Optimal control with time varying parameters and fixed control in Gekko

Question

Consider an optimal control problem as follows:

min int(0,1) x(t).u(t)+u(t)**2
x.dt()= x(t)+u(t), x(0)=1
0 <= u(t) <= t**2+(1-t)**3 for 0<=t<=1

My first question is how to define the upper bound of control in Gekko. Also, suppose we want to compare this problem with the case the control is constant during the planning horizon, i.e., u(0)=...=u(t)=...=u(1). How can we define it?

In another case, how is it possible to have fixed but unknown control in different sub-intervals? For example, in [0,t1], control should be fixed, in [t1,t2] control should be fixed but can be different from control in [0,t1] (e.g. t1=0.5, t2=1, Tf=t2=1).

I would be thankful to know if it is possible to study a case where t1, t2, ... are also control and should be determined?

Answer 1

Here is code that gives a solution to the problem:

# min int(0,1) x(t).u(t)+u(t)**2
# x.dt()= x(t)+u(t), x(0)=1
# 0 <= u(t) <= t**2+(1-t)**3 for 0<=t<=1
import numpy as np
from gekko import GEKKO
m = GEKKO(remote=False); m.time=np.linspace(0,1,101)
t = m.Var(0); m.Equation(t.dt()==1)
ub = m.Intermediate(t**2+(1-t)**3)
u = m.MV(0,lb=0,ub=1); m.Equation(u<=ub)
u.STATUS=1; u.DCOST=0; m.free_initial(u)
x = m.Var(0); m.Equation(x.dt()==x+u)
p = np.zeros(101); p[-1]=1; final=m.Param(p)
m.Minimize(m.integral(x*u+u**2)*final)
m.options.IMODE=6; m.options.NODES=3; m.solve()
print(m.options.OBJFCNVAL)

import matplotlib.pyplot as plt
plt.plot(m.time,x.value,'b--',label='x')
plt.plot(m.time,u.value,'k-',label='u')
plt.plot(m.time,ub.value,'r--',label='ub')
plt.legend()
plt.show()

The solution isn't very interesting because the optimal objective is u(t)=0 and x(t)=0. If you add a final condition like x(1)=0.75 then the solution is more interesting.

m.Equation(final*(x-0.75)==0)

If you want all of the interval to be one value then I recommend that you use a u=m.FV() type. The u=m.MV() type is adjustable by the optimizer at every interval when you set u.STATUS=1. You can also reduce degrees of freedom with the m.options.MV_STEP_HOR=5 as a global option in Gekko for all MVs or else u.MV_STEP_HOR=5 to adjust it just for that MV. There is more information on the different Gekko types.

You can set the final time by using m.time = [0,...,1] and then scale it with time final tf. The derivatives in your problem need to be divided by tf as well. Here is a related rocket launch problem or the Jennings optimal control problem that minimize the final time. You can also set up multiple time intervals and then connect them with m.Connection().