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:

$\begin{array}{llclr} \displaystyle \max_{x, u} & x_3(t_f) \\[1.5ex] \mbox{s.t.} & \dot{x}_1(t) & = & (1 - x_2(t)^2) x_1(t) - x_2(t) + u(t) \qquad & \forall t \in [t_0, t_f], \\ & \dot{x}_2(t) & = & x_1(t) \qquad & \forall t \in [t_0, t_f], \\ & \dot{x}_3(t) & = & x_1(t)^2 + x_2(t)^2 + u(t)^2 \qquad & \forall t \in [t_0, t_f], \\[1.5ex] & x(0) &=& (0, 1, 0)^T, \\ & u(t) &\in& [-0.3, 1] \qquad & \forall t \in [t_0, t_f]. \end{array}$

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).