mintOC - User contributions [en]

Compartmental OED

2026-03-26T10:18:55Z

RobertLampel: /* Mathematical formulation */

{{Dimensions
|nd = 1
|nx = 12
|nw = 1
}}

The '''Compartmental OED problem''' looks for an optimal measurement strategy to determine three parameters in a one-dimensional [[:Category:ODE model|ODE model]], where we can directly measure the single state. The following description is taken from [[#compartmental| [1]]].

Here we consider the open one-compartment model with first-order absorption input fitted by Button (unpublished Ph.D. thesis, Texas A&M University, 1979) to the results of an experiment in which six horses each received 15 mg/kg of theophylline as aminophylline by intragastric administration.

The optimal integer control functions shows [[:Category:Chattering|bang bang]] behavior.

== Mathematical formulation ==
For the three-dimensional parameter <math>p = (\theta_1, \theta_2, \theta_3)</math> the original initial value problem is given by
<math>
\dot{y}(t) = f(t, p) = \theta_3 \cdot (-\theta_1 \cdot \exp(-\theta_1 \cdot t) + \theta_2 \cdot \exp(-\theta_2 \cdot t)), \quad y(0) = 0.
</math>

We assume both <math>x_0</math> and <math>t_f</math> to be fixed and are only interested in when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the state directly, i.e. <math>h(x(t)) = x(t)</math>.

Now we formulate the OED problem:

<math>
\begin{array}{lll}
\displaystyle \min_{y,G,F,z,w} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{y}(t) & = & f(t, p) \\
\quad \dot{G}(t) & = & f_p(y(t),p) \\
\quad \dot{F}(t) & = & w(t)(h_y(y(t))G(t))^T(h_y(y(t))G(t)) \\
\quad \dot{z}(t) & = & w(t), \\
\quad y(0) & = & y_0 \\
\quad G(0) & = & 0 \\
\quad F(0) & = & 0, \\
\quad z(0) & = & 0 \\
\quad w(t) & \in & \mathcal{W} \\
\quad z(t_f) & \leq & M
\end{array}
</math>


== Parameters ==
These fixed values are used within the model:

<math>
y_0 = 0; \quad t_f = 40; \quad \mathcal{W} = [0,1]; \quad M = 5; \quad p = (0.05884,4.298,21.80)
</math>


== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Compartmental OED.png| Sensitivities and measurement control for <math>\theta=(0.05884, 4.298, 21.80)</math>.
</gallery>

== References ==
[1] Optimum Experimental Designs for Properties of a Compartmental Model, A. Atkinson, K. Chaloner, A. Herzberg, J. Juritz, https://www.jstor.org/stable/pdf/2532547.pdf 


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]

Jackson OED

2026-03-26T10:17:14Z

RobertLampel: /* Mathematical formulation */

{{Dimensions
|nd = 1
|nx = 13
|nw = 3
}}

The '''Jackson OED problem''' is a variation of the [[:Jackson]] problem. It looks for optimal time intervals to measure the three states in order to minimize the uncertainty of a follow-up parameter estimation problem for the two unknown parameters.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. It also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

== Mathematical formulation ==

We are interested in estimating the parameters <math>k_1</math> and <math>k_2</math> of the initial value problem

<math>
\begin{array}{rcl}
\dot{x_1}(t) &=& -u(t) (k_1 x_1(t) - k_2 x_2(t)), && t \in [0,t_f], \quad x_1(0) = 1, \\
\dot{x_2}(t) &=& u(t) (k_1 x_1(t) - k_2 x_2(t)) - (1-u(t)) k_3 x_2(t), && t \in [0,t_f], \quad x_2(0) = 0, \\
\dot{x_3}(t) &=& (1-u(t)) k_3 x_2(t), && t \in [0,t_f], \quad x_3(0) = 0.
\end{array}
</math>


The initial values and <math>t_f</math> are fixed. We are interested in how to choose the control <math>u</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states <math>x_1</math> and <math>x_2</math> directly, <math>h^1(x(t)) = x_1(t)</math> and <math>h^2(x(t)) = x_2(t)</math>. We use two different sampling functions, <math>w_1(\cdot)</math> and <math>w_2(\cdot)</math> in the same experimental setting. This can be seen either as a two-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (k_1, k_2)</math>:

<math>
\begin{array}{lll}
\displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=1}^{n_o} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t), \\
\quad x(0) & = & x_0 \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
\quad z(0) & = & 0 \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
\end{array}
</math>


The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{2 \times 2}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 1,2,</math> for each observed function of states.

== Parameters ==

These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>k_1</math> || align=right | 1 || Interaction between <math>x_1</math> and <math>x_2</math>
|-
| align=center | <math>k_2</math> || align=right | 10 || Interaction between <math>x_1</math> and <math>x_2</math>
|-
| align=center | <math>k_3</math> || align=right | 1 || Growth of <math>x_3</math> under complementary control
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 1 || Horizon of the control problem
|-
| align=center | <math>\varepsilon_\mathrm{reg}</math> || align=right | 0.01 || Regularization of Fisher matrix
|-
| align=center | <math>\mathcal{U}</math> || align=right | [0,1] || Bounds of control function
|-
| align=center | <math>\mathcal{W}</math> || align=right | [0,1] || Bounds of measurement function
|-
| align=center | <math>M_1, M_2</math> || align=right | 0.2 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Jackson_OED.png| States, control, and sampling functions for a local optimum. Both measurement functions overlap.
</gallery>

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]

Jackson OED

2026-03-26T10:16:59Z

RobertLampel: /* Mathematical formulation */

{{Dimensions
|nd = 1
|nx = 13
|nw = 3
}}

The '''Jackson OED problem''' is a variation of the [[:Jackson]] problem. It looks for optimal time intervals to measure the three states in order to minimize the uncertainty of a follow-up parameter estimation problem for the two unknown parameters.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. It also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

== Mathematical formulation ==

We are interested in estimating the parameters <math>k_1</math> and <math>k_2</math> of the initial value problem

<math>
\begin{array}{rcl}
\dot{x_1}(t) &=& -u(t) (k_1 x_1(t) - k_2 x_2(t)), && t \in [0,t_f], \quad x_1(0) = 1, \\
\dot{x_2}(t) &=& u(t) (k_1 x_1(t) - k_2 x_2(t)) - (1-u(t)) k_3 x_2(t), && t \in [0,t_f], \quad x_2(0) = 0, \\
\dot{x_3}(t) &=& (1-u(t)) k_3 x_2(t), && t \in [0,t_f], \quad x_3(0) = 0.
\end{array}
</math>


The initial values and <math>t_f</math> are fixed. We are interested in how to choose the control <math>u</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states <math>x_1</math> and <math>x_2</math> directly, <math>h^1(x(t)) = x_1(t)</math> and <math>h^2(x(t)) = x_2(t)</math>. We use two different sampling functions, <math>w_1(\cdot)</math> and <math>w_2(\cdot)</math> in the same experimental setting. This can be seen either as a two-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (k_1, k_2)</math>:

<math>
\begin{array}{lll}
\displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=1}^{n_o} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t), \\
\quad x(0) & = & x_0 \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
\quad z(0) & = & 0 \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
\end{array}
</math>


The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{2 \times 2}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 1,2</math> for each observed function of states.

== Parameters ==

These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>k_1</math> || align=right | 1 || Interaction between <math>x_1</math> and <math>x_2</math>
|-
| align=center | <math>k_2</math> || align=right | 10 || Interaction between <math>x_1</math> and <math>x_2</math>
|-
| align=center | <math>k_3</math> || align=right | 1 || Growth of <math>x_3</math> under complementary control
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 1 || Horizon of the control problem
|-
| align=center | <math>\varepsilon_\mathrm{reg}</math> || align=right | 0.01 || Regularization of Fisher matrix
|-
| align=center | <math>\mathcal{U}</math> || align=right | [0,1] || Bounds of control function
|-
| align=center | <math>\mathcal{W}</math> || align=right | [0,1] || Bounds of measurement function
|-
| align=center | <math>M_1, M_2</math> || align=right | 0.2 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Jackson_OED.png| States, control, and sampling functions for a local optimum. Both measurement functions overlap.
</gallery>

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]

Lotka Shared OED

2026-03-26T10:16:25Z

RobertLampel: /* Mathematical formulation */

{{Dimensions
|nd = 1
|nx = 21
|nw = 4}}

The '''Lotka Shared Experimental Design problem''' looks for an optimal strategy to be performed on a fixed time horizon to control the biomasses multiple species as described in [[:LV Shared Resource]]. The goal here, however, is to minimize the uncertainty of a follow-up parameter estimation problem. When measurements of the three state variables are performed becomes a degree of freedom.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. The ODE from [[LV Shared Resource]] is extended such that it also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

The optimal integer control functions shows [[:Category:Chattering|bang bang]] behavior.

== Mathematical formulation ==

We are interested in estimating the parameters <math>\alpha_0, \alpha_1,</math> and <math>\alpha_2</math> of the Lotka-Volterra type predator-prey initial value problem

<math>
\begin{array}{rcl}
\dot{x}_0(t) & = & x_0(t) - \alpha_0 x_0(t) x_1(t) - x_0(t) x_2(t), && t \in [0,t_f], \quad x_0(0) = 1.5, \\
\dot{x}_1(t) & = & - x_1(t) + \alpha_1 x_0(t) x_1(t) - c_1 x_1(t) u(t), && t \in [0,t_f], \quad x_1(0) = 0.5, \\
\dot{x}_2(t) & = & -x_2(t) + \alpha_2 x_0(t) x_2(t) - c_2 x_2(t) u(t), && t \in [0,t_f], \quad x_2(0) = 1,
\end{array}
</math>

where <math>u(\cdot)</math> is a control that may or may not be fixed. The other parameters, the initial values and <math>t_f = 20</math> are fixed. We are interested in how to choose <math>u</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=0,1,2</math>. We use three different sampling functions, <math>w_i(\cdot), i=0,1,2,</math> in the same experimental setting. This can be seen either as a three-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (\alpha_0, \alpha_1, \alpha_2)</math>:

<math>
\begin{array}{lll}
\displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=0}^{2} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t) \\
\quad x(0) & = & (1.5, 0.5, 1)^T \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
\quad z(0) & = & 0 \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
\end{array}
</math>


The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{3 \times 3}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 0,1,2,</math> for each observed function of states.

== Parameters ==

These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>\alpha_0</math> || align=right | 1.0 ||
|-
| align=center | <math>\alpha_1</math> || align=right | 1.0 ||
|-
| align=center | <math>\alpha_2</math> || align=right | 1.2 ||
|-
| align=center | <math>c_1</math> || align=right | 0.1 ||
|-
| align=center | <math>c_2</math> || align=right | 0.4 ||
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 20 || Horizon of the control problem
|-
| align=center | <math>\varepsilon_\mathrm{reg}</math> || align=right | 0.1 || Regularization of Fisher matrix
|-
| align=center | <math>\mathcal{U}</math> || align=right | <math>[0,1]^3</math> || Bounds of control function
|-
| align=center | <math>\mathcal{W}</math> || align=right | <math>[0,1]^3</math> || Bounds of measurement function
|-
| align=center | <math>M_0, M_1, M_2</math> || align=right | 4 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Lotka_Shared_OED.png| States, control, and sampling functions for a local optimum.
</gallery>

== Source Code ==

Model descriptions are available in

* [[:Category:Casadi | CasADi]] at https://github.com/rlampel/PySHeRLOC

== Miscellaneous and Further Reading ==
The Lotka Volterra fishing problem was introduced by Sebastian Sager in a proceedings paper <bib id="Sager2006" /> and revisited in his PhD thesis <bib id="Sager2005" />. These are also the references to look for more details. The experimental design problem has been described in the habilitation thesis of Sager, <bib id="Sager2011d" />.

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]
[[Category:Population dynamics]]

Three Tank OED

2026-03-26T10:15:58Z

RobertLampel: /* Mathematical formulation */

{{Dimensions
|nd = 1
|nx = 21
|nw = 6
}}

The '''Three Tank OED problem''' is a variation of the [[:Three Tank multimode problem]]. It looks for optimal time intervals to measure the two states in order to minimize the uncertainty of a follow-up parameter estimation problem for the two unknown parameters.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. It also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

== Mathematical formulation ==

We are interested in estimating the parameters <math>c_1, c_2,</math> and <math>c_3</math> of the initial value problem

<math>
\begin{array}{rcl}
\dot{x}_1(t) & =& -\sqrt{x_1(t)}+c_1 u_1(t) + c_2 u_2(t) - u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_1(0) = 2, \\
\dot{x}_2(t) & =& \sqrt{x_1(t)}-\sqrt{x_2(t)}, && t \in [0,t_f], \quad x_2(0) = 2, \\
\dot{x}_3(t) & =& \sqrt{x_2(t)}-\sqrt{x_3(t)} + u_3(t) \sqrt{c_3 x_1(t)}, && t \in [0,t_f], \quad x_3(0) = 2.
\end{array}
</math>

Additionally, the controls <math>u_1,u_2,</math> and <math>u_3</math> are constrained by:

<math>
\sum_{i=1}^3 u_i(t) = 1 \quad \forall t\in [0,t_f]
</math>

The initial values and <math>t_f = 12</math> are fixed. We are interested in how to choose the controls <math>u_i, \ i=1,2,3,</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=1,2,3</math>. We use three different sampling functions, <math>w_i(\cdot), \ i=1,2,3,</math> in the same experimental setting. This can be seen either as a three-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (p_1, p_2)</math>:

<math>
\begin{array}{lll}
\displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=1}^{3} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t) \\
\quad x(0) & = & (2,2,2)^T \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
\quad z(0) & = & 0 \\
\quad \sum_{i=1}^3 u_i(t) & = & 1 \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
\end{array}
</math>


The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{3 \times 3}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 1,2,3,</math> for each observed function of states.

== Parameters ==
These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>c_1</math> || align=right | 1 || Unknown parameter
|-
| align=center | <math>c_2</math> || align=right | 2 || Unknown parameter
|-
| align=center | <math>c_3</math> || align=right | 0.8 || Unknown parameter
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 12 || Horizon of the control problem
|-
| align=center | <math>\varepsilon_{\mathrm{reg}}</math> || align=right | 0.1 || Regularization of Fisher matrix
|-
| align=center | <math>\mathcal{U}</math> || align=right | <math>[0,1]^3</math> || Bounds of control function
|-
| align=center | <math>\mathcal{W}</math> || align=right | <math>[0,1]^3</math> || Bounds of measurement function
|-
| align=center | <math>M_1, M_2, M_3</math> || align=right | 2 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Three_Tank_OED.png| States, control, and sampling functions for a local optimum.
</gallery>

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]

File:Three Tank OED.png

2026-03-26T10:14:48Z

RobertLampel: RobertLampel uploaded a new version of File:Three Tank OED.png

2026-03-26T09:13:13Z

RobertLampel: /* Source Code */

{{Dimensions
|nd = 1
|nx = 21
|nw = 4}}

The '''Lotka Shared Experimental Design problem''' looks for an optimal strategy to be performed on a fixed time horizon to control the biomasses multiple species as described in [[:LV Shared Resource]]. The goal here, however, is to minimize the uncertainty of a follow-up parameter estimation problem. When measurements of the three state variables are performed becomes a degree of freedom.

The mathematical equations form a small-scale [[:Category:ODE model|ODE model]]. The ODE from [[LV Shared Resource]] is extended such that it also includes state sensitivities, the Fisher information matrix entries and integrated sampling states.

The optimal integer control functions shows [[:Category:Chattering|bang bang]] behavior.

== Mathematical formulation ==

We are interested in estimating the parameters <math>p_2</math> and <math>p_4</math> of the Lotka-Volterra type predator-prey fish initial value problem

<math>
\begin{array}{rcl}
\dot{x}_0(t) & = & x_0(t) - \alpha_0 x_0(t) x_1(t) - x_0(t) x_2(t), \\
\dot{x}_1(t) & = & - x_1(t) + \alpha_1 x_0(t) x_1(t) - c_1 x_1(t) u(t), \\
\dot{x}_2(t) & = & -x_2(t) + \alpha_2 x_0(t) x_2(t) - c_2 x_2(t) u(t),
\end{array}
</math>

where <math>u(\cdot)</math> is a control that may or may not be fixed. The other parameters, the initial values and <math>t_f = 20</math> are fixed. We are interested in how to choose <math>u</math> and when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the states directly, i.e., <math>h^i(x(t)) = x_i(t), \ i=0,1,2</math>. We use three different sampling functions, <math>w^i(\cdot), i=0,1,2,</math> in the same experimental setting. This can be seen either as a three-dimensional measurement function <math>h(x(t))</math>, or as a special case of a multiple experiment, in which <math>u(\cdot), x(\cdot)</math>, and <math>G(\cdot)</math> are identical.

Now we formulate the OED problem with <math>\theta := (\alpha_0, \alpha_1, \alpha_2)</math>:

<math>
\begin{array}{lll}
\displaystyle \min_{x,G,F,z,w,u} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{x}(t) & = & f(x(t),u(t),\theta) \\
\quad \dot{G}(t) & = & f_x(x(t),u(t),\theta) G(t) + f_\theta(x(t),u(t),\theta) \\
\quad \dot{F}(t) & = & \sum_{i=0}^{2} w_i(t)(h^i_x(x(t))G(t))^T(h^i_x(x(t))G(t)) \\
\quad \dot{z}(t) & = & w(t), \\
\quad x(0) & = & x_0 \\
\quad G(0) & = & \frac{\partial x(0)}{\partial \theta} \\
\quad F(0) & = & I \cdot \varepsilon_{\mathrm{reg}}, \\
\quad z(0) & = & 0 \\
\quad u(t) & \in & \mathcal{U} \\
\quad w(t) & \in & \mathcal{W} \\
\quad z_i(t_f) & \leq & M_i
\end{array}
</math>


The evolution of the symmetric matrix <math>F: \left[0,t_f \right] \rightarrow \mathbb{R}^{3 \times 3}</math> is given by the weighted sum of observability Gramians
<math>h^i_x (x(t)) G(t), \ i = 0,1,2,</math> for each observed function of states.

== Parameters ==

These fixed values are used within the model:

{| border="1" align="center" cellpadding="5" cellspacing="0"
|- bgcolor=#c7c7c7
! Symbol !! Value !! Description
|-
| align=center | <math>\alpha_0</math> || align=right | 1.0 ||
|-
| align=center | <math>\alpha_1</math> || align=right | 1.0 ||
|-
| align=center | <math>\alpha_2</math> || align=right | 1.2 ||
|-
| align=center | <math>c_1</math> || align=right | 0.1 ||
|-
| align=center | <math>c_2</math> || align=right | 0.4 ||
|-
| align=center | <math>t_\mathrm{f}</math> || align=right | 20 || Horizon of the control problem
|-
| align=center | <math>\varepsilon_\mathrm{reg}</math> || align=right | 0.1 || Regularization of Fisher matrix
|-
| align=center | <math>\mathcal{U}</math> || align=right | [0,1] || Bounds of control function
|-
| align=center | <math>\mathcal{W}</math> || align=right | [0,1] || Bounds of measurement function
|-
| align=center | <math>M_1, M_2, M_3</math> || align=right | 4 || Maximum measurement time
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Lotka_Shared_OED.png| States, control, and sampling functions for a local optimum.
</gallery>

== Source Code ==

Model descriptions are available in

* [[:Category:Casadi | CasADi]] at https://github.com/rlampel/PySHeRLOC

== Variants ==

There are several alternative formulations and variants of the above problem, in particular

* a prescribed time grid for the control function <bib id="Sager2006" />, see also [[Lotka Experimental Design (AMPL)]],
* no fishing, i.e., <math>u \equiv 0</math>,
* different fishing control functions for the two species,
* different parameters and start values.

== Miscellaneous and Further Reading ==
The Lotka Volterra fishing problem was introduced by Sebastian Sager in a proceedings paper <bib id="Sager2006" /> and revisited in his PhD thesis <bib id="Sager2005" />. These are also the references to look for more details. The experimental design problem has been described in the habilitation thesis of Sager, <bib id="Sager2011d" />.

== References ==
<biblist />


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]
[[Category:Population dynamics]]

RobertLampel:

{{Dimensions
|nd = 1
|nx = 12
|nw = 1
}}

The '''Compartmental OED problem''' looks for an optimal measurement strategy to determine three parameters in a one-dimensional [[:Category:ODE model|ODE model]], where we can directly measure the single state. The following description is taken from [[#compartmental| [1]]].

Here we consider the open one-compartment model with first-order absorption input fitted by Button (unpublished Ph.D. thesis, Texas A&M University, 1979) to the results of an experiment in which six horses each received 15 mg/kg of theophylline as aminophylline by intragastric administration.

The optimal integer control functions shows [[:Category:Chattering|bang bang]] behavior.

== Mathematical formulation ==
For the three-dimensional parameter <math>p = (\theta_1, \theta_2, \theta_3)</math> the original initial value problem is given by
<math>
\dot{y}(t) =: f(t, p) = \theta_3 \cdot (-\theta_1 \cdot \exp(-\theta_1 \cdot t) + \theta_2 \cdot \exp(-\theta_2 \cdot t)), \quad y(0) = 0.
</math>

We assume both <math>x_0</math> and <math>t_f</math> to be fixed and are only interested in when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the state directly, i.e. <math>h(x(t)) = x(t)</math>.

Now we formulate the OED problem:

<math>
\begin{array}{lll}
\displaystyle \min_{y,G,F,z,w} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{y}(t) & = & f(t, p) \\
\quad \dot{G}(t) & = & f_p(y(t),p) \\
\quad \dot{F}(t) & = & w(t)(h_y(y(t))G(t))^T(h_y(y(t))G(t)) \\
\quad \dot{z}(t) & = & w(t), \\
\quad y(0) & = & y_0 \\
\quad G(0) & = & 0 \\
\quad F(0) & = & 0, \\
\quad z(0) & = & 0 \\
\quad w(t) & \in & \mathcal{W} \\
\quad z(t_f) & \leq & M
\end{array}
</math>


== Parameters ==
These fixed values are used within the model:

<math>
y_0 = 0; \quad t_f = 40; \quad \mathcal{W} = [0,1]; \quad M = 5; \quad p = (0.05884,4.298,21.80)
</math>


== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Compartmental OED.png| Sensitivities and measurement control for <math>\theta=(0.05884, 4.298, 21.80)</math>.
</gallery>

== References ==
[1] Optimum Experimental Designs for Properties of a Compartmental Model, A. Atkinson, K. Chaloner, A. Herzberg, J. Juritz, https://www.jstor.org/stable/pdf/2532547.pdf 


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]

Compartmental OED

2026-02-13T13:41:14Z

RobertLampel:

{{Dimensions
|nd = 1
|nx = 12
|nw = 1
}}

The '''Compartmental OED problem''' looks for an optimal measurement strategy to determine a three parameters in a one-dimensional [[:Category:ODE model|ODE model]], where we can directly measure the single state. The following description is taken from [[#compartmental| [1]]].

Here we consider the open one-compartment model with first-order absorption input fitted by Button (unpublished Ph.D. thesis, Texas A&M University, 1979) to the results of an experiment in which six horses each received 15 mg/kg of theophylline as aminophylline by intragastric administration.

The optimal integer control functions shows [[:Category:Chattering|bang bang]] behavior.

== Mathematical formulation ==
For the three-dimensional parameter <math>p = (\theta_1, \theta_2, \theta_3)</math> the original initial value problem is given by
<math>
\dot{y}(t) =: f(t, p) = \theta_3 \cdot (-\theta_1 \cdot \exp(-\theta_1 \cdot t) + \theta_2 \cdot \exp(-\theta_2 \cdot t)), \quad y(0) = 0.
</math>

We assume both <math>x_0</math> and <math>t_f</math> to be fixed and are only interested in when to measure, with an upper bound <math>M</math> on the measuring time. We can measure the state directly, i.e. <math>h(x(t)) = x(t)</math>.

Now we formulate the OED problem:

<math>
\begin{array}{lll}
\displaystyle \min_{y,G,F,z,w} && \text{trace} \; \left( F^{-1}(t_f) \right) \\
\text{subject to} \\
\quad \dot{y}(t) & = & f(t, p) \\
\quad \dot{G}(t) & = & f_p(y(t),p) \\
\quad \dot{F}(t) & = & w(t)(h_y(y(t))G(t))^T(h_y(y(t))G(t)) \\
\quad \dot{z}(t) & = & w(t), \\
\quad y(0) & = & y_0 \\
\quad G(0) & = & 0 \\
\quad F(0) & = & 0, \\
\quad z(0) & = & 0 \\
\quad w(t) & \in & \mathcal{W} \\
\quad z(t_f) & \leq & M
\end{array}
</math>


== Parameters ==
These fixed values are used within the model:

<math>
y_0 = 0; \quad t_f = 40; \quad \mathcal{W} = [0,1]; \quad M = 5; \quad p = (0.05884,4.298,21.80)
</math>


== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
Image:Compartmental OED.png| Sensitivities and measurement control for <math>\theta=(0.05884, 4.298, 21.80)</math>.
</gallery>

== References ==
[1] Optimum Experimental Designs for Properties of a Compartmental Model, A. Atkinson, K. Chaloner, A. Herzberg, J. Juritz, https://www.jstor.org/stable/pdf/2532547.pdf 


[[Category:MIOCP]]
[[Category:Optimum Experimental Design]]
[[Category:ODE model]]
[[Category:Bang bang]]