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

$\begin{array}{lll}\min {}_{x,y,w}& \underset{t_0}{\overset{t_f}{\int }}& (x(t{)}^2+y(t{)}^{2}dt\\ s.t.& \dot{x}& =y,\\ & \dot{y}& =\sum _{i=1}^3{c}_i{w}_i(1-x^2)y-x,\\ & x(0)& =1,\\ & y(0)& =0,\\ & 1& =\sum _{i=1}^3{w}_i(t),\\ & w_i(t)& \in \{0,1\},i=1\dots 3.\end{array}$

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.

Source Code

Model description is available in

• AMPL code at Van der Pol Oscillator binary variant(AMPL)