Fermenter - Revision history

RobertLampel: /* Reference Solutions */

2025-11-28T13:45:11Z

Reference Solutions

← Older revision		Revision as of 13:45, 28 November 2025
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	Here is one local solution to the above control problem.		Here is one local solution to the above control problem.

	<gallery caption="Reference solution plots" widths="~~180px~~" heights="~~140px~~" perrow="1">		<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
	Image:Fermenter.png\| States and discretized control for a local optimum.		Image:Fermenter.png\| States and discretized control for a local optimum.
	</gallery>		</gallery>

RobertLampel: /* Parameters */

2025-11-24T12:22:53Z

Parameters

← Older revision		Revision as of 12:22, 24 November 2025
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	== Parameters ==		== Parameters ==

	{\| ~~class~~="~~wikitable~~"		{\| border="1" align="center" cellpadding="5" cellspacing="0"
	~~\|+Parameters~~		\|- bgcolor=#c7c7c7
	\|-		!Symbol !! Value
	\|Symbol
	\|Value
	\|-		\|-
	\|<math>t_f</math>		\|<math>t_f</math>
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	\|<math>5 \cdot 10^4</math>		\|<math>5 \cdot 10^4</math>
	\|}		\|}


	== Reference Solutions ==		== Reference Solutions ==

RobertLampel: /* Mathematical formulation */

2025-08-22T08:39:53Z

Mathematical formulation

← Older revision		Revision as of 08:39, 22 August 2025
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	\quad \dot{V}(t) &=& u_{S_1}(t) + u_{S_2}(t) - u_p(t) \\		\quad \dot{V}(t) &=& u_{S_1}(t) + u_{S_2}(t) - u_p(t) \\
	\quad \dot{G}(t) &=& -\gamma_{x,g} \cdot E(t) \cdot S_1(t) \cdot S_2(t) \cdot G(t) - G(t) \cdot \frac{u_{S_1} + u_{S_2}(t)}{25 \cdot V(t)} \\		\quad \dot{G}(t) &=& -\gamma_{x,g} \cdot E(t) \cdot S_1(t) \cdot S_2(t) \cdot G(t) - G(t) \cdot \frac{u_{S_1} + u_{S_2}(t)}{25 \cdot V(t)} \\
	\quad &=& u_P(t) \cdot P(t) + \frac{u_{S_1}(t) + u_{S_2}(t) - u_P(t)}{25} \cdot P(t) + V(t) \cdot \dot{P}(t) \\		\quad \dot{P_{\text{acc}}}(t) &=& u_P(t) \cdot P(t) + \frac{u_{S_1}(t) + u_{S_2}(t) - u_P(t)}{25} \cdot P(t) + V(t) \cdot \dot{P}(t) \\
	\quad \dot{S_{1, \text{acc}}}(t) &=& 0.0168 \cdot u_{S_1}(t) \\		\quad \dot{S_{1, \text{acc}}}(t) &=& 0.0168 \cdot u_{S_1}(t) \\
	\quad \dot{S_{2, \text{acc}}}(t) &=& 0.01332 \cdot u_{S_2}(t)		\quad \dot{S_{2, \text{acc}}}(t) &=& 0.01332 \cdot u_{S_2}(t)

RobertLampel at 07:54, 22 August 2025

2025-08-22T07:54:54Z

Show changes

RobertLampel: Created page with "{{Dimensions |nd = 1 |nx = 9 |nw = 3 }} The '''Fermenter problem''' describes a fermentation process with two substrates $S_1$ and $S_2$ , and two products $P$ and $G$ . Enzyme biomass concentration is modeled by a state $E$ . Further states are the fermentation volume $V$ and the accumulated product $P_{\text{acc}}$ and substrates $S_{1,\text{acc}}$ and $S_{2..."$

2025-08-22T07:01:46Z

Created page with "{{Dimensions |nd = 1 |nx = 9 |nw = 3 }} The '''Fermenter problem''' describes a fermentation process with two substrates <math>S_1</math> and <math>S_2</math> , and two products <math>P</math> and <math>G</math>. Enzyme biomass concentration is modeled by a state <math>E</math>. Further states are the fermentation volume <math>V</math> and the accumulated product <math>P_{\text{acc}}</math> and substrates <math>S_{1,\text{acc}}</math> and <math>S_{2..."

New page

{{Dimensions
|nd = 1
|nx = 9
|nw = 3
}}

The '''Fermenter problem''' describes a fermentation process with two substrates <math>S_1</math> and <math>S_2</math> , and two products <math>P</math> and <math>G</math>. Enzyme biomass concentration is modeled by a state <math>E</math>. Further states are the fermentation volume <math>V</math> and the accumulated product <math>P_{\text{acc}}</math> and substrates <math>S_{1,\text{acc}}</math> and <math>S_{2,\text{acc}}</math>.
<math>S_1<math> and <math>S_2</math> can be fed into the reactor. This is described by two controls <math>u_{S_1}</math> and <math>u_{S_2}</math>. Furthermore, <math>P</math> can be harvested with rate <math>u_P<math> . The dynamics are given by an [[:Category:ODE model|ODE model]].

The optimal control function exhibits a [[:Category:Sensitivity-seeking arcs|singular arc]].

== Mathematical formulation ==

<math>
\begin{array}{lll}
\displaystyle \min_{w} && y(t_f) \\
\text{subject to} \\
\quad \dot{y}(t) & = & \exp(-\rho \cdot t) \cdot (U(t)- A(t) - u_1(t) \cdot C(t) - D(t)),\\
\quad \dot{S}(t) & = & u_1(t) - u_2(t) - \gamma \cdot (S(t) - \omega \cdot D_L(t)),\\
\quad \dot{R}(t) & = & -u_1(t) \\
\quad y(0) &=& 0, \\
\quad S(0) &=& 2 \cdot 10^3, \\
\quad R(0) &=& 10^4 \\
\quad S(t), R(t) & \in & [0,10^5], \\
\quad u_1(t), u_2(t) & \in & [0,40] \\
\end{array}
</math>


with auxiliary functions


<math>
\begin{align}
&U(t) = b \cdot u_1(t) - \mu \cdot u_1(t)^2, \quad \quad
&& D(t) = \nu \cdot (0.3 \cdot S(t) - S_{\text{preind}})^2, \\
&A(t) = a_1 \cdot u_2(t) + a_2 \cdot u_2(t)^2, \quad \quad
&& D_L(t) = D_{L, 0} + R_0 + S_0 - R(t) - S(t), \\
&C(t) = c_1 - c_2 \cdot R(t). &
\end{align}
</math>


== Parameters ==

{| class="wikitable"
|+Parameters
|-
|Symbol
|Value
|-
|<math>\rho</math>
|<math>0.03</math>
|-
|<math>\gamma</math>
|<math>0.001</math>
|-
|<math>\omega</math>
|<math>0.1</math>
|-
|<math>b</math>
|<math>50</math>
|-
|<math>\mu</math>
|<math>0.5</math>
|-
|<math>a_1</math>
|<math>2</math>
|-
|<math>a_2</math>
|<math>2</math>
|-
|<math>\nu</math>
|<math>1</math>
|-
|<math>c_1</math>
|<math>50</math>
|-
|<math>c_2</math>
|<math>0.004</math>
|-
|<math>S_{\text{preind}}</math>
|<math>600</math>
|-
|<math>S_0</math>
|<math>2000</math>
|-
|<math>R_0</math>
|<math>10^4</math>
|-
|<math>D_{L,0}</math>
|<math>2.3 \cdot 10^4</math>
|}

== Reference Solutions ==

Here is one local solution to the above control problem.

<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">
Image:Ocean.png| States and discretized control for a local optimum. Due to the explicit time dependence the time <math>t</math> was added as an additional state.
</gallery>

== Miscellaneous and Further Reading ==
The problem description and further references can be found in the PhD thesis of Dennis Janka [[#JankaPhD|[2]]].

== References ==
[1] W. Rickels and S. Sager. Personal communication. 2015. 
[2] Janka, D.: Sequential quadratic programming with indefinite Hessian approximations for nonlinear optimum experimental design for parameter estimation in differential-algebraic equations. Ph.D. thesis, Ruprecht-Karls-Universität Heidelberg (2015). URL https://mathopt.de/publications/Janka2015.pdf 

[[Category:MIOCP]]
[[Category:Sensitivity-seeking arcs]]