Van der Pol Oscillator (binary variant): Difference between revisions

Van der Pol Oscillator (binary variant)
State dimension:	1
Differential states:	2
Discrete control functions:	3
Interior point equalities:	2

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Revision as of 16:58, 10 January 2018

This site describes a Van der Pol Oscillator variant with three binary controls instead of only one continuous control.

Mathematical formulation

The mixed-integer optimal control problem is given by

$\begin{array}{lll} \min_{x, y, w} & \int_{t_{0}}^{t_{f}} & (x (t)^{2} + y (t)^{2} d t \\ s . t . & \dot{x} & = y, \\ \dot{y} & = \sum_{i = 1}^{3} c_{i} w_{i} (1 - x^{2}) y - x, \\ x (0) & = 1, \\ y (0) & = 0, \\ 1 & = \sum_{i = 1}^{3} w_{i} (t), \\ w_{i} (t) & \in {0, 1}, i = 1 \dots 3 . \end{array}$

Parameters

These fixed values are used within the model:

$[t_{0}, t_{f}] = [0, 20], c_{1} = - 1, c_{2} = 0.75, c_{3} = - 2 .$

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 $n_{t} = 6000, n_{u} = 60$ is $1.30167235$ . The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is $1.30273681$ .

Reference solution plots
Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and $n_{t} = 6000, n_{u} = 60$ .
Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and $n_{t} = 6000, n_{u} = 60$ . The relaxed controls were approximated by Combinatorial Integral Approximation.

@@ Line 14: / Line 14: @@
 <math>
 \begin{array}{lll}
-\min\limits_{x,y,w}  & \int\limits_{t_0}^{t_f} & (x(t)^2+y(t)^2+u(t)^2) dt\\
+\min\limits_{x,y,w}  & \int\limits_{t_0}^{t_f} & (x(t)^2+y(t)^2 dt\\
 s.t. & 	 \dot x & = y,\\
 &	\dot y & =  \sum\limits_{i=1}^{3} c_{i}\;  w_i \;(1-x^2) y-x,\\
@@ Line 32: / Line 32: @@
 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=6000, \, n_u=60  </math> is <math>1.30167235</math>. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is <math>1.30273681</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:VanderpolCIA_6000_100_1.pdf| Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation)  and <math>n_t=6000, \, n_u=60</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:MmlotkaCIA 12000 30 1.png| Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and <math>n_t=6000, \, n_u=60</math>. The relaxed controls were approximated by Combinatorial Integral Approximation.
 </gallery>