Difference between revisions of "Bioreactor"
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|nx = 3 | |nx = 3 | ||
|nu = 1 | |nu = 1 | ||
| − | |nc = | + | |nc = 2 |
|nre = 3 | |nre = 3 | ||
}} | }} | ||
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<p> | <p> | ||
<math> | <math> | ||
| − | \begin{array}{ | + | \begin{array}{clcl} |
\displaystyle \min_{x,S_f} & J(x,S_f)\\[1.5ex] | \displaystyle \min_{x,S_f} & J(x,S_f)\\[1.5ex] | ||
\mbox{s.t.} | \mbox{s.t.} | ||
| − | & \dot{x} | + | & \dot{x} & = & f(x,S_f)\\ |
| − | & x(0) = (6.5,12,22)^T \\ | + | & x(0) & = & (6.5,12,22)^T \\ |
| − | & S_f | + | & S_f & \in &[28.7,40]. |
\end{array} | \end{array} | ||
</math> | </math> | ||
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<!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here --> | <!--List of all categories this page is part of. List characterization of solution behavior, model properties, ore presence of implementation details (e.g., AMPL for AMPL model) here --> | ||
| − | [[Category:MIOCP]] [[Category: ODE model]] [[Chemical engineering]] | + | [[Category:MIOCP]] [[Category: ODE model]] [[Category:Chemical engineering]] |
| − | [[Bang bang]] | + | [[Category:Bang bang]] |
Latest revision as of 09:27, 27 July 2016
| Bioreactor | |
|---|---|
| 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.
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.
Contents
Model Formulation
The dynamic model is an ODE model:
The right-hand side of these equations will be summed up in 
.
The three states describe the concentration of the biomass (
), the substrate (
), and the product (
) in the reactor. In steady state the feed and outlet are equal and dilute all three concentrations with a ratio 
. The biomass grows with a rate
, while it eats up the substrate with the rate 
and produces product at a rate 
. The rate is given by:
The fixed parameters (constants) of the model are as follows.
| Name | Symbol | Value | Unit |
| Dilution |
|
0.15 | [-] |
| Rate coefficient | 
|
22 | [-] |
| Rate coefficient | 
|
1.2 | [-] |
| Rate coefficient | 
|
50 | [-] |
| Substrate to Biomass rate | 
|
0.4 | [-] |
| Linear slope | 
|
2.2 | [-] |
| Linear intercept | 
|
0.2 | [-] |
| Maximal growth rate | 
|
0.48 | [-] |
Mathematical formulation
Writing shortly for the states in vector notation 
the OCP reads:
![\begin{array}{clcl}
\displaystyle \min_{x,S_f} & J(x,S_f)\\[1.5ex]
\mbox{s.t.}
& \dot{x} & = & f(x,S_f)\\
& x(0) & = & (6.5,12,22)^T \\
& S_f & \in &[28.7,40].
\end{array}](https://mintoc.de/images/math/8/8/e/88e448fa43ddbb22c56b967ef453bc16.png)
Objective
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.
- Reference solution
Optimal solution for the ACADO example.
Source Code
Model descriptions are available in
