Bioreactor

The bioreactor example is an easy bioreactor with an substrate that is converted to a product by the biomass in the reactor. It has three states and a control that is describing the feed concentration of substrate. It is taken from the examples folder of the ACADO toolkit described in:

Houska, Boris, Hans Joachim Ferreau, and Moritz Diehl. "ACADO toolkit—An open‐source framework for automatic control and dynamic optimization."
Optimal Control Applications and Methods 32.3 (2011): 298-312.

Originally the problem seems to be motivated by:

VERSYCK, KARINA J., and JAN F. VAN IMPE. "Feed rate optimization for fed-batch bioreactors: From optimal process performance to optimal parameter estimation."
Chemical Engineering Communications 172.1 (1999): 107-124.

Model Formulation

The dynamic model is an ODE model:

$\begin{array}{rcl} \dot{X}&=&-DX+\mu X \\ \dot{S}&=& D(S_{f}-S)-\mu /Y_{xs} X \\ \dot{P}&=&-DP+ (\alpha \mu +\beta) X. \end{array}$

The three states describe the concentration of the biomass ($X$), the substrate ($S$), and the product ($P$) in the reactor. In steady state the feed and outlet are equal and dilute all three concentrations with a ratio $D$. The biomass grows with a rate $\mu$, while it eats up the substrate with the rate $\mu/Y_{xs}$ and produces product at a rate $(\alpha \mu +\beta)$. The rate $\mu$ is given by:

$\mu = \mu_{m}*(1-P/P_{m})*S/(K_m+S+S^2/K_i)$

The fixed parameters (constants) of the model are as follows.

Parameters

Name	Symbol	Value	Unit
Dilution	$D$	0.15	[-]
Rate coefficient	$K_i$	22	[-]
Rate coefficient	$K_m$	1.2	[-]
Rate coefficient	$P_m$	50	[-]
Substrate to Biomass rate	$Y_{xs}$	0.4	[-]
Linear slope	$\alpha$	2.2	[-]
Linear intercept	$\beta$	0.2	[-]
Maximal growth rate	$\mu_m$	0.48	[-]

Optimal Control Problem

Writing shortly for the states in vector notation $x=(X,S,P)^T$ the OCP reads:

$\begin{array}{cl} \displaystyle \min_{x,S_f} & J(x,S_f)\\[1.5ex] \mbox{s.t.} & \dot{x} = f(x,S_f), \forall \, t \in [0,48]\\ & x(0) = (6.5,12,22)^T \\ & x \in \R^3,\,S_f \in [28.7,40]. \end{array}$

Objective

$J(x,S_f)=\int_0^{48}D(S_f-P)^2dt$