Difference between revisions of "Catalyst mixing problem"

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{{Dimensions
 
{{Dimensions
 
|nd        = 1
 
|nd        = 1
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|nx        = 3
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|nx        = 2
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|nw       = 1
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|nu       = 1
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|nre      = 3
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|nc        = 2
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|nre      = 2
 
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<math>
 
<math>
 
\begin{array}{llcl}
 
\begin{array}{llcl}
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  \displaystyle \min_{x, u} &-1 + x_1(t_f) + x_2(t_f)  \\[1.5ex]
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  \displaystyle \min_{x, w} &-1 + x_1(t_f) + x_2(t_f)  \\[1.5ex]
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  \mbox{s.t.} & \dot{x}_1 & = &  u ( 10 x_2 - x_1), \\
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  \mbox{s.t.}  
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  & \dot{x}_2 & = & u ( x_1 - 10 x_2) - (1 - u \, x_2) ,  \\
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& \dot{x}_1 & = &  w(t) ( 10 x_2(t) - x_1(t)), \\
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  & \dot{x}_2 & = & w(t) ( x_1(t) - 10 x_2(t)) - (1 - w(t)) \, x_2(t) ,  \\
 
  & x(t_0) &=& (1, 0)^T, \\
 
  & x(t_0) &=& (1, 0)^T, \\
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  & u(t) &\in&  \[0,1\].
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  & w(t) &\in&  \{0,1\}.
 
\end{array}  
 
\end{array}  
 
</math>
 
</math>
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In this model the parameters used are <math> t_0 = 0, \, \, t_f = 1 </math>.
 
In this model the parameters used are <math> t_0 = 0, \, \, t_f = 1 </math>.
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== Reference Solution ==
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If the problem is relaxed, i.e., we demand that w(t) be in the continuous interval [0, 1] instead of the binary choice \{0,1\}, the optimal solution can be determined by means of direct optimal control.
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<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
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Image:Catalyst_Mixing_Problem_Performance.png| Results with relaxed controls and collocation from the [http://www.mcs.anl.gov/~more/cops/ COPS library]
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Image:Catalyst Mixing Controls.png| Optimal relaxed controls showing a bang-bang structure.
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</gallery>
  
 
== Source Code ==
 
== Source Code ==
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[[Category:MIOCP]]
 
[[Category:MIOCP]]
 
[[Category:ODE model]]
 
[[Category:ODE model]]
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[[Category:Chemical engineering]]

Latest revision as of 20:15, 12 January 2018

Catalyst mixing problem
State dimension: 1
Differential states: 2
Continuous control functions: 1
Path constraints: 2
Interior point equalities: 2

The Catalyst mixing problem seeks an optimal policy for mixing two catalysts "along the length of a tubular plug ow reactor involving several reactions". (Cite and problem taken from the COPS library)


Mathematical formulation

The problem is given by

\begin{array}{llcl} \displaystyle \min_{x, w} &-1 + x_1(t_f) + x_2(t_f) \\[1.5ex] \mbox{s.t.} & \dot{x}_1 & = & w(t) ( 10 x_2(t) - x_1(t)), \\ & \dot{x}_2 & = & w(t) ( x_1(t) - 10 x_2(t)) - (1 - w(t)) \, x_2(t) , \\ & x(t_0) &=& (1, 0)^T, \\ & w(t) &\in& \{0,1\}. \end{array}

Parameters

In this model the parameters used are t_0 = 0, \, \, t_f = 1 .

Reference Solution

If the problem is relaxed, i.e., we demand that w(t) be in the continuous interval [0, 1] instead of the binary choice \{0,1\}, the optimal solution can be determined by means of direct optimal control.

  • Reference solution plots

  • Results with relaxed controls and collocation from the COPS library

  • Optimal relaxed controls showing a bang-bang structure.

Source Code

Model descriptions are available in

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