Fermenter

Fermenter
State dimension:	1
Differential states:	9
Discrete control functions:	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 $S_{2,\text{acc}}$. $S_1$ 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

$\begin{array}{lll}\underset{u_{S_1},u_{S_2},u_P}{\min }& & \frac{2\cdot S_{1,\text{acc}}(t_f)\cdot S_{2,\text{acc}}(t_F)}{P_{\text{acc}}(t_f)}\\ \text{subject to}\\ \dot{P}(t)& =& {\mu }_p\cdot E(t)\cdot S_1(t)\cdot S_2(t)-P(t)\cdot \frac{u_{S_1}(t)+u_{S_2}(t)}{25\cdot V(t)}\\ \dot{S_1}(t)& =& -{\gamma }_{x,1}\cdot E(t)\cdot S_1(t)\cdot S_2(t)\cdot G(t)-{\gamma }_{p,1}\cdot E(t)\cdot S_1(t)\cdot S_2(t)\\ & & +0.42\cdot \frac{u_{S_1}(t)}{25\cdot V(t)}-S_1(t)\cdot \frac{u_{S_1}(t)+u_{S_2}(t)}{25\cdot V(t)}\\ \dot{S_2}(t)& =& -{\gamma }_{x,2}\cdot E(t)\cdot S_1(t)\cdot S_2(t)\cdot G(t)-{\gamma }_{p,2}\cdot E(t)\cdot S_1(t)\cdot S_2(t)\\ & & +0.333\cdot \frac{u_{S_2}(t)}{25\cdot V(t)}-S_2(t)\cdot \frac{u_{S_1}(t)+u_{S_2}(t)}{25\cdot V(t)}\\ \dot{E}(t)& =& {\mu }_x\cdot E(t)\cdot S_1(t)\cdot S_2(t)\cdot G(t)-E(t)\cdot \frac{u_{S_1}+u_{S_2}}{25\cdot V(t)}\\ \dot{V}(t)& =& u_{S_1}(t)+u_{S_2}(t)-u_p(t)\\ \dot{G}(t)& =& -{\gamma }_{x,g}\cdot E(t)\cdot S_1(t)\cdot S_2(t)\cdot G(t)-G(t)\cdot \frac{u_{S_1}+u_{S_2}(t)}{25\cdot V(t)}\\ \dot{P_{\text{acc}}}(t)& =& u_P(t)\cdot P(t)+\frac{u_{S_1}(t)+u_{S_2}(t)-u_P(t)}{25}\cdot P(t)+V(t)\cdot \dot{P}(t)\\ \dot{S_{1,\text{acc}}}(t)& =& 0.0168\cdot u_{S_1}(t)\\ \dot{S_{2,\text{acc}}}(t)& =& 0.01332\cdot u_{S_2}(t)\end{array}$

with bounds for the control functions given by

$\begin{array}{r}{u}_{S_1}\in [0,15],u_{S_2}\in [0,1],u_P\in [0,30]\end{array}$

and bounds for the states given by

$\begin{array}{rlrlrl}& P(t)\in [0,0.1],& & S_1(t)\in [0,0.04],& & S_2(t)\in [0,0.03],\\ & E(t)\in [0,0.1],& & V(t)\in [0.3,0.45],& & G(t)\in [0,0.1],\\ & P_{\text{acc}}(t)\in [0,0.05],& & S_{1,\text{acc}}(t)\in [0,0.2],& & S_{2,\text{acc}}(t)\in [0,0.025].\end{array}$

Parameters

Symbol	Value
$t_f$	$1$
${\mu }_x$	$2\cdot 10^5$
${\mu }_p$	$5000$
${\gamma }_{x,g}$	$5\cdot 10^4$
${\gamma }_{x,1}$	$10^5$
${\gamma }_{p,1}$	$2\cdot 10^4$
${\gamma }_{x,2}$	$1500$
${\gamma }_{p,2}$	$5\cdot 10^4$

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