0 0
u1t440=10.56
u1t441q=0.3 0.0 1.0 0 0
u1t441=10.584
u1t442q=0.3 0.0 1.0 0 0
u1t442=10.608
u1t443q=0.3 0.0 1.0 0 0
u1t443=10.632
u1t444q=0.3 0.0 1.0 0 0
u1t444=10.656
u1t445q=0.3 0.0 1.0 0 0
u1t445=10.68
u1t446q=0.3 0.0 1.0 0 0
u1t446=10.704
u1t447q=0.3 0.0 1.0 0 0
u1t447=10.728
u1t448q=0.3 0.0 1.0 0 0
u1t448=10.752
u1t449q=0.3 0.0 1.0 0 0
u1t449=10.776
u1t450q=0.3 0.0 1.0 0 0
u1t450=10.8
u1t451q=0.3 0.0 1.0 0 0
u1t451=10.824
u1t452q=0.3 0.0 1.0 0 0
u1t452=10.848
u1t453q=0.3 0.0 1.0 0 0
u1t453=10.872
u1t454q=0.3 0.0 1.0 0 0
u1t454=10.896
u1t455q=0.3 0.0 1.0 0 0
u1t455=10.92
u1t456q=0.3 0.0 1.0 0 0
u1t456=10.944
u1t457q=0.3 0.0 1.0 0 0
u1t457=10.968
u1t458q=0.3 0.0 1.0 0 0
u1t458=10.992
u1t459q=0.3 0.0 1.0 0 0
u1t459=11.016
u1t460q=0.3 0.0 1.0 0 0
u1t460=11.04
u1t461q=0.3 0.0 1.0 0 0
u1t461=11.064
u1t462q=0.3 0.0 1.0 0 0
u1t462=11.088
u1t463q=0.3 0.0 1.0 0 0
u1t463=11.112
u1t464q=0.3 0.0 1.0 0 0
u1t464=11.136
u1t465q=0.3 0.0 1.0 0 0
u1t465=11.16
u1t466q=0.3 0.0 1.0 0 0
u1t466=11.184
u1t467q=0.3 0.0 1.0 0 0
u1t467=11.208
u1t468q=0.3 0.0 1.0 0 0
u1t468=11.232
u1t469q=0.3 0.0 1.0 0 0
u1t469=11.256
u1t470q=0.3 0.0 1.0 0 0
u1t470=11.28
u1t471q=0.3 0.0 1.0 0 0
u1t471=11.304
u1t472q=0.3 0.0 1.0 0 0
u1t472=11.328
u1t473q=0.3 0.0 1.0 0 0
u1t473=11.352
u1t474q=0.3 0.0 1.0 0 0
u1t474=11.376
u1t475q=0.3 0.0 1.0 0 0
u1t475=11.4
u1t476q=0.3 0.0 1.0 0 0
u1t476=11.424
u1t477q=0.3 0.0 1.0 0 0
u1t477=11.448
u1t478q=0.3 0.0 1.0 0 0
u1t478=11.472
u1t479q=0.3 0.0 1.0 0 0
u1t479=11.496
u1t480q=0.3 0.0 1.0 0 0
u1t480=11.52
u1t481q=0.3 0.0 1.0 0 0
u1t481=11.544
u1t482q=0.3 0.0 1.0 0 0
u1t482=11.568
u1t483q=0.3 0.0 1.0 0 0
u1t483=11.592
u1t484q=0.3 0.0 1.0 0 0
u1t484=11.616
u1t485q=0.3 0.0 1.0 0 0
u1t485=11.64
u1t486q=0.3 0.0 1.0 0 0
u1t486=11.664
u1t487q=0.3 0.0 1.0 0 0
u1t487=11.688
u1t488q=0.3 0.0 1.0 0 0
u1t488=11.712
u1t489q=0.3 0.0 1.0 0 0
u1t489=11.736
u1t490q=0.3 0.0 1.0 0 0
u1t490=11.76
u1t491q=0.3 0.0 1.0 0 0
u1t491=11.784
u1t492q=0.3 0.0 1.0 0 0
u1t492=11.808
u1t493q=0.3 0.0 1.0 0 0
u1t493=11.832
u1t494q=0.3 0.0 1.0 0 0
u1t494=11.856
u1t495q=0.3 0.0 1.0 0 0
u1t495=11.88
u1t496q=0.3 0.0 1.0 0 0
u1t496=11.904
u1t497q=0.3 0.0 1.0 0 0
u1t497=11.928
u1t498q=0.3 0.0 1.0 0 0
u1t498=11.952
u1t499q=0.3 0.0 1.0 0 0
u1t499=11.976
u1t500q=0.3 0 0 0 0
u1t500=tend

[Messungen]
tAnzahl=12

t1=1
t1Anzahl=2
t1m1=mfcn1 1.0 1e-06 1
t1m2=mfcn2 1.0 1e-06 1
t1minmax=0 1e+10

t2=2.000
t2Anzahl=2
t2m1=mfcn1 1.0 1e-06 1
t2m2=mfcn2 1.0 1e-06 1
t2minmax=0 1e+10

t3=3
t3Anzahl=2
t3m1=mfcn1 1.0 1e-06 1
t3m2=mfcn2 1.0 1e-06 1
t3minmax=0 1e+10

t4=4
t4Anzahl=2
t4m1=mfcn1 1.0 1e-06 1
t4m2=mfcn2 1.0 1e-06 1
t4minmax=0 1e+10

t5=5
t5Anzahl=2
t5m1=mfcn1 1.0 1e-06 1
t5m2=mfcn2 1.0 1e-06 1
t5minmax=0 1e+10

t6=6
t6Anzahl=2
t6m1=mfcn1 1.0 1e-06 1
t6m2=mfcn2 1.0 1e-06 1
t6minmax=0 1e+10

t7=7
t7Anzahl=2
t7m1=mfcn1 1.0 1e-06 1
t7m2=mfcn2 1.0 1e-06 1
t7minmax=0 1e+10

t8=8
t8Anzahl=2
t8m1=mfcn1 1.0 1e-06 1
t8m2=mfcn2 1.0 1e-06 1
t8minmax=0 1e+10

t9=9
t9Anzahl=2
t9m1=mfcn1 1.0 1e-06 1
t9m2=mfcn2 1.0 1e-06 1
t9minmax=0 1e+10

t10=10.000
t10Anzahl=2
t10m1=mfcn1 1.0 1e-06 1
t10m2=mfcn2 1.0 1e-06 1
t10minmax=0 1e+10

t11=11.000
t11Anzahl=2
t11m1=mfcn1 1.0 1e-06 1
t11m2=mfcn2 1.0 1e-06 1
t11minmax=0 1e+10

t12=12.000
t12Anzahl=2
t12m1=mfcn1 1.0 1e-06 1
t12m2=mfcn2 1.0 1e-06 1
t12minmax=0 1e+10
[NebenbedingungenSteuergroessen]
cAnzahl=0
[Messverfahren]
mAnzahl=2
m1=mfcn1 1 0 1e+10 0
m1f1=mess3 sigma3 1
m2=mfcn2 1 0 1e+10 0
m2f1=mess4 sigma4 1
mminmaxges=0 8

[OptionenIntegration]
teps=1e-08
rtol=1e-08
atol=1e-07
stepsize=0.0001
maxorder=6
maxstepnumber=4000
minstepsize=-1
maxstepsize=-1
maxitNewton=3
realworkspace=1700000
integerworkspace=5000
printlevel=0
mcnonlinearflag=0
mcDAEflag=0
mctol=1e-07
mcmaxit=50
mclinesearch=1
mcalpha0=1
rndmethod=-1
rndeps=1e-05
rndverbose=0

</source>

Lotka Experimental Design (VPLAN)

2016-01-19T16:20:46Z

FelixJost: /* VPLAN */

== VPLAN ==

Differential equations:

<source lang="fortran">

c RHS of the differential equations

subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag )
implicit none

real*8 x(*), f(*), p(*), q(*), rwh(*), t
integer*4 iwh(*), iflag

real*8 p1,p3,p5,p6, myu

c fixed parameters
p1 = 1.0
p3 = 1.0
p5 = 0.4
p6 = 0.2

c DISCRETIZE1( myu, rwh, iwh )

f(1) = p1*x(1) - p(1)*x(1)*x(2) - p5*myu*x(1)
f(2) = (-1.0)*p3*x(2) + p(2)*x(1)*x(2) - p6*myu*x(2)

end

</source>

First Measurement function:

<source lang="fortran">

c Messfunktion

subroutine mess3( t, x, h, p, q, rwh, iwh, iflag )
implicit none

real*8 t, x(*), h, p(*), q(*), rwh(*)
integer*4 iwh(*), iflag

h = x(1)

iflag = 0

end

</source>

Second Measurement function:

<source lang="fortran">

c Messfunktion

subroutine mess4( t, x, h, p, q, rwh, iwh, iflag )
implicit none

real*8 t, x(*), h, p(*), q(*), rwh(*)
integer*4 iwh(*), iflag

h = x(2)

iflag = 0

end

</source>

Standard deviation of first measurement function

<source lang="fortran">

c Standardabweichung der Messfunktion

subroutine sigma3( t, x, s, p, q, rwh, iwh, iflag )
implicit none

real*8 rwh(*)
integer*4 iwh(*), iflag

real*8 t, x(*), p(*), q(*)
real*8 s
real*8 h

s = 1.0d+0

iflag = 0

end

</source>

Standard deviation of second measurement function:

<source lang="fortran">
c Standardabweichung der Messfunktion

subroutine sigma4( t, x, s, p, q, rwh, iwh, iflag )
implicit none

real*8 rwh(*)
integer*4 iwh(*), iflag

real*8 t, x(*), p(*), q(*)
real*8 s(*)

s(1) = 1.0

iflag = 0

end

</source>

VPLAN specific experimental setup:

<source lang="vplan">

; ini-File fuer Experiment

[Flags]
switch=1

[Kosten]
costs=0 -1e+10 1e+10

[Integrationsintervall]
t0=0
tend=12

[Modellfunktionen]
ffcn=ffcn
gfcn=gfcn

[Zustandsvariablen]
yAnzahl=2
y1=x1 0.5 -1e+10 1e+10
y2=x2 0.7 -1e+10 1e+10

zAnzahl=0

[Mehrzielknoten]
tAnzahl=12
t1=1
t2=2
t3=3
t4=4
t5=5
t6=6
t7=7
t8=8
t9=9
t10=10
t11=11
t12=12

[DynamischeNebenbedingungen]
bAnzahl=0

[GitterUeberpruefungNebenbedingungen]
tAnzahl=0

[Steuergroessen]
qAnzahl=0

[Steuerfunktionen]
uAnzahl=1
u1=myu 0 0.0 1.0
u1tAnzahl=500
u1t0=t0
u1t1q=0.3 0.0 1.0 0 0
u1t1=0.024
u1t2q=0.3 0.0 1.0 0 0
u1t2=0.048
u1t3q=0.3 0.0 1.0 0 0
u1t3=0.072
u1t4q=0.3 0.0 1.0 0 0
u1t4=0.096
u1t5q=0.3 0.0 1.0 0 0
u1t5=0.12
u1t6q=0.3 0.0 1.0 0 0
u1t6=0.144
u1t7q=0.3 0.0 1.0 0 0
u1t7=0.168
u1t8q=0.3 0.0 1.0 0 0
u1t8=0.192
u1t9q=0.3 0.0 1.0 0 0
u1t9=0.216
u1t10q=0.3 0.0 1.0 0 0
u1t10=0.24
u1t11q=0.3 0.0 1.0 0 0
u1t11=0.264
u1t12q=0.3 0.0 1.0 0 0
u1t12=0.288
u1t13q=0.3 0.0 1.0 0 0
u1t13=0.312
u1t14q=0.3 0.0 1.0 0 0
u1t14=0.336
u1t15q=0.3 0.0 1.0 0 0
u1t15=0.36
u1t16q=0.3 0.0 1.0 0 0
u1t16=0.384
u1t17q=0.3 0.0 1.0 0 0
u1t17=0.408
u1t18q=0.3 0.0 1.0 0 0
u1t18=0.432
u1t19q=0.3 0.0 1.0 0 0
u1t19=0.456
u1t20q=0.3 0.0 1.0 0 0
u1t20=0.48
u1t21q=0.3 0.0 1.0 0 0
u1t21=0.504
u1t22q=0.3 0.0 1.0 0 0
u1t22=0.528
u1t23q=0.3 0.0 1.0 0 0
u1t23=0.552
u1t24q=0.3 0.0 1.0 0 0
u1t24=0.576
u1t25q=0.3 0.0 1.0 0 0
u1t25=0.6
u1t26q=0.3 0.0 1.0 0 0
u1t26=0.624
u1t27q=0.3 0.0 1.0 0 0
u1t27=0.648
u1t28q=0.3 0.0 1.0 0 0
u1t28=0.672
u1t29q=0.3 0.0 1.0 0 0
u1t29=0.696
u1t30q=0.3 0.0 1.0 0 0
u1t30=0.72
u1t31q=0.3 0.0 1.0 0 0
u1t31=0.744
u1t32q=0.3 0.0 1.0 0 0
u1t32=0.768
u1t33q=0.3 0.0 1.0 0 0
u1t33=0.792
u1t34q=0.3 0.0 1.0 0 0
u1t34=0.816
u1t35q=0.3 0.0 1.0 0 0
u1t35=0.84
u1t36q=0.3 0.0 1.0 0 0
u1t36=0.864
u1t37q=0.3 0.0 1.0 0 0
u1t37=0.888
u1t38q=0.3 0.0 1.0 0 0
u1t38=0.912
u1t39q=0.3 0.0 1.0 0 0
u1t39=0.936
u1t40q=0.3 0.0 1.0 0 0
u1t40=0.96
u1t41q=0.3 0.0 1.0 0 0
u1t41=0.984
u1t42q=0.3 0.0 1.0 0 0
u1t42=1.008
u1t43q=0.3 0.0 1.0 0 0
u1t43=1.032
u1t44q=0.3 0.0 1.0 0 0
u1t44=1.056
u1t45q=0.3 0.0 1.0 0 0
u1t45=1.08
u1t46q=0.3 0.0 1.0 0 0
u1t46=1.104
u1t47q=0.3 0.0 1.0 0 0
u1t47=1.128
u1t48q=0.3 0.0 1.0 0 0
u1t48=1.152
u1t49q=0.3 0.0 1.0 0 0
u1t49=1.176
u1t50q=0.3 0.0 1.0 0 0
u1t50=1.2
u1t51q=0.3 0.0 1.0 0 0
u1t51=1.224
u1t52q=0.3 0.0 1.0 0 0
u1t52=1.248
u1t53q=0.3 0.0 1.0 0 0
u1t53=1.272
u1t54q=0.3 0.0 1.0 0 0
u1t54=1.296
u1t55q=0.3 0.0 1.0 0 0
u1t55=1.32
u1t56q=0.3 0.0 1.0 0 0
u1t56=1.344
u1t57q=0.3 0.0 1.0 0 0
u1t57=1.368
u1t58q=0.3 0.0 1.0 0 0
u1t58=1.392
u1t59q=0.3 0.0 1.0 0 0
u1t59=1.416
u1t60q=0.3 0.0 1.0 0 0
u1t60=1.44
u1t61q=0.3 0.0 1.0 0 0
u1t61=1.464
u1t62q=0.3 0.0 1.0 0 0
u1t62=1.488
u1t63q=0.3 0.0 1.0 0 0
u1t63=1.512
u1t64q=0.3 0.0 1.0 0 0
u1t64=1.536
u1t65q=0.3 0.0 1.0 0 0
u1t65=1.56
u1t66q=0.3 0.0 1.0 0 0
u1t66=1.584
u1t67q=0.3 0.0 1.0 0 0
u1t67=1.608
u1t68q=0.3 0.0 1.0 0 0
u1t68=1.632
u1t69q=0.3 0.0 1.0 0 0
u1t69=1.656
u1t70q=0.3 0.0 1.0 0 0
u1t70=1.68
u1t71q=0.3 0.0 1.0 0 0
u1t71=1.704
u1t72q=0.3 0.0 1.0 0 0
u1t72=1.728
u1t73q=0.3 0.0 1.0 0 0
u1t73=1.752
u1t74q=0.3 0.0 1.0 0 0
u1t74=1.776
u1t75q=0.3 0.0 1.0 0 0
u1t75=1.8
u1t76q=0.3 0.0 1.0 0 0
u1t76=1.824
u1t77q=0.3 0.0 1.0 0 0
u1t77=1.848
u1t78q=0.3 0.0 1.0 0 0
u1t78=1.872
u1t79q=0.3 0.0 1.0 0 0
u1t79=1.896
u1t80q=0.3 0.0 1.0 0 0
u1t80=1.92
u1t81q=0.3 0.0 1.0 0 0
u1t81=1.944
u1t82q=0.3 0.0 1.0 0 0
u1t82=1.968
u1t83q=0.3 0.0 1.0 0 0
u1t83=1.992
u1t84q=0.3 0.0 1.0 0 0
u1t84=2.016
u1t85q=0.3 0.0 1.0 0 0
u1t85=2.04
u1t86q=0.3 0.0 1.0 0 0
u1t86=2.064
u1t87q=0.3 0.0 1.0 0 0
u1t87=2.088
u1t88q=0.3 0.0 1.0 0 0
u1t88=2.112
u1t89q=0.3 0.0 1.0 0 0
u1t89=2.136
u1t90q=0.3 0.0 1.0 0 0
u1t90=2.16
u1t91q=0.3 0.0 1.0 0 0
u1t91=2.184
u1t92q=0.3 0.0 1.0 0 0
u1t92=2.208
u1t93q=0.3 0.0 1.0 0 0
u1t93=2.232
u1t94q=0.3 0.0 1.0 0 0
u1t94=2.256
u1t95q=0.3 0.0 1.0 0 0
u1t95=2.28
u1t96q=0.3 0.0 1.0 0 0
u1t96=2.304
u1t97q=0.3 0.0 1.0 0 0
u1t97=2.328
u1t98q=0.3 0.0 1.0 0 0
u1t98=2.352
u1t99q=0.3 0.0 1.0 0 0
u1t99=2.376
u1t100q=0.3 0.0 1.0 0 0
u1t100=2.4
u1t101q=0.3 0.0 1.0 0 0
u1t101=2.424
u1t102q=0.3 0.0 1.0 0 0
u1t102=2.448
u1t103q=0.3 0.0 1.0 0 0
u1t103=2.472
u1t104q=0.3 0.0 1.0 0 0
u1t104=2.496
u1t105q=0.3 0.0 1.0 0 0
u1t105=2.52
u1t106q=0.3 0.0 1.0 0 0
u1t106=2.544
u1t107q=0.3 0.0 1.0 0 0
u1t107=2.568
u1t108q=0.3 0.0 1.0 0 0
u1t108=2.592
u1t109q=0.3 0.0 1.0 0 0
u1t109=2.616
u1t110q=0.3 0.0 1.0 0 0
u1t110=2.64
u1t111q=0.3 0.0 1.0 0 0
u1t111=2.664
u1t112q=0.3 0.0 1.0 0 0
u1t112=2.688
u1t113q=0.3 0.0 1.0 0 0
u1t113=2.712
u1t114q=0.3 0.0 1.0 0 0
u1t114=2.736
u1t115q=0.3 0.0 1.0 0 0
u1t115=2.76
u1t116q=0.3 0.0 1.0 0 0
u1t116=2.784
u1t117q=0.3 0.0 1.0 0 0
u1t117=2.808
u1t118q=0.3 0.0 1.0 0 0
u1t118=2.832
u1t119q=0.3 0.0 1.0 0 0
u1t119=2.856
u1t120q=0.3 0.0 1.0 0 0
u1t120=2.88
u1t121q=0.3 0.0 1.0 0 0
u1t121=2.904
u1t122q=0.3 0.0 1.0 0 0
u1t122=2.928
u1t123q=0.3 0.0 1.0 0 0
u1t123=2.952
u1t124q=0.3 0.0 1.0 0 0
u1t124=2.976
u1t125q=0.3 0.0 1.0 0 0
u1t125=3.0
u1t126q=0.3 0.0 1.0 0 0
u1t126=3.024
u1t127q=0.3 0.0 1.0 0 0
u1t127=3.048
u1t128q=0.3 0.0 1.0 0 0
u1t128=3.072
u1t129q=0.3 0.0 1.0 0 0
u1t129=3.096
u1t130q=0.3 0.0 1.0 0 0
u1t130=3.12
u1t131q=0.3 0.0 1.0 0 0
u1t131=3.144
u1t132q=0.3 0.0 1.0 0 0
u1t132=3.168
u1t133q=0.3 0.0 1.0 0 0
u1t133=3.192
u1t134q=0.3 0.0 1.0 0 0
u1t134=3.216
u1t135q=0.3 0.0 1.0 0 0
u1t135=3.24
u1t136q=0.3 0.0 1.0 0 0
u1t136=3.264
u1t137q=0.3 0.0 1.0 0 0
u1t137=3.288
u1t138q=0.3 0.0 1.0 0 0
u1t138=3.312
u1t139q=0.3 0.0 1.0 0 0
u1t139=3.336
u1t140q=0.3 0.0 1.0 0 0
u1t140=3.36
u1t141q=0.3 0.0 1.0 0 0
u1t141=3.384
u1t142q=0.3 0.0 1.0 0 0
u1t142=3.408
u1t143q=0.3 0.0 1.0 0 0
u1t143=3.432
u1t144q=0.3 0.0 1.0 0 0
u1t144=3.456
u1t145q=0.3 0.0 1.0 0 0
u1t145=3.48
u1t146q=0.3 0.0 1.0 0 0
u1t146=3.504
u1t147q=0.3 0.0 1.0 0 0
u1t147=3.528
u1t148q=0.3 0.0 1.0 0 0
u1t148=3.552
u1t149q=0.3 0.0 1.0 0 0
u1t149=3.576
u1t150q=0.3 0.0 1.0 0 0
u1t150=3.6
u1t151q=0.3 0.0 1.0 0 0
u1t151=3.624
u1t152q=0.3 0.0 1.0 0 0
u1t152=3.648
u1t153q=0.3 0.0 1.0 0 0
u1t153=3.672
u1t154q=0.3 0.0 1.0 0 0
u1t154=3.696
u1t155q=0.3 0.0 1.0 0 0
u1t155=3.72
u1t156q=0.3 0.0 1.0 0 0
u1t156=3.744
u1t157q=0.3 0.0 1.0 0 0
u1t157=3.768
u1t158q=0.3 0.0 1.0 0 0
u1t158=3.792
u1t159q=0.3 0.0 1.0 0 0
u1t159=3.816
u1t160q=0.3 0.0 1.0 0 0
u1t160=3.84
u1t161q=0.3 0.0 1.0 0 0
u1t161=3.864
u1t162q=0.3 0.0 1.0 0 0
u1t162=3.888
u1t163q=0.3 0.0 1.0 0 0
u1t163=3.912
u1t164q=0.3 0.0 1.0 0 0
u1t164=3.936
u1t165q=0.3 0.0 1.0 0 0
u1t165=3.96
u1t166q=0.3 0.0 1.0 0 0
u1t166=3.984
u1t167q=0.3 0.0 1.0 0 0
u1t167=4.008
u1t168q=0.3 0.0 1.0 0 0
u1t168=4.032
u1t169q=0.3 0.0 1.0 0 0
u1t169=4.056
u1t170q=0.3 0.0 1.0 0 0
u1t170=4.08
u1t171q=0.3 0.0 1.0 0 0
u1t171=4.104
u1t172q=0.3 0.0 1.0 0 0
u1t172=4.128
u1t173q=0.3 0.0 1.0 0 0
u1t173=4.152
u1t174q=0.3 0.0 1.0 0 0
u1t174=4.176
u1t175q=0.3 0.0 1.0 0 0
u1t175=4.2
u1t176q=0.3 0.0 1.0 0 0
u1t176=4.224
u1t177q=0.3 0.0 1.0 0 0
u1t177=4.248
u1t178q=0.3 0.0 1.0 0 0
u1t178=4.272
u1t179q=0.3 0.0 1.0 0 0
u1t179=4.296
u1t180q=0.3 0.0 1.0 0 0
u1t180=4.32
u1t181q=0.3 0.0 1.0 0 0
u1t181=4.344
u1t182q=0.3 0.0 1.0 0 0
u1t182=4.368
u1t183q=0.3 0.0 1.0 0 0
u1t183=4.392
u1t184q=0.3 0.0 1.0 0 0
u1t184=4.416
u1t185q=0.3 0.0 1.0 0 0
u1t185=4.44
u1t186q=0.3 0.0 1.0 0 0
u1t186=4.464
u1t187q=0.3 0.0 1.0 0 0
u1t187=4.488
u1t188q=0.3 0.0 1.0 0 0
u1t188=4.512
u1t189q=0.3 0.0 1.0 0 0
u1t189=4.536
u1t190q=0.3 0.0 1.0 0 0
u1t190=4.56
u1t191q=0.3 0.0 1.0 0 0
u1t191=4.584
u1t192q=0.3 0.0 1.0 0 0
u1t192=4.608
u1t193q=0.3 0.0 1.0 0 0
u1t193=4.632
u1t194q=0.3 0.0 1.0 0 0
u1t194=4.656
u1t195q=0.3 0.0 1.0 0 0
u1t195=4.68
u1t196q=0.3 0.0 1.0 0 0
u1t196=4.704
u1t197q=0.3 0.0 1.0 0 0
u1t197=4.728
u1t198q=0.3 0.0 1.0 0 0
u1t198=4.752
u1t199q=0.3 0.0 1.0 0 0
u1t199=4.776
u1t200q=0.3 0.0 1.0 0 0
u1t200=4.8
u1t201q=0.3 0.0 1.0 0 0
u1t201=4.824
u1t202q=0.3 0.0 1.0 0 0
u1t202=4.848
u1t203q=0.3 0.0 1.0 0 0
u1t203=4.872
u1t204q=0.3 0.0 1.0 0 0
u1t204=4.896
u1t205q=0.3 0.0 1.0 0 0
u1t205=4.92
u1t206q=0.3 0.0 1.0 0 0
u1t206=4.944
u1t207q=0.3 0.0 1.0 0 0
u1t207=4.968
u1t208q=0.3 0.0 1.0 0 0
u1t208=4.992
u1t209q=0.3 0.0 1.0 0 0
u1t209=5.016
u1t210q=0.3 0.0 1.0 0 0
u1t210=5.04
u1t211q=0.3 0.0 1.0 0 0
u1t211=5.064
u1t212q=0.3 0.0 1.0 0 0
u1t212=5.088
u1t213q=0.3 0.0 1.0 0 0
u1t213=5.112
u1t214q=0.3 0.0 1.0 0 0
u1t214=5.136
u1t215q=0.3 0.0 1.0 0 0
u1t215=5.16
u1t216q=0.3 0.0 1.0 0 0
u1t216=5.184
u1t217q=0.3 0.0 1.0 0 0
u1t217=5.208
u1t218q=0.3 0.0 1.0 0 0
u1t218=5.232
u1t219q=0.3 0.0 1.0 0 0
u1t219=5.256
u1t220q=0.3 0.0 1.0 0 0
u1t220=5.28
u1t221q=0.3 0.0 1.0 0 0
u1t221=5.304
u1t222q=0.3 0.0 1.0 0 0
u1t222=5.328
u1t223q=0.3 0.0 1.0 0 0
u1t223=5.352
u1t224q=0.3 0.0 1.0 0 0
u1t224=5.376
u1t225q=0.3 0.0 1.0 0 0
u1t225=5.4
u1t226q=0.3 0.0 1.0 0 0
u1t226=5.424
u1t227q=0.3 0.0 1.0 0 0
u1t227=5.448
u1t228q=0.3 0.0 1.0 0 0
u1t228=5.472
u1t229q=0.3 0.0 1.0 0 0
u1t229=5.496
u1t230q=0.3 0.0 1.0 0 0
u1t230=5.52
u1t231q=0.3 0.0 1.0 0 0
u1t231=5.544
u1t232q=0.3 0.0 1.0 0 0
u1t232=5.568
u1t233q=0.3 0.0 1.0 0 0
u1t233=5.592
u1t234q=0.3 0.0 1.0 0 0
u1t234=5.616
u1t235q=0.3 0.0 1.0 0 0
u1t235=5.64
u1t236q=0.3 0.0 1.0 0 0
u1t236=5.664
u1t237q=0.3 0.0 1.0 0 0
u1t237=5.688
u1t238q=0.3 0.0 1.0 0 0
u1t238=5.712
u1t239q=0.3 0.0 1.0 0 0
u1t239=5.736
u1t240q=0.3 0.0 1.0 0 0
u1t240=5.76
u1t241q=0.3 0.0 1.0 0 0
u1t241=5.784
u1t242q=0.3 0.0 1.0 0 0
u1t242=5.808
u1t243q=0.3 0.0 1.0 0 0
u1t243=5.832
u1t244q=0.3 0.0 1.0 0 0
u1t244=5.856
u1t245q=0.3 0.0 1.0 0 0
u1t245=5.88
u1t246q=0.3 0.0 1.0 0 0
u1t246=5.904
u1t247q=0.3 0.0 1.0 0 0
u1t247=5.928
u1t248q=0.3 0.0 1.0 0 0
u1t248=5.952
u1t249q=0.3 0.0 1.0 0 0
u1t249=5.976
u1t250q=0.3 0.0 1.0 0 0
u1t250=6.0
u1t251q=0.3 0.0 1.0 0 0
u1t251=6.024
u1t252q=0.3 0.0 1.0 0 0
u1t252=6.048
u1t253q=0.3 0.0 1.0 0 0
u1t253=6.072
u1t254q=0.3 0.0 1.0 0 0
u1t254=6.096
u1t255q=0.3 0.0 1.0 0 0
u1t255=6.12
u1t256q=0.3 0.0 1.0 0 0
u1t256=6.144
u1t257q=0.3 0.0 1.0 0 0
u1t257=6.168
u1t258q=0.3 0.0 1.0 0 0
u1t258=6.192
u1t259q=0.3 0.0 1.0 0 0
u1t259=6.216
u1t260q=0.3 0.0 1.0 0 0
u1t260=6.24
u1t261q=0.3 0.0 1.0 0 0
u1t261=6.264
u1t262q=0.3 0.0 1.0 0 0
u1t262=6.288
u1t263q=0.3 0.0 1.0 0 0
u1t263=6.312
u1t264q=0.3 0.0 1.0 0 0
u1t264=6.336
u1t265q=0.3 0.0 1.0 0 0
u1t265=6.36
u1t266q=0.3 0.0 1.0 0 0
u1t266=6.384
u1t267q=0.3 0.0 1.0 0 0
u1t267=6.408
u1t268q=0.3 0.0 1.0 0 0
u1t268=6.432
u1t269q=0.3 0.0 1.0 0 0
u1t269=6.456
u1t270q=0.3 0.0 1.0 0 0
u1t270=6.48
u1t271q=0.3 0.0 1.0 0 0
u1t271=6.504
u1t272q=0.3 0.0 1.0 0 0
u1t272=6.528
u1t273q=0.3 0.0 1.0 0 0
u1t273=6.552
u1t274q=0.3 0.0 1.0 0 0
u1t274=6.576
u1t275q=0.3 0.0 1.0 0 0
u1t275=6.6
u1t276q=0.3 0.0 1.0 0 0
u1t276=6.624
u1t277q=0.3 0.0 1.0 0 0
u1t277=6.648
u1t278q=0.3 0.0 1.0 0 0
u1t278=6.672
u1t279q=0.3 0.0 1.0 0 0
u1t279=6.696
u1t280q=0.3 0.0 1.0 0 0
u1t280=6.72
u1t281q=0.3 0.0 1.0 0 0
u1t281=6.744
u1t282q=0.3 0.0 1.0 0 0
u1t282=6.768
u1t283q=0.3 0.0 1.0 0 0
u1t283=6.792
u1t284q=0.3 0.0 1.0 0 0
u1t284=6.816
u1t285q=0.3 0.0 1.0 0 0
u1t285=6.84
u1t286q=0.3 0.0 1.0 0 0
u1t286=6.864
u1t287q=0.3 0.0 1.0 0 0
u1t287=6.888
u1t288q=0.3 0.0 1.0 0 0
u1t288=6.912
u1t289q=0.3 0.0 1.0 0 0
u1t289=6.936
u1t290q=0.3 0.0 1.0 0 0
u1t290=6.96
u1t291q=0.3 0.0 1.0 0 0
u1t291=6.984
u1t292q=0.3 0.0 1.0 0 0
u1t292=7.008
u1t293q=0.3 0.0 1.0 0 0
u1t293=7.032
u1t294q=0.3 0.0 1.0 0 0
u1t294=7.056
u1t295q=0.3 0.0 1.0 0 0
u1t295=7.08
u1t296q=0.3 0.0 1.0 0 0
u1t296=7.104
u1t297q=0.3 0.0 1.0 0 0
u1t297=7.128
u1t298q=0.3 0.0 1.0 0 0
u1t298=7.152
u1t299q=0.3 0.0 1.0 0 0
u1t299=7.176
u1t300q=0.3 0.0 1.0 0 0
u1t300=7.2
u1t301q=0.3 0.0 1.0 0 0
u1t301=7.224
u1t302q=0.3 0.0 1.0 0 0
u1t302=7.248
u1t303q=0.3 0.0 1.0 0 0
u1t303=7.272
u1t304q=0.3 0.0 1.0 0 0
u1t304=7.296
u1t305q=0.3 0.0 1.0 0 0
u1t305=7.32
u1t306q=0.3 0.0 1.0 0 0
u1t306=7.344
u1t307q=0.3 0.0 1.0 0 0
u1t307=7.368
u1t308q=0.3 0.0 1.0 0 0
u1t308=7.392
u1t309q=0.3 0.0 1.0 0 0
u1t309=7.416
u1t310q=0.3 0.0 1.0 0 0
u1t310=7.44
u1t311q=0.3 0.0 1.0 0 0
u1t311=7.464
u1t312q=0.3 0.0 1.0 0 0
u1t312=7.488
u1t313q=0.3 0.0 1.0 0 0
u1t313=7.512
u1t314q=0.3 0.0 1.0 0 0
u1t314=7.536
u1t315q=0.3 0.0 1.0 0 0
u1t315=7.56
u1t316q=0.3 0.0 1.0 0 0
u1t316=7.584
u1t317q=0.3 0.0 1.0 0 0
u1t317=7.608
u1t318q=0.3 0.0 1.0 0 0
u1t318=7.632
u1t319q=0.3 0.0 1.0 0 0
u1t319=7.656
u1t320q=0.3 0.0 1.0 0 0
u1t320=7.68
u1t321q=0.3 0.0 1.0 0 0
u1t321=7.704
u1t322q=0.3 0.0 1.0 0 0
u1t322=7.728
u1t323q=0.3 0.0 1.0 0 0
u1t323=7.752
u1t324q=0.3 0.0 1.0 0 0
u1t324=7.776
u1t325q=0.3 0.0 1.0 0 0
u1t325=7.8
u1t326q=0.3 0.0 1.0 0 0
u1t326=7.824
u1t327q=0.3 0.0 1.0 0 0
u1t327=7.848
u1t328q=0.3 0.0 1.0 0 0
u1t328=7.872
u1t329q=0.3 0.0 1.0 0 0
u1t329=7.896
u1t330q=0.3 0.0 1.0 0 0
u1t330=7.92
u1t331q=0.3 0.0 1.0 0 0
u1t331=7.944
u1t332q=0.3 0.0 1.0 0 0
u1t332=7.968
u1t333q=0.3 0.0 1.0 0 0
u1t333=7.992
u1t334q=0.3 0.0 1.0 0 0
u1t334=8.016
u1t335q=0.3 0.0 1.0 0 0
u1t335=8.04
u1t336q=0.3 0.0 1.0 0 0
u1t336=8.064
u1t337q=0.3 0.0 1.0 0 0
u1t337=8.088
u1t338q=0.3 0.0 1.0 0 0
u1t338=8.112
u1t339q=0.3 0.0 1.0 0 0
u1t339=8.136
u1t340q=0.3 0.0 1.0 0 0
u1t340=8.16
u1t341q=0.3 0.0 1.0 0 0
u1t341=8.184
u1t342q=0.3 0.0 1.0 0 0
u1t342=8.208
u1t343q=0.3 0.0 1.0 0 0
u1t343=8.232
u1t344q=0.3 0.0 1.0 0 0
u1t344=8.256
u1t345q=0.3 0.0 1.0 0 0
u1t345=8.28
u1t346q=0.3 0.0 1.0 0 0
u1t346=8.304
u1t347q=0.3 0.0 1.0 0 0
u1t347=8.328
u1t348q=0.3 0.0 1.0 0 0
u1t348=8.352
u1t349q=0.3 0.0 1.0 0 0
u1t349=8.376
u1t350q=0.3 0.0 1.0 0 0
u1t350=8.4
u1t351q=0.3 0.0 1.0 0 0
u1t351=8.424
u1t352q=0.3 0.0 1.0 0 0
u1t352=8.448
u1t353q=0.3 0.0 1.0 0 0
u1t353=8.472
u1t354q=0.3 0.0 1.0 0 0
u1t354=8.496
u1t355q=0.3 0.0 1.0 0 0
u1t355=8.52
u1t356q=0.3 0.0 1.0 0 0
u1t356=8.544
u1t357q=0.3 0.0 1.0 0 0
u1t357=8.568
u1t358q=0.3 0.0 1.0 0 0
u1t358=8.592
u1t359q=0.3 0.0 1.0 0 0
u1t359=8.616
u1t360q=0.3 0.0 1.0 0 0
u1t360=8.64
u1t361q=0.3 0.0 1.0 0 0
u1t361=8.664
u1t362q=0.3 0.0 1.0 0 0
u1t362=8.688
u1t363q=0.3 0.0 1.0 0 0
u1t363=8.712
u1t364q=0.3 0.0 1.0 0 0
u1t364=8.736
u1t365q=0.3 0.0 1.0 0 0
u1t365=8.76
u1t366q=0.3 0.0 1.0 0 0
u1t366=8.784
u1t367q=0.3 0.0 1.0 0 0
u1t367=8.808
u1t368q=0.3 0.0 1.0 0 0
u1t368=8.832
u1t369q=0.3 0.0 1.0 0 0
u1t369=8.856
u1t370q=0.3 0.0 1.0 0 0
u1t370=8.88
u1t371q=0.3 0.0 1.0 0 0
u1t371=8.904
u1t372q=0.3 0.0 1.0 0 0
u1t372=8.928
u1t373q=0.3 0.0 1.0 0 0
u1t373=8.952
u1t374q=0.3 0.0 1.0 0 0
u1t374=8.976
u1t375q=0.3 0.0 1.0 0 0
u1t375=9.0
u1t376q=0.3 0.0 1.0 0 0
u1t376=9.024
u1t377q=0.3 0.0 1.0 0 0
u1t377=9.048
u1t378q=0.3 0.0 1.0 0 0
u1t378=9.072
u1t379q=0.3 0.0 1.0 0 0
u1t379=9.096
u1t380q=0.3 0.0 1.0 0 0
u1t380=9.12
u1t381q=0.3 0.0 1.0 0 0
u1t381=9.144
u1t382q=0.3 0.0 1.0 0 0
u1t382=9.168
u1t383q=0.3 0.0 1.0 0 0
u1t383=9.192
u1t384q=0.3 0.0 1.0 0 0
u1t384=9.216
u1t385q=0.3 0.0 1.0 0 0
u1t385=9.24
u1t386q=0.3 0.0 1.0 0 0
u1t386=9.264
u1t387q=0.3 0.0 1.0 0 0
u1t387=9.288
u1t388q=0.3 0.0 1.0 0 0
u1t388=9.312
u1t389q=0.3 0.0 1.0 0 0
u1t389=9.336
u1t390q=0.3 0.0 1.0 0 0
u1t390=9.36
u1t391q=0.3 0.0 1.0 0 0
u1t391=9.384
u1t392q=0.3 0.0 1.0 0 0
u1t392=9.408
u1t393q=0.3 0.0 1.0 0 0
u1t393=9.432
u1t394q=0.3 0.0 1.0 0 0
u1t394=9.456
u1t395q=0.3 0.0 1.0 0 0
u1t395=9.48
u1t396q=0.3 0.0 1.0 0 0
u1t396=9.504
u1t397q=0.3 0.0 1.0 0 0
u1t397=9.528
u1t398q=0.3 0.0 1.0 0 0
u1t398=9.552
u1t399q=0.3 0.0 1.0 0 0
u1t399=9.576
u1t400q=0.3 0.0 1.0 0 0
u1t400=9.6
u1t401q=0.3 0.0 1.0 0 0
u1t401=9.624
u1t402q=0.3 0.0 1.0 0 0
u1t402=9.648
u1t403q=0.3 0.0 1.0 0 0
u1t403=9.672
u1t404q=0.3 0.0 1.0 0 0
u1t404=9.696
u1t405q=0.3 0.0 1.0 0 0
u1t405=9.72
u1t406q=0.3 0.0 1.0 0 0
u1t406=9.744
u1t407q=0.3 0.0 1.0 0 0
u1t407=9.768
u1t408q=0.3 0.0 1.0 0 0
u1t408=9.792
u1t409q=0.3 0.0 1.0 0 0
u1t409=9.816
u1t410q=0.3 0.0 1.0 0 0
u1t410=9.84
u1t411q=0.3 0.0 1.0 0 0
u1t411=9.864
u1t412q=0.3 0.0 1.0 0 0
u1t412=9.888
u1t413q=0.3 0.0 1.0 0 0
u1t413=9.912
u1t414q=0.3 0.0 1.0 0 0
u1t414=9.936
u1t415q=0.3 0.0 1.0 0 0
u1t415=9.96
u1t416q=0.3 0.0 1.0 0 0
u1t416=9.984
u1t417q=0.3 0.0 1.0 0 0
u1t417=10.008
u1t418q=0.3 0.0 1.0 0 0
u1t418=10.032
u1t419q=0.3 0.0 1.0 0 0
u1t419=10.056
u1t420q=0.3 0.0 1.0 0 0
u1t420=10.08
u1t421q=0.3 0.0 1.0 0 0
u1t421=10.104
u1t422q=0.3 0.0 1.0 0 0
u1t422=10.128
u1t423q=0.3 0.0 1.0 0 0
u1t423=10.152
u1t424q=0.3 0.0 1.0 0 0
u1t424=10.176
u1t425q=0.3 0.0 1.0 0 0
u1t425=10.2
u1t426q=0.3 0.0 1.0 0 0
u1t426=10.224
u1t427q=0.3 0.0 1.0 0 0
u1t427=10.248
u1t428q=0.3 0.0 1.0 0 0
u1t428=10.272
u1t429q=0.3 0.0 1.0 0 0
u1t429=10.296
u1t430q=0.3 0.0 1.0 0 0
u1t430=10.32
u1t431q=0.3 0.0 1.0 0 0
u1t431=10.344
u1t432q=0.3 0.0 1.0 0 0
u1t432=10.368
u1t433q=0.3 0.0 1.0 0 0
u1t433=10.392
u1t434q=0.3 0.0 1.0 0 0
u1t434=10.416
u1t435q=0.3 0.0 1.0 0 0
u1t435=10.44
u1t436q=0.3 0.0 1.0 0 0
u1t436=10.464
u1t437q=0.3 0.0 1.0 0 0
u1t437=10.488
u1t438q=0.3 0.0 1.0 0 0
u1t438=10.512
u1t439q=0.3 0.0 1.0 0 0
u1t439=10.536
u1t440q=0.3 0.0 1.0 0 0
u1t440=10.56
u1t441q=0.3 0.0 1.0 0 0
u1t441=10.584
u1t442q=0.3 0.0 1.0 0 0
u1t442=10.608
u1t443q=0.3 0.0 1.0 0 0
u1t443=10.632
u1t444q=0.3 0.0 1.0 0 0
u1t444=10.656
u1t445q=0.3 0.0 1.0 0 0
u1t445=10.68
u1t446q=0.3 0.0 1.0 0 0
u1t446=10.704
u1t447q=0.3 0.0 1.0 0 0
u1t447=10.728
u1t448q=0.3 0.0 1.0 0 0
u1t448=10.752
u1t449q=0.3 0.0 1.0 0 0
u1t449=10.776
u1t450q=0.3 0.0 1.0 0 0
u1t450=10.8
u1t451q=0.3 0.0 1.0 0 0
u1t451=10.824
u1t452q=0.3 0.0 1.0 0 0
u1t452=10.848
u1t453q=0.3 0.0 1.0 0 0
u1t453=10.872
u1t454q=0.3 0.0 1.0 0 0
u1t454=10.896
u1t455q=0.3 0.0 1.0 0 0
u1t455=10.92
u1t456q=0.3 0.0 1.0 0 0
u1t456=10.944
u1t457q=0.3 0.0 1.0 0 0
u1t457=10.968
u1t458q=0.3 0.0 1.0 0 0
u1t458=10.992
u1t459q=0.3 0.0 1.0 0 0
u1t459=11.016
u1t460q=0.3 0.0 1.0 0 0
u1t460=11.04
u1t461q=0.3 0.0 1.0 0 0
u1t461=11.064
u1t462q=0.3 0.0 1.0 0 0
u1t462=11.088
u1t463q=0.3 0.0 1.0 0 0
u1t463=11.112
u1t464q=0.3 0.0 1.0 0 0
u1t464=11.136
u1t465q=0.3 0.0 1.0 0 0
u1t465=11.16
u1t466q=0.3 0.0 1.0 0 0
u1t466=11.184
u1t467q=0.3 0.0 1.0 0 0
u1t467=11.208
u1t468q=0.3 0.0 1.0 0 0
u1t468=11.232
u1t469q=0.3 0.0 1.0 0 0
u1t469=11.256
u1t470q=0.3 0.0 1.0 0 0
u1t470=11.28
u1t471q=0.3 0.0 1.0 0 0
u1t471=11.304
u1t472q=0.3 0.0 1.0 0 0
u1t472=11.328
u1t473q=0.3 0.0 1.0 0 0
u1t473=11.352
u1t474q=0.3 0.0 1.0 0 0
u1t474=11.376
u1t475q=0.3 0.0 1.0 0 0
u1t475=11.4
u1t476q=0.3 0.0 1.0 0 0
u1t476=11.424
u1t477q=0.3 0.0 1.0 0 0
u1t477=11.448
u1t478q=0.3 0.0 1.0 0 0
u1t478=11.472
u1t479q=0.3 0.0 1.0 0 0
u1t479=11.496
u1t480q=0.3 0.0 1.0 0 0
u1t480=11.52
u1t481q=0.3 0.0 1.0 0 0
u1t481=11.544
u1t482q=0.3 0.0 1.0 0 0
u1t482=11.568
u1t483q=0.3 0.0 1.0 0 0
u1t483=11.592
u1t484q=0.3 0.0 1.0 0 0
u1t484=11.616
u1t485q=0.3 0.0 1.0 0 0
u1t485=11.64
u1t486q=0.3 0.0 1.0 0 0
u1t486=11.664
u1t487q=0.3 0.0 1.0 0 0
u1t487=11.688
u1t488q=0.3 0.0 1.0 0 0
u1t488=11.712
u1t489q=0.3 0.0 1.0 0 0
u1t489=11.736
u1t490q=0.3 0.0 1.0 0 0
u1t490=11.76
u1t491q=0.3 0.0 1.0 0 0
u1t491=11.784
u1t492q=0.3 0.0 1.0 0 0
u1t492=11.808
u1t493q=0.3 0.0 1.0 0 0
u1t493=11.832
u1t494q=0.3 0.0 1.0 0 0
u1t494=11.856
u1t495q=0.3 0.0 1.0 0 0
u1t495=11.88
u1t496q=0.3 0.0 1.0 0 0
u1t496=11.904
u1t497q=0.3 0.0 1.0 0 0
u1t497=11.928
u1t498q=0.3 0.0 1.0 0 0
u1t498=11.952
u1t499q=0.3 0.0 1.0 0 0
u1t499=11.976
u1t500q=0.3 0 0 0 0
u1t500=tend

[Messungen]
tAnzahl=12

t1=1
t1Anzahl=2
t1m1=mfcn1 1.0 1e-06 1
t1m2=mfcn2 1.0 1e-06 1
t1minmax=0 1e+10

t2=2.000
t2Anzahl=2
t2m1=mfcn1 1.0 1e-06 1
t2m2=mfcn2 1.0 1e-06 1
t2minmax=0 1e+10

t3=3
t3Anzahl=2
t3m1=mfcn1 1.0 1e-06 1
t3m2=mfcn2 1.0 1e-06 1
t3minmax=0 1e+10

t4=4
t4Anzahl=2
t4m1=mfcn1 1.0 1e-06 1
t4m2=mfcn2 1.0 1e-06 1
t4minmax=0 1e+10

t5=5
t5Anzahl=2
t5m1=mfcn1 1.0 1e-06 1
t5m2=mfcn2 1.0 1e-06 1
t5minmax=0 1e+10

t6=6
t6Anzahl=2
t6m1=mfcn1 1.0 1e-06 1
t6m2=mfcn2 1.0 1e-06 1
t6minmax=0 1e+10

t7=7
t7Anzahl=2
t7m1=mfcn1 1.0 1e-06 1
t7m2=mfcn2 1.0 1e-06 1
t7minmax=0 1e+10

t8=8
t8Anzahl=2
t8m1=mfcn1 1.0 1e-06 1
t8m2=mfcn2 1.0 1e-06 1
t8minmax=0 1e+10

t9=9
t9Anzahl=2
t9m1=mfcn1 1.0 1e-06 1
t9m2=mfcn2 1.0 1e-06 1
t9minmax=0 1e+10

t10=10.000
t10Anzahl=2
t10m1=mfcn1 1.0 1e-06 1
t10m2=mfcn2 1.0 1e-06 1
t10minmax=0 1e+10

t11=11.000
t11Anzahl=2
t11m1=mfcn1 1.0 1e-06 1
t11m2=mfcn2 1.0 1e-06 1
t11minmax=0 1e+10

t12=12.000
t12Anzahl=2
t12m1=mfcn1 1.0 1e-06 1
t12m2=mfcn2 1.0 1e-06 1
t12minmax=0 1e+10
[NebenbedingungenSteuergroessen]
cAnzahl=0
[Messverfahren]
mAnzahl=2
m1=mfcn1 1 0 1e+10 0
m1f1=mess3 sigma3 1
m2=mfcn2 1 0 1e+10 0
m2f1=mess4 sigma4 1
mminmaxges=0 8

[OptionenIntegration]
teps=1e-08
rtol=1e-08
atol=1e-07
stepsize=0.0001
maxorder=6
maxstepnumber=4000
minstepsize=-1
maxstepsize=-1
maxitNewton=3
realworkspace=1700000
integerworkspace=5000
printlevel=0
mcnonlinearflag=0
mcDAEflag=0
mctol=1e-07
mcmaxit=50
mclinesearch=1
mcalpha0=1
rndmethod=-1
rndeps=1e-05
rndverbose=0

</source>

Lotka Experimental Design (VPLAN)

2016-01-19T16:18:02Z

FelixJost: Created page with " == VPLAN == Differential equations: <source lang="fortran"> c RHS of the differential equations subroutine ffcn( t, x, f, p, q, rwh, iwh, iflag ) impl..."

Diels-Alder Reaction Experimental Design

2015-12-09T08:14:49Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\
\vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\
\vartheta_{up} + 273 & t \in [8,t_{end}]
\end{array} \right. \\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T08:13:18Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
L(x, 1) &=& \left\{ \begin{array}{cl} b & b \\
b & b \\
&
\end{array} \right. \\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T08:13:03Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
\vartheta(t) &=& \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\ \vartheta_{up} + 273 & t \in [8,t_{end}] \end{array} \\
\\

L(x, 1) &=& \left\{ \begin{array}{cl} b & b \\
b & b \\
&
\end{array} \right. \\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T08:10:50Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
\vartheta(t) &=& \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\ \vartheta_{up} + 273 & t \in [8,t_{end}] \end{array} \\
\\

& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T08:10:39Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
\vartheta(t) &=& \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\ \vartheta_{up} + 273 & t \in [8,t_{end}] \end{array} \\
\\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T08:10:23Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
\vartheta(t) &=& \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\ \vartheta_{up} + 273 & t \in [8,t_{end}] \\
\\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T08:10:13Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
\vartheta(t) &=& \left\{ \begin{array}{cl} \vartheta_{lo} + 273 & t \in [t_0,2] \\ \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 & t \in [2,8] \\ \vartheta_{up} + 273 & t \in [8,t_{end}] \right. \\
\\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T08:08:42Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
L(x, 1) &=& \left\{ \begin{array}{cl} e \; p_1 & \mbox{if } \sigma_1 \ge 0 \\ e \; p_2 & \mbox{else if } \sigma_2 \ge 0 \\ e \; \sum_{i=0}^{5} c_i (1) \left( \frac{1}{10} \gamma\ x_1 \right)^{-i} \quad & \mbox{else} \end{array} \right. \\
\\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T08:07:25Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T08:07:09Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array}
</math>


\begin{equation}
\begin{cases}
2x^{2} & \text{f"ur } x \textless 4 \\
2x^{3} + 4^{2} & \text{f"ur } 4 \ge x \textless 27 \\
3x^{2} \cdot sin(x) & \text{f"ur } x \ge 27
\end{cases}
\end{equation}

{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T08:06:56Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array}
</math>


<math>
\begin{equation}
\begin{cases}
2x^{2} & \text{f"ur } x \textless 4 \\
2x^{3} + 4^{2} & \text{f"ur } 4 \ge x \textless 27 \\
3x^{2} \cdot sin(x) & \text{f"ur } x \ge 27
\end{cases}
\end{equation}
</math>

{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T08:06:21Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array}
</math>


<math>
\begin{cases}
2x^{2} & \text{f"ur } x \textless 4 \\
2x^{3} + 4^{2} & \text{f"ur } 4 \ge x \textless 27 \\
3x^{2} \cdot sin(x) & \text{f"ur } x \ge 27
\end{cases}
</math>

{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T08:06:03Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
\vartheta(t) & = & \begin{cases}
x(n),\\
x(n-1)\\
x(n-1)
\end{cases}
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array}
</math>


<math>
\begin{cases}
2x^{2} & \text{f"ur } x \textless 4 \\
2x^{3} + 4^{2} & \text{f"ur } 4 \ge x \textless 27 \\
3x^{2} \cdot sin(x) & \text{f"ur } x \ge 27
\end{cases}
</math>

{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T08:05:05Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
\vartheta(t) & = & \begin{cases}
x(n),\\
x(n-1)\\
x(n-1)
\end{cases}
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T08:04:32Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
\vartheta(t) & = & \begin{cases}
test \\
test
\end{cases}

& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T08:03:19Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
\vartheta(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
\vartheta(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
\vartheta(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T08:02:03Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
T(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Fixed parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T08:01:25Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
T(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
& & x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P .
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T08:01:10Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
T(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
x \in \mathcal{X},\,u \in \mathcal{U},\, p \in P & &.
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T08:00:35Z

FelixJost: /* Constraints */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights (initial mass):

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content (fraction of active substances):


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
T(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
x & \in & \mathcal{X},\,u \in \mathcal{U},\, p \in P.
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T07:56:56Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights:

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content:


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
T(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
x & \in & \mathcal{X},\,u \in \mathcal{U},\, p \in P.
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[2,8]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[8,t_{end}]</math>
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T07:55:59Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights:

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content:


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
T(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
x & \in & \mathcal{X},\,u \in \mathcal{U},\, p \in P.
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[t_0,2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|[2,8]
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|[8,math>\t_{end}</math>]
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T07:55:38Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights:

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content:


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
T(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
x & \in & \mathcal{X},\,u \in \mathcal{U},\, p \in P.
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>t_0,2</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|[2,8]
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|[8,math>\t_{end}</math>]
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T07:54:51Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights:

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content:


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
T(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
x & \in & \mathcal{X},\,u \in \mathcal{U},\, p \in P.
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|<math>[\t_{0},2]</math>
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|[2,8]
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|[8,math>\t_{end}</math>]
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T07:53:23Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights:

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content:


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x,\ G,\ F,\ Tc,\ n_{a1},\ n_{a2},\ n_{a4},\ c_{kat},\ \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
T(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
x & \in & \mathcal{X},\,u \in \mathcal{U},\, p \in P.
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|[math>\t_{0}</math>,2]
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|[2,8]
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|[8,math>\t_{end}</math>]
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]

Diels-Alder Reaction Experimental Design

2015-12-09T07:52:53Z

FelixJost: /* Optimum Experimental Design Problem */

The '''Diels-Alder Reaction''' is an organic chemical reaction.
A conjugated diene and a substituted alkene react and form a substituted cyclohexene system.
Stefan Körkel used this model in his PhD thesis to compute optimum experimental designs.

== Model Formulation ==

The reactionkinetics can be modelled by the following differential equation system:

<math>
\begin{array}{rcl}
\dot{n_1}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_2}(t) &=& -k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}}, \\
& & \\
\dot{n_3}(t) &=& \ \ k \cdot \frac{n_1(t) \ \cdot \ n_2(t)}{m_{tot}} \\
& & \\
\dot{n_4}(t) &=& 0
\end{array}
</math>

The reaction velocity constant <math>k</math> consists of two parts. One part reflects the non-catalytic and the other the catalytic reaction. The velocity law follows the Arrhenius relation

<math>
k = k_1 \ \cdot \ exp(- \frac{E_1}{R} \ \cdot \ (\frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) ) \ + \ k_{cat} \ \cdot \ c_{cat} \ \cdot \ exp(-\lambda \ \cdot \ t) \ \cdot \ exp( - \frac{E_{cat}}{R} \ \cdot \ (\ \frac{1}{T(t)} \ - \ \frac{1}{T_{ref}}) )
</math>

Total mass:

<math>
m_{tot} = n_1 \ \cdot \ M_1 \ + \ n_2 \ \cdot \ M_2 \ + \ n_3 \ \cdot \ M_3 \ + \ n_4 \ \cdot \ M_4
</math>

Temperature in Kelvin:

<math>
T(t) = \vartheta (t) + 273
</math>

The ODE system is summarized to:

<math>
\begin{array}{rcl}
\dot{x}(t) &=& f(x(t), u(t), p)
\end{array}
</math>

== Constraints ==

The control variables are constrained with respect to the mass of sample weights:

<math>
\begin{array}{cll}
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \le 10
\end{array}
</math>

and to the mass of active ingredient content:


<math>
\begin{array}{cll}
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \le 0.7
\end{array}
</math>

== Optimum Experimental Design Problem ==

The aim is to compute an optimal experimental design <math>\xi = (q,w)</math> which minimizes the uncertainties of the parameters <math>k_1, k_{cat}, E_1, E_{cat}, \lambda</math>. So, we have to solve the following optimum experimental design problem:

<math>
\begin{array}{cll}
\displaystyle \min_{x, G, F, Tc, n_{a1}, n_{a2}, n_{a4}, c_{kat}, \vartheta(t)} && trace(F^{-1} (t_{end})) \\[1.5ex]
\mbox{s.t.} \\
\dot{x}(t) & = & f(x(t), u(t),p), \\
\\
h(t) & = & \frac{n_3(t) \ \cdot \ M_3}{m_{tot}} \ \cdot \ 100 \\
\\
\dot{G}(t) & = & f_x(x(t),u(t),p)G(t) \ + \ f_p(x(t),u(t),p) \\
\\
\dot{F}(t) & = & w(t) (h_x(x(t),u(t),p)G(t))^T (h_x(x(t),u(t),p)G(t)) \\
\\
0.1 & \le & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
10 & \ge & n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 \\
\\
0.1 & \le & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
0.7 & \ge & \frac{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 }{ n_{a1} \ \cdot \ M_1 \ + \ n_{a2} \ \cdot \ M_2 \ + \ n_{a4} \ \cdot \ M_4 } \\
\\
T(t) & = & \vartheta_{lo} + 273, \quad \forall \, t \in [t_0,2] \\
\\
T(t) & = & \vartheta_{lo} + \frac{t-2}{6} ( \vartheta_{up} - \vartheta_{lo} ) + 273 , \quad \forall \, t \in [2,8] \\
\\
T(t) & = & \vartheta_{up} + 273, \quad \forall \, t \in [8,t_{end}] \\
\\
x & \in & \mathcal{X},\,u \in \mathcal{U},\, p \in P.
\end{array}
</math>


{| class="wikitable"
|+State variables
|-
|Name
|Symbol
|Initial value (<math>t_0</math>)
|-
|Molar number 1
|<math>n_1(t)</math>
|<math>n_1(t_0) = n_{a1} </math>
|-
|Molar number 2
|<math>n_2(t)</math>
|<math>n_2(t_0) = n_{a2} </math>
|-
|Molar number 3
|<math>n_3(t)</math>
|<math>n_3(t_0) = 0 </math>
|-
|Solvent
|<math>n_4(t)</math>
|<math>n_4(t_0) = n_{a4} </math>
|}

{| class="wikitable"
|+Constants
|-
|Name
|Symbol
|Value
|-
|Molar Mass
|<math>M_1</math>
|0.1362
|-
|Molar Mass
|<math>M_2</math>
|0.09806
|-
|Molar Mass
|<math>M_3</math>
|0.23426
|-
|Molar Mass
|<math>M_4</math>
|0.236
|-
|Universal gas constant
|<math>R</math>
|8.314
|-
|Reference temperature
|<math>T_{ref}</math>
|293
|-
|St.dev of measurement error
|<math>\sigma</math>
|1
|}

Remember, in an optimum experimental design problem the parameters of the model are fixed. But, we minimize the parameter's uncertainties by optimizing over the control variables and functions.

{| class="wikitable"
|+Parameters
|-
|Name
|Symbol
|Value
|-
|Steric factor
|<math>k_1</math>
|<math>p_1 \cdot 0.01</math>
|-
|Steric factor
|<math>k_{kat}</math>
|<math>p_2 \cdot 0.10</math>
|-
|Activation energie
|<math>E_1</math>
|<math>p_3 \cdot 60000</math>
|-
|Activation energie
|<math>E_{kat}</math>
|<math>p_4 \cdot 40000</math>
|-
|Catalyst deactivation coefficient
|<math>\lambda</math>
|<math>p_5 \cdot 0.25</math>
|}
with <math>p_j = 1, \ j =1, \dots, 5</math>

{| class="wikitable"
|+Optimization/control variables
|-
|Name
|Symbol
|Interval
|-
|Initial molar number 1
|<math>n_{a1}</math>
|[0.4,9.0]
|-
|Initial molar number 2
|<math>n_{a2}</math>
|[0.4,9.0]
|-
|Initial molar number 4
|<math>n_{a4}</math>
|[0.4,9.0]
|-
|Concentration of the catalyst
|<math>c_{kat}</math>
|[0.0,6.0]
|}

{| class="wikitable"
|+Control function
|-
|Name
|Symbol
|Time interval
|Value interval
|Initial value
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|[math>\t_{0}</math>,2]
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|[2,8]
|[20.0,100.0]
|20.0
|-
|Initial molar number 1
|<math>\vartheta(t)</math>
|[8,math>\t_{end}</math>]
|[20.0,100.0]
|20.0
|}

'''Measurement grid'''

<math>
\begin{array}{llll}
t_0 = 0 & & & \\
t_{end} = 20 & & & \\
t_j = j/3, & j = 1,\dots, 15, & t_j = j - 10, & j = 16, \dots, 20.
\end{array}
</math>

== References ==

R. T. Morrison and R.N. Boyd. Organic Chemistry. Allyn and Bacon, Inc., 4th edition, 1983
S. Körkel. Numerische Methoden für Optimale Versuchsplanungsprobleme bei nichtlinearen DAE-Modellen.PhD thesis, Universität Heidelberg, Heidelber,2002

[[Category:VPLAN]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]