Fermenter: Difference between revisions

Fermenter
State dimension:	1
Differential states:	9
Discrete control functions:	3

Latest revision as of 13:45, 28 November 2025

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 $P_{acc}$ and substrates $S_{1, acc}$ and $S_{2, acc}$ . $S_{1}$ and $S_{2}$ can be fed into the reactor. This is described by two controls $u_{S_{1}}$ and $u_{S_{2}}$ . Furthermore, $P$ can be harvested with rate $u_{P}$ . The dynamics are given by an ODE model.

This model description is taken from the PhD thesis of Dennis Janka [1].

The optimal control function exhibits a singular arc.

Mathematical formulation

$\begin{array}{lll} \min_{u_{S_{1}}, u_{S_{2}}, u_{P}} & \frac{2 \cdot S_{1, acc} (t_{f}) \cdot S_{2, acc} (t_{F})}{P_{acc} (t_{f})} \\ subject to \\ \dot{P} (t) & = & μ_{p} \cdot E (t) \cdot S_{1} (t) \cdot S_{2} (t) - P (t) \cdot \frac{u_{S_{1}} (t) + u_{S_{2}} (t)}{25 \cdot V (t)} \\ \dot{S_{1}} (t) & = & - γ_{x, 1} \cdot E (t) \cdot S_{1} (t) \cdot S_{2} (t) \cdot G (t) - γ_{p, 1} \cdot E (t) \cdot S_{1} (t) \cdot S_{2} (t) \\ + 0.42 \cdot \frac{u_{S_{1}} (t)}{25 \cdot V (t)} - S_{1} (t) \cdot \frac{u_{S_{1}} (t) + u_{S_{2}} (t)}{25 \cdot V (t)} \\ \dot{S_{2}} (t) & = & - γ_{x, 2} \cdot E (t) \cdot S_{1} (t) \cdot S_{2} (t) \cdot G (t) - γ_{p, 2} \cdot E (t) \cdot S_{1} (t) \cdot S_{2} (t) \\ + 0.333 \cdot \frac{u_{S_{2}} (t)}{25 \cdot V (t)} - S_{2} (t) \cdot \frac{u_{S_{1}} (t) + u_{S_{2}} (t)}{25 \cdot V (t)} \\ \dot{E} (t) & = & μ_{x} \cdot E (t) \cdot S_{1} (t) \cdot S_{2} (t) \cdot G (t) - E (t) \cdot \frac{u_{S_{1}} + u_{S_{2}}}{25 \cdot V (t)} \\ \dot{V} (t) & = & u_{S_{1}} (t) + u_{S_{2}} (t) - u_{p} (t) \\ \dot{G} (t) & = & - γ_{x, g} \cdot E (t) \cdot S_{1} (t) \cdot S_{2} (t) \cdot G (t) - G (t) \cdot \frac{u_{S_{1}} + u_{S_{2}} (t)}{25 \cdot V (t)} \\ \dot{P_{acc}} (t) & = & u_{P} (t) \cdot P (t) + \frac{u_{S_{1}} (t) + u_{S_{2}} (t) - u_{P} (t)}{25} \cdot P (t) + V (t) \cdot \dot{P} (t) \\ \dot{S_{1, acc}} (t) & = & 0.0168 \cdot u_{S_{1}} (t) \\ \dot{S_{2, acc}} (t) & = & 0.01332 \cdot u_{S_{2}} (t) \end{array}$

with bounds for the control functions given by

$\begin{aligned} u_{S_{1}} \in [0, 15], u_{S_{2}} \in [0, 1], u_{P} \in [0, 30] \end{aligned}$

and bounds for the states given by

$\begin{aligned} P (t) \in [0, 0.1], & S_{1} (t) \in [0, 0.04], & S_{2} (t) \in [0, 0.03], \\ E (t) \in [0, 0.1], & V (t) \in [0.3, 0.45], & G (t) \in [0, 0.1], \\ P_{acc} (t) \in [0, 0.05], & S_{1, acc} (t) \in [0, 0.2], & S_{2, acc} (t) \in [0, 0.025] . \end{aligned}$

Parameters

Symbol	Value
$t_{f}$	$1$
$μ_{x}$	$2 \cdot 1 0^{5}$
$μ_{p}$	$5000$
$γ_{x, g}$	$5 \cdot 1 0^{4}$
$γ_{x, 1}$	$1 0^{5}$
$γ_{p, 1}$	$2 \cdot 1 0^{4}$
$γ_{x, 2}$	$1500$
$γ_{p, 2}$	$5 \cdot 1 0^{4}$

Reference Solutions

Here is one local solution to the above control problem.

Reference solution plots
States and discretized control for a local optimum.

References

[1] 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

@@ Line 6: / Line 6: @@
 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]].
+<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]].
+This model description is taken from the PhD thesis of Dennis Janka [[#JankaPhD|[1]]].
 The optimal control function exhibits a [[:Category:Sensitivity-seeking arcs|singular arc]].
@@ Line 14: / Line 16: @@
 <math>
   \begin{array}{lll}
-  \displaystyle \min_{w} && y(t_f) \\
+  \displaystyle \min_{u_{S_1}, u_{S_2}, u_P} && \frac{2 \cdot S_{1,\text{acc}}(t_f) \cdot S_{2,\text{acc}}(t_F)}{P_{\text{acc}}(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{P}(t) &=& \mu_p \cdot E(t) \cdot S_1(t) \cdot S_2(t) - P(t) \cdot \frac{u_{S_1}(t) + u_{S_2}(t)}{25 \cdot V(t)} \\
-\quad \dot{S}(t) & = &  u_1(t) - u_2(t) - \gamma \cdot (S(t) - \omega \cdot D_L(t)),\\
+  \quad \dot{S_1}(t) &=& -\gamma_{x,1} \cdot E(t) \cdot S_1(t) \cdot S_2(t) \cdot G(t) - \gamma_{p,1} \cdot E(t) \cdot S_1(t) \cdot S_2(t) \\
-\quad \dot{R}(t) & = & -u_1(t) \\
+ \quad & & + 0.42 \cdot \frac{u_{S_1}(t)}{25 \cdot V(t)} - S_1(t) \cdot \frac{u_{S_1}(t) + u_{S_2}(t)}{25 \cdot V(t)} \\
-\quad y(0) &=& 0, \\
+ \quad \dot{S_2}(t) &=& -\gamma_{x,2} \cdot E(t) \cdot S_1(t) \cdot S_2(t) \cdot G(t) - \gamma_{p,2} \cdot E(t) \cdot S_1(t) \cdot S_2(t) \\
-\quad S(0) &=& 2 \cdot 10^3, \\
+ \quad  & & + 0.333 \cdot \frac{u_{S_2}(t)}{25 \cdot V(t)} - S_2(t) \cdot \frac{u_{S_1}(t) + u_{S_2}(t)}{25 \cdot V(t)} \\
-\quad R(0) &=& 10^4 \\
+ \quad \dot{E}(t) &=& \mu_x \cdot E(t) \cdot S_1(t) \cdot S_2(t) \cdot G(t) - E(t) \cdot \frac{u_{S_1} + u_{S_2}}{25 \cdot V(t)} \\
-\quad S(t), R(t) & \in & [0,10^5], \\
+ \quad \dot{V}(t) &=& u_{S_1}(t) + u_{S_2}(t) - u_p(t) \\
-\quad u_1(t), u_2(t) & \in & [0,40] \\
+ \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{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_{2, \text{acc}}}(t) &=& 0.01332 \cdot u_{S_2}(t)
    \end{array}
 </math>
 </p>
-with auxiliary functions
+with bounds for the control functions given by
+<p>
+<math>
+ \begin{align}
+  u_{S_1} \in [0,15], \quad \quad
+  u_{S_2} \in [0,1], \quad \quad
+  u_{P} \in [0,30]
+  \end{align}
+</math>
+</p>
+and bounds for the states given by
 <p>
 <math>
   \begin{align}
-   &U(t) = b \cdot u_1(t) - \mu \cdot u_1(t)^2, \quad \quad
+   &  P(t) \in [0, 0.1], \quad \quad
-   && D(t) = \nu \cdot (0.3 \cdot S(t) - S_{\text{preind}})^2, \\
+  && S_1(t) \in [0, 0.04],\quad \quad
-   &A(t) = a_1 \cdot u_2(t) + a_2 \cdot u_2(t)^2, \quad \quad
+   && S_2(t) \in [0, 0.03], \\
-   && D_L(t) = D_{L, 0} + R_0 + S_0 - R(t) - S(t), \\
+  &  E(t) \in [0, 0.1], \quad \quad
-   &C(t) = c_1 - c_2 \cdot R(t). &
+   && V(t) \in [0.3, 0.45],\quad \quad
+  && G(t) \in [0, 0.1], \\
+  &  P_{\text{acc}}(t) \in [0, 0.05], \quad \quad
+   && S_{1,\text{acc}}(t) \in [0, 0.2],\quad \quad
+   && S_{2,\text{acc}}(t) \in [0, 0.025].
   \end{align}
 </math>
@@ Line 44: / Line 65: @@
 == Parameters ==
-{| class="wikitable"
+{| border="1" align="center" cellpadding="5" cellspacing="0"
-|+Parameters
+|- bgcolor=#c7c7c7
+!Symbol !! Value
 |-
-|Symbol
+|<math>t_f</math>
-|Value
+|<math>1</math>
 |-
-|<math>\rho</math>
+|<math>\mu_x</math>
-|<math>0.03</math>
+|<math>2\cdot 10^5</math>
 |-
-|<math>\gamma</math>
+|<math>\mu_p</math>
-|<math>0.001</math>
+|<math>5000</math>
 |-
-|<math>\omega</math>
+|<math>\gamma_{x,g}</math>
-|<math>0.1</math>
+|<math>5 \cdot 10^4</math>
 |-
-|<math>b</math>
+|<math>\gamma_{x,1}</math>
-|<math>50</math>
+|<math>10^5</math>
 |-
-|<math>\mu</math>
+|<math>\gamma_{p,1}</math>
-|<math>0.5</math>
+|<math>2 \cdot 10^4</math>
 |-
-|<math>a_1</math>
+|<math>\gamma_{x,2}</math>
-|<math>2</math>
+|<math>1500</math>
 |-
-|<math>a_2</math>
+|<math>\gamma_{p,2}</math>
-|<math>2</math>
+|<math>5 \cdot 10^4</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 ==
@@ Line 98: / Line 98: @@
 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: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.
+  Image:Fermenter.png| States and discretized control for a local optimum.
 </gallery>
-== Miscellaneous and Further Reading ==
-The problem description and further references can be found in the PhD thesis of Dennis Janka [[#JankaPhD|[2]]].
 == References ==
-<span id="PersonalComm">[1]</span> W. Rickels and S. Sager. Personal communication. 2015. <br>
+<span id="JankaPhD">[1]</span> 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 <br>
-<span id="JankaPhD">[2]</span> 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 <br>
 [[Category:MIOCP]]
 [[Category:Sensitivity-seeking arcs]]