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

← Older edit

Latest revision as of 19:15, 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

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

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

AMPL/TACO code at Catalyst mixing problem (TACO)

@@ Line 1: / Line 1: @@
 {{Dimensions
 |nd        = 1
-|nx        = 3
+|nx        = 2
-|nw        = 1
+|nu        = 1
-|nre       = 3
+|nc        = 2
+|nre       = 2
 }}<!-- Do not insert line break here or Dimensions Box moves up in the layout...
@@ Line 18: / Line 19: @@
 <math>
 \begin{array}{llcl}
-  \displaystyle \min_{x, u} &-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 & = &  u ( 10 x_2 - x_1), \\
+  \mbox{s.t.}
-  & \dot{x}_2 & = & u ( x_1 - 10 x_2) - (1 - u \, 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, \\
-  & u(t) &\in&  [0,1].
+  & w(t) &\in&  \{0,1\}.
 \end{array}
 </math>
@@ Line 30: / Line 32: @@
 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 ==
@@ Line 41: / Line 51: @@
 [[Category:MIOCP]]
 [[Category:ODE model]]
+[[Category:Chemical engineering]]