More Optimal Control Problems

Exercise

Objective: Set up and solve several optimal control benchmark problems. Create a program to optimize and display the results. Estimated Time (each): 30 minutes


Aly-Chan Singular Control Problem

  • Solve the following nonlinear and constrained problem.
  • The objective is to minimize final state x3(pi/2) by adjusting the value of u.
  • Compare to the exact solution of u(t)= -sin(t).

import numpy as np
import matplotlib.pyplot as plt
from gekko import GEKKO

m = GEKKO()

nt = 1001
t = np.linspace(0,np.pi/2,nt)
m.time = t

# Variables
x1 = m.Var(value=0)
x2 = m.Var(value=1)
x3 = m.Var(value=0)

u = m.MV(value=0,ub=1,lb=-1)
u.STATUS = 1
u.DCOST = 0

p = np.zeros(nt)
p[-1] = 1.0
final = m.Param(value=p)

# Equations
m.Equation(x1.dt()==x2)
m.Equation(x2.dt()==u)
m.Equation(2*x3.dt()==x2**2-x1**2)

# Objective Function
m.Obj(x3*final)

m.options.IMODE = 6
m.options.NODES = 4
m.solve()

plt.figure(1)
plt.subplot(2,1,1)
plt.plot(m.time,x1.value,'k:',LineWidth=2,label=r'$x_1$')
plt.plot(m.time,x2.value,'b-',LineWidth=2,label=r'$x_2$')
plt.plot(m.time,x2.value,'k-',LineWidth=2,label=r'$x_3$')
plt.subplot(2,1,2)
plt.plot(m.time,u.value,'r--',LineWidth=2,label=r'$u$')
plt.plot(t,-np.sin(t),'k:',LineWidth=2,label='Exact')
plt.legend(loc='best')
plt.xlabel('Time')
plt.ylabel('Value')
plt.show()

Aly Singular Control Problem

See estimation example using the same model to explain estimator objectives.


Catalyzed Reaction


Maximize Profit (Commercial Fishery)


Minimize Final Time (Jennings)


Integral Objective (Luus)


Bryson-Denham Problem


References

  1. Beal, L.D.R., Hill, D., Martin, R.A., and Hedengren, J. D., GEKKO Optimization Suite, Processes, Volume 6, Number 8, 2018, doi: 10.3390/pr6080106. Article
  2. Hedengren, J. D. and Asgharzadeh Shishavan, R., Powell, K.M., and Edgar, T.F., Nonlinear Modeling, Estimation and Predictive Control in APMonitor, Computers and Chemical Engineering, Volume 70, pg. 133–148, 2014. Article
  3. Aly G.M. and Chan W.C. Application of a modified quasilinearization technique to totally singular optimal problems. International Journal of Control, 17(4): 809-815, 1973.