GEKKO for minimum time solution optimal control problem - controls

Question

This is a standard benchmark problem for minimum flight time.
This is a very standard problem. I am trying to solve it in gekko, but it is neither converging to local minima nor global, here is the code. I followed the set up from the Jennings problem but still, if anybody can help, that would be very nice.
from gekko import GEKKO
import numpy as np
import matplotlib.pyplot as plt
import math
m = GEKKO()
nt = 501
tm = np.linspace(0,1,nt)
m.time = tm
x1=m.Var(value=-2.5)
x2=m.Var(value=0)
u=m.MV(value=1,lb=0,ub=2*math.pi)
p = np.zeros(nt)
p[-1] = 1.0
final = m.Param(value=p)
tf = m.FV(value=0,lb=0.1,ub=100.0)
tf.STATUS = 1
if x2.value>1:
m.Equation(x1.dt()==((1+(x2-1)**2)*m.cos(u)*tf))
m.Equation(x2.dt()==((1+(x2-1)**2)*m.sin(u)*tf))
else:
m.Equation(x1.dt()==(m.cos(u)*tf))
m.Equation(x2.dt()==(m.sin(u)*tf))
#m.Equation(x1*final<=3)
#m.Equation(x2*final<=0)
m.Minimize(tf)
m.options.IMODE = 6
m.solve()
tm = tm * tf.value[0]
plt.figure(1)
plt.plot(tm,x1.value,'k-',lw=2,label=r'$x_1$')
plt.plot(tm,x2.value,'b-',lw=2,label=r'$x_2$')
plt.plot(tm,u.value,'r--',lw=2,label=r'$u$')
plt.legend(loc='best')
plt.xlabel('Time')
plt.ylabel('Value')
plt.show()

Answer 1

Use the m.if3() function for the conditional statement. Here is the local solution that they discussed on pg 332 of the Cristiani and Martinon publication.
from gekko import GEKKO
import numpy as np
import matplotlib.pyplot as plt
import math
m = GEKKO()
nt = 101; pi = math.pi
tm = np.linspace(0,1,nt); m.time = tm
x1=m.Var(value=-2.5,lb=-100,ub=100)
x2=m.Var(value=0,lb=-100,ub=100)
u=m.MV(value=0,lb=-pi,ub=pi); u.STATUS=1; u.DCOST=0.1
p = np.zeros(nt); p[-1] = 1.0
final = m.Param(value=p)
tf = m.FV(value=10,lb=0.1,ub=100.0); tf.STATUS = 1
c = m.if3(x2-1,1,(x2-1)**2+1)
m.Equation(x1.dt()==c*m.cos(u)*tf)
m.Equation(x2.dt()==c*m.sin(u)*tf)
# hard constraints (fix endpoint)
#m.fix_final(x1,3)
#m.fix_final(x2,0)
# soft constraints (objective)
m.Minimize(100*final*(x1-3)**2)
m.Minimize(100*final*(x2-0)**2)
# minimize final time
# initialize with IPOPT Solver
m.Minimize(tf)
m.options.IMODE = 6
m.options.SOLVER=3
m.solve()
# find MINLP solution with APOPT Solver
m.options.SOLVER=1
m.options.TIME_SHIFT=0
m.solve()
tm = tm * tf.value[0]
plt.figure(figsize=(8,5))
plt.plot(tm,x1.value,'k-',lw=2,label=r'$x_1$')
plt.plot(tm,x2.value,'b-',lw=2,label=r'$x_2$')
plt.plot(tm,u.value,'r--',lw=2,label=r'$u$')
plt.legend(loc='best'); plt.grid()
plt.xlabel('Time'); plt.ylabel('Value')
plt.savefig('results.png',dpi=300); plt.show()
The global solution is shown in the paper.
The solvers in Gekko (APOPT, BPOPT, IPOPT) are local solvers. You need to add constraints or use different initial guess values to find the global optimum.