Difference between revisions of "Bioreactor"

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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:
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{{Dimensions
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|nd        = 1
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|nx        = 3
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|nu        = 1
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|nc        = 2
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|nre      = 3
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}}
  
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  Houska, Boris, Hans Joachim Ferreau, and Moritz Diehl. "ACADO toolkit—An open‐source framework for automatic control and dynamic optimization."
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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:
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<bib id="Houska2011a" />
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  Houska, Boris, Hans Joachim Ferreau, and Moritz Diehl.  
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"ACADO toolkit—An open‐source framework for automatic control and dynamic optimization."
 
  Optimal Control Applications and Methods 32.3 (2011): 298-312.
 
  Optimal Control Applications and Methods 32.3 (2011): 298-312.
  
 
Originally the problem seems to be motivated by:
 
Originally the problem seems to be motivated by:
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<bib id="Versyck1999" />
  
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  VERSYCK, KARINA J., and JAN F. VAN IMPE. "Feed rate optimization for fed-batch bioreactors: From optimal process performance to optimal parameter estimation."
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  VERSYCK, KARINA J., and JAN F. VAN IMPE.  
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"Feed rate optimization for fed-batch bioreactors: From optimal process performance to optimal parameter estimation."
 
  Chemical Engineering Communications 172.1 (1999): 107-124.
 
  Chemical Engineering Communications 172.1 (1999): 107-124.
  
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</math>
 
</math>
 
</p>
 
</p>
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The right-hand side of these equations will be summed up in <math> f(x, S_f) </math>.
  
 
The three states describe the concentration of the biomass (<math>X</math>), the substrate (<math>S</math>), and the product (<math>P</math>) in the reactor. In steady state the feed and outlet are equal and dilute all three concentrations with a ratio <math>D</math>. The biomass grows with a rate
 
The three states describe the concentration of the biomass (<math>X</math>), the substrate (<math>S</math>), and the product (<math>P</math>) in the reactor. In steady state the feed and outlet are equal and dilute all three concentrations with a ratio <math>D</math>. The biomass grows with a rate
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|}
 
|}
  
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== Optimal Control Problem ==
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== Mathematical formulation ==
  
 
Writing shortly for the states in vector notation <math>x=(X,S,P)^T</math> the OCP reads:
 
Writing shortly for the states in vector notation <math>x=(X,S,P)^T</math> the OCP reads:
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<p>
 
<p>
 
<math>
 
<math>
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\begin{array}{cl}
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\begin{array}{clcl}
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  \displaystyle \min_{x,Sf} & J(x,Sf)\\[1.5ex]
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  \displaystyle \min_{x,S_f} & J(x,S_f)\\[1.5ex]
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  \mbox{s.t.} & \dot{x} = f(x,Sf), \forall \, t \in [0,48]\\
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  \mbox{s.t.}  
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  & 0 = g(x(t_o)) \\
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& \dot{x} & = & f(x,S_f)\\
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  & x \in \R^3,\,Sf \in [28.7,40].
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  & x(0) & = & (6.5,12,22)^T \\
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  & S_f & \in &[28.7,40].
 
\end{array}  
 
\end{array}  
 
</math>
 
</math>
 
</p>
 
</p>
  
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This is the most general formulation, where the initial value, parameters, and inputs are free. The states and inputs are vectors (or matrices) of functions while the parameters are just a vector of variables. If the initial (or final) state is fixed, one just adds a constraint <math>x(t_0)=x_0</math> and specifies the value(s) of <math>x_0</math> in a list or table.
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=== Objective ===
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If the parameters are fixed, then one just removes them from the list of optimization variables and states their values in a list (same for point conditions inside of the interval <math>I</math>). If the OCP is a parameter estimation problem without inputs,
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<p>
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the inputs just dont appear in the problem above at all, neither as optimization variable nor as part of the constraints. In contrast if they are just fixed this is done by the equality constraints. In this setup the above problem is very general and by
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<math>
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stating the different objective functions and constraint functions one can cover a wide range of OCPs.
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J(x,S_f)=\int_0^{48}D(S_f-P)^2dt
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</math>
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</p>
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== Reference Solution ==
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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.
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<gallery caption="Reference solution" widths="551px" heights="390px" perrow="1">
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Image:ACADO_bioreactor.png| Optimal solution for the ACADO example.
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</gallery>
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== Source Code ==
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Model descriptions are available in
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* [[:Category:ACADO | ACADO code]] at [[Bioreactor (ACADO)]]
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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 -->
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[[Category:MIOCP]] [[Category: ODE model]] [[Category:Chemical engineering]]
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[[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.

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 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.

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 [-]

Mathematical formulation

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

Objective

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

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

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