Three Tank multimode problem: Difference between revisions

Three Tank multimode problem
State dimension:	1
Differential states:	3
Discrete control functions:	3
Interior point equalities:	2

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Revision as of 09:12, 14 March 2020

This site describes a Double tank problem variant with three binary controls instead of only one control and three tanks, i.e., three differential states representing different compartments.

Mathematical formulation

The mixed-integer optimal control problem is given by

$\begin{array}{llll} \min_{x, w} & \int_{0}^{T} & k_{1} (x_{2} - k_{2})^{2} + k_{3} (x_{3} - k_{4})^{2} d t \\ s.t. & {\dot{x}}_{1} & = - \sqrt{x_{1}} + c_{1} w_{1} + c_{2} w_{2} - w_{3} \sqrt{c_{3} x_{3}}, \\ {\dot{x}}_{2} & = \sqrt{x_{1}} - \sqrt{x_{2}}, \\ {\dot{x}}_{3} & = \sqrt{x_{2}} - \sqrt{x_{3}} + w_{3} \sqrt{c_{3} x_{3}}, \\ x (0) & = (2, 2, 2)^{T}, \\ 1 & = \sum_{i = 1}^{3} w_{i} (t), \\ w_{i} (t) & \in {0, 1}, i = 1 \dots 3 . \end{array}$

Parameters

These fixed values are used within the model.

$T = 12, c_{1} = 1, c_{2} = 2, c_{3} = 0.8, k_{1} = 2, k_{2} = 3, k_{3} = 1, k_{4} = 3 .$

Reference Solutions

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 and the CIA decomposition. We denote the relaxed control values with $a (t) \in [0, 1]$ .

The optimal objective value of the relaxed problem with $n_{t} = 100, n_{u} = 100$ is $8.775979$ . The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is $8.789487$ .

Reference solution plots
Optimal relaxed controls and states determined by an direct approach with python 3.6 and CasADi, applied Multiple Shooting, 4th order Runge Kutta scheme and $n_{u} = 100$ .
According optimal binary controls and states determined by the direct approach. The relaxed controls were approximated by Combinatorial Integral Approximation.
Binary and relaxed control values as part of the Combinatorial Integral Approximation problem

@@ Line 33: / Line 33: @@
 == Reference Solutions ==
-If the problem is relaxed, i.e., we demand that <math>w(t)</math> be in the continuous interval <math>[0, 1]</math> instead of the binary choice <math>\{0,1\}</math>, the optimal solution can be determined by means of direct optimal control.
+If the problem is relaxed, i.e., we demand that <math>w(t)</math> be in the continuous interval <math>[0, 1]</math> instead of the binary choice <math>\{0,1\}</math>, the optimal solution can be determined by means of direct optimal control and the CIA decomposition. We denote the relaxed control values with <math>a(t)\in[0, 1]</math>.
-The optimal objective value of the relaxed problem with  <math> n_t=12000, \, n_u=100  </math> is <math>2.59106823</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>2.59121008</math>.
+The optimal objective value of the relaxed problem with  <math> n_t=100, \, n_u=100  </math> is <math>8.775979</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>8.789487</math>.
 <gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
-  Image:MmdoubletankRelaxed 12000 120 1.png| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation)  and <math>n_t=12000, \, n_u=100</math>.
+  Image:three_tank_relaxed_solution.png| Optimal relaxed controls and states determined by an direct approach with python 3.6 and CasADi, applied Multiple Shooting, 4th order Runge Kutta scheme and <math>n_u=100</math>.
-  Image:MmdoubletankCIA_12000_120_1.png| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=100</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
+  Image:three_tank_binary_solution.png| According optimal binary controls and states determined by the direct approach. The relaxed controls were approximated by Combinatorial Integral Approximation.
+ Image:three_tank_rounding_solution.png| Binary and relaxed control values as part of the Combinatorial Integral Approximation problem
 </gallery>