Time optimal car problem (GEKKO)

This page contains a solution of the Time optimal car problem in GEKKO Python format. The GEKKO package is available with pip install gekko. The Python code uses orthogonal collocation and a simultaneous optimization method. The end point constraints are imposed as hard constraints.

from gekko import GEKKO
import matplotlib.pyplot as plt
import numpy as np
 
# set up the gekko model
m = GEKKO()
 
# set up the time (minimize the time with time scaling)
m.time = np.linspace(0, 1, 100)
 
# set up the variables
Z1 = m.Var(value=0, ub=330, lb=0)
Z2 = m.Var(value=0, ub=33, lb=0)
m.fix(Z2, len(m.time)-1, 0)
m.fix(Z1, len(m.time)-1, 300)
 
# set up the value we modify over the horizon
tf = m.FV(value=500, lb=0.1)
tf.STATUS = 1
 
# set up the MV
u = m.MV(integer=True, lb=-2, ub=1)
u.STATUS = 1
 
# set up the equations
m.Equation(Z1.dt() / tf == Z2)
m.Equation(Z2.dt() / tf == u)
 
# set the objective
m.Obj(tf)
 
# set up the options
m.options.IMODE = 6
m.options.SOLVER = 1
 
# solve
m.solve(disp=False)
 
# print the time
print("Total time taken: " + str(tf.NEWVAL))
 
# plot the results
plt.figure()
plt.subplot(211)
plt.plot(np.linspace(0,1,100)*tf.NEWVAL, Z1, label=r'$Z_1$')
plt.plot(np.linspace(0,1,100)*tf.NEWVAL, Z2, label=r'$Z_2$')
plt.ylabel('Z')
plt.legend()
plt.subplot(212)
plt.plot(np.linspace(0,1,100)*tf.NEWVAL, u, label=r'$u$')
plt.ylabel('u')
plt.xlabel('Time')
plt.legend()
plt.show()

Results with APOPT (MINLP)

An MINLP solution is calculated with APOPT with an objective function value of $30.3077$. Thanks to Tanner Jasperson for providing a solution.