Difference between revisions of "Catalyst mixing problem"
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{{Dimensions | {{Dimensions | ||
|nd = 1 | |nd = 1 | ||
− | |nx = | + | |nx = 2 |
− | | | + | |nu = 1 |
− | |nre = | + | |nc = 2 |
+ | |nre = 2 | ||
}}<!-- Do not insert line break here or Dimensions Box moves up in the layout... | }}<!-- Do not insert line break here or Dimensions Box moves up in the layout... | ||
Line 17: | Line 18: | ||
<p> | <p> | ||
<math> | <math> | ||
− | \begin{array}{ | + | \begin{array}{llcl} |
\displaystyle \min_{x, w} &-1 + x_1(t_f) + x_2(t_f) \\[1.5ex] | \displaystyle \min_{x, w} &-1 + x_1(t_f) + x_2(t_f) \\[1.5ex] | ||
− | \mbox{s.t.} & \dot{x}_1 & = & | + | \mbox{s.t.} |
− | & \dot{x}_2 & = & | + | & \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, \\ | & x(t_0) &=& (1, 0)^T, \\ | ||
− | & | + | & 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>. | ||
+ | |||
+ | == 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. | ||
+ | |||
+ | <gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2"> | ||
+ | Image:Catalyst_Mixing_Problem_Performance.png| Results with relaxed controls and collocation from the [http://www.mcs.anl.gov/~more/cops/ COPS library] | ||
+ | Image:Catalyst Mixing Controls.png| Optimal relaxed controls showing a bang-bang structure. | ||
+ | </gallery> | ||
== Source Code == | == Source Code == | ||
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[[Category:MIOCP]] | [[Category:MIOCP]] | ||
[[Category:ODE model]] | [[Category:ODE model]] | ||
+ | [[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
Parameters
In this model the parameters used are .
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.
Results with relaxed controls and collocation from the COPS library
Source Code
Model descriptions are available in