Van der Pol Oscillator (binary variant)

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

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

$\begin{array}{lll} \min\limits_{x,y,w} & \int\limits_{t_0}^{t_f} & (x(t)^2+y(t)^2+u(t)^2) dt\\ s.t. & \dot x & = y,\\ & \dot y & = \sum\limits_{i=1}^{3} c_{i}\; w_i \;(1-x^2) y-x,\\ & x(0) & =1,\\ & y(0) & =0,\\ & 1 &= \sum\limits_{i=1}^{3}w_i(t), \\ & w_i(t) &\in \{0, 1\}, \quad i=1\ldots 3. \end{array}$

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 $n_t=12000, \, n_u=400$ is $x_2(t_f) =1.82875272$. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is $x_2(t_f) =1.82878681$.

• Reference solution plots
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  Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and $n_t=12000, \, n_u=400$.

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  Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and $n_t=12000, \, n_u=400$. The relaxed controls were approximated by Combinatorial Integral Approximation.