Catalyst mixing problem: Difference between revisions

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

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Revision as of 18:47, 12 January 2018

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

$\begin{array}{llcl} \min_{x, u} & - 1 + x_{1} (t_{f}) + x_{2} (t_{f}) \\ s.t. & {\dot{x}}_{1} & = & u (10 x_{2} (t) - x_{1} (t)), \\ {\dot{x}}_{2} & = & u (x_{1} (t) - 10 x_{2} (t)) - (1 - u (t)) x_{2} (t), \\ x (t_{0}) & = & (1, 0)^{T}, \\ u (t) & \in & {0, 1} . \end{array}$

Parameters

In this model the parameters used are $t_{0} = 0, t_{f} = 1$ .

Source Code

Model descriptions are available in

AMPL/TACO code at Catalyst mixing problem (TACO)

@@ Line 21: / Line 21: @@
   \displaystyle \min_{x, u} &-1 + x_1(t_f) + x_2(t_f)   \\[1.5ex]
   \mbox{s.t.}
-  & \dot{x}_1 & = &  u ( 10 x_2 - x_1), \\
+  & \dot{x}_1 & = &  u ( 10 x_2(t) - x_1(t)), \\
-  & \dot{x}_2 & = & u ( x_1 - 10 x_2) - (1 - u \, x_2) ,  \\
+  & \dot{x}_2 & = & u ( x_1(t) - 10 x_2(t)) - (1 - u(t)) \, x_2(t) ,  \\
   & x(t_0) &=& (1, 0)^T, \\
-  & u(t) &\in&  [0,1].
+  & u(t) &\in&  \{0,1\}.
 \end{array}
 </math>