Double Tank multimode problem: Difference between revisions

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

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Revision as of 15:25, 10 January 2018

This site describes a Double tank problem variant with three binary controls instead of only one control.

Mathematical formulation

The mixed-integer optimal control problem is given by

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

Parameters

These fixed values are used within the model.

$T = 10, c_{1} = 1, c_{2} = 0.5, c_{3} = 2, k_{1} = 2, k_{2} = 3 .$

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.

The optimal objective value of the relaxed problem with $n_{t} = 12000, n_{u} = 100$ is $2.59106823$ . The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is $2.59121008$ .

Reference solution plots
Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and nt=12000,nu=100.

Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and $n_{t} = 12000, n_{u} = 100$ .
Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and $n_{t} = 12000, n_{u} = 100$ . The relaxed controls were approximated by Combinatorial Integral Approximation.

@@ Line 23: / Line 23: @@
 </math>
-Here the differential states <math>(x_0, x_1)</math> describe the biomasses of prey and predator, respectively. The third differential state is used here to transform the objective, an integrated deviation, into the Mayer formulation <math>\min \; x_2(t_f)</math>. This problem variant allows to choose between three different fishing options.
 == Parameters ==
@@ Line 30: / Line 28: @@
 These fixed values are used within the model.
-<math>
+<math> T=10, c_1=1, c_2=0.5, c_3=2, k_1=2, k_2=3. </math>
-\begin{array}{rcl}
-[t_0, t_f] &=& [0, 12],\\
-(c_{0,1}, c_{1,1}) &=& (0.2, 0.1),\\
-(c_{0,2}, c_{1,2}) &=& (0.4, 0.2),\\
-(c_{0,3}, c_{1,3}) &=& (0.01, 0.1).
-\end{array}
-</math>
 == Reference Solutions ==
@@ Line 43: / Line 34: @@
 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.
-The optimal objective value of the relaxed problem with  <math> n_t=12000, \, n_u=400  </math> is <math>x_2(t_f) =1.82875272</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>x_2(t_f) =1.82878681</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>.
 <gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="2">
-  Image:MmlotkaRelaxed_12000_30_1.png| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation)  and <math>n_t=12000, \, n_u=400</math>.
+  Image:MmlotkaRelaxed_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:MmlotkaCIA 12000 30 1.png| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=12000, \, n_u=400</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
+  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.
 </gallery>