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State dimension: 1
Differential states: 3
Continuous control functions: 1
Path constraints: 2
Interior point equalities: 3

The bioreactor problem describes 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 the substrate. The problem is taken from the examples folder of the ACADO toolkit described in: [Houska2011a]The entry doesn't exist yet.

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: [Versyck1999]The entry doesn't exist yet.

"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:

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

The right-hand side of these equations will be summed up in  f(x, S_f) .

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.

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 [-]

Mathematical formulation

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

 \displaystyle \min_{x,S_f} & J(x,S_f)\\[1.5ex]
 & \dot{x} & = & f(x,S_f)\\
 & x(0) & = & (6.5,12,22)^T \\
 & S_f & \in  &[28.7,40].



Reference Solution

Here we present the reference solution of the reimplemented example in the ACADO code generation with matlab. The source code is given in the next section.

Source Code

Model descriptions are available in