Van der Pol Oscillator (Jump)

This is an implementation of a slightly modified form of the Van der Pol Oscillator problem using JuMP.

The problem in question can be stated as follows:

$\begin{array}{llclr} \max_{x, u} & x_{3} (t_{f}) \\ s.t. & {\dot{x}}_{1} & = & (1 - x_{2}^{2}) x_{1} - x_{2} + u, \\ {\dot{x}}_{2} & = & x_{1}, \\ {\dot{x}}_{3} & = & x_{1}^{2} + x_{2}^{2} + u^{2}, \\ x (0) & = & (0, 1, 0)^{T}, \\ u (t) & \in & [- 0.3, 1] . \end{array}$

where $[t_{0}, t_{f}] = [0, 5]$ .

The problem was discretized and the ODEs are solved using the explicit Euler method. Although not necessary in JuMP the code was divided into three parts (following AMPL) - model file, data file and run file. The run file calls the other files and performs additional tasks such as printing results.

A reference solution including plots done with JuMP can be found below the code!

Model file ("vdposc_mod.jl"):

#JuMP implementation of Van der Pol oscillator using collocation
#mod file

#declaring the model
m = Model(solver=IpoptSolver())

#defining variables
@defVar(m, x[ii=1:n_x, tt=1:N])
@defVar(m, L_control <= u[jj = 1:n_u, tt=1:N] <= U_control)

#set objective function
@setObjective(m, Min, x[3,N])

#setting constraints
#starting values
@addConstraint(m, starting_value[ii=1:n_x], x[ii,1] == x_start[ii])

#ODE - solved with explicit euler method (i.e. x_k+1 = x_k + stepsize * f(x_k, t_k))
@addNLConstraint(m, ODE_nonlin[ii=1:1, tt=1:N-1],  x[ii,tt+1] - x[ii,tt] - step_size * ((1 - x[2,tt]^2) * x[1,tt]
 - x[2,tt] + u[1,tt]) == 0)
@addConstraint(m, ODE[ii=2:n_x, tt=1:N-1],  x[ii,tt+1] - x[ii,tt] - step_size * ode_rhs(time_disc[tt], x[:,tt], u[:,tt])[ii]  >= 0)
@addConstraint(m, ODE[ii=2:n_x, tt=1:N-1],  x[ii,tt+1] - x[ii,tt] - step_size * ode_rhs(time_disc[tt], x[:,tt], u[:,tt])[ii]  <= 0)

Data file ("vdposc_dat.jl"):

#JuMP implementation of Van der Pol oscillator using collocation
#dat file

#number of states
n_x = 3;

#number of controls
n_u = 1;

##discretization
#number of shooting intervals / discretization points
N = 300;
#starting / end time
t_start = 0;
t_end = 5;
#time discretization
time_disc = linspace(t_start,t_end, N+1);
step_size = (t_end - t_start)/N;

#starting value
x_start = [0, 1, 0];

#bounds for control
L_control = -0.3;
U_control = 1;


##right hand side of ODE
function ode_rhs(time, state, control)
#give in form f1, f2, f3,...
0,
state[1],
state[1]^2 + state[2]^2 + control[1]^2
end

Run file ("vdposc_run.jl"):

#JuMP implementation of Van der Pol oscillator using collocation
#run file

using JuMP;
using Ipopt;


println("----------------------------------------------------")
println("Time used for data")
@time include("vdposc_dat.jl")
println("----------------------------------------------------")
println("Time used for modeling")
@time include("vdposc_mod.jl")
println("----------------------------------------------------")
println("Time used for solving")
@time solve(m);

println("----------------------------------------------------")
println("----------------------------------------------------")


println("Optimal objective value is: ", getObjectiveValue(m))
println("Optimal Solution is: \n", getValue(x), getValue(u))

Reference solutions

This solution was computed using JuMP with a collocation method and 300 discretization points (using the code above). The differential equations were solved using the explicit Euler Method. The optimal objective value is 2.902143.

Reference solution plots
Optimal states x1(t), x2(t) and x3(t).
Optimal control u(t).