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Python GEKKO Solution

The GEKKO package is available through the package manager pip in Python.

 python -m pip install gekko

GEKKO Python is designed for large-scale optimization and accesses solvers of constrained, unconstrained, continuous, and discrete problems.

(:source lang=python:) import numpy as np import matplotlib.pyplot as plt from gekko import GEKKO

1. create GEKKO model

m = GEKKO()

1. scale 0-1 time with tf

m.time = np.linspace(0,1,101)

1. options

m.options.NODES = 6 m.options.SOLVER = 3 m.options.IMODE = 6 m.options.MAX_ITER = 500 m.options.MV_TYPE = 0 m.options.DIAGLEVEL = 0

1. final time

tf = m.FV(value=1.0,lb=0.1,ub=100) tf.STATUS = 1

1. force

u = m.MV(value=0,lb=-1.1,ub=1.1) u.STATUS = 1 u.DCOST = 1e-5

1. variables

s = m.Var(value=0) v = m.Var(value=0,lb=0,ub=1.7) mass = m.Var(value=1,lb=0.2)

1. differential equations scaled by tf

m.Equation(s.dt()==tf*v) m.Equation(mass*v.dt()==tf*(u-0.2*v**2)) m.Equation(mass.dt()==tf*(-0.01*u**2))

1. specify endpoint conditions

m.fix(s, pos=len(m.time)-1,val=10.0) m.fix(v, pos=len(m.time)-1,val=0.0)

1. minimize final time

m.Obj(tf)

1. Optimize launch

m.solve()

print('Optimal Solution (final time): ' + str(tf.value[0]))

1. scaled time

ts = m.time * tf.value[0]

1. plot results

plt.figure(1) plt.subplot(4,1,1) plt.plot(ts,s.value,'r-',linewidth=2) plt.ylabel('Position') plt.legend(['s (Position)'])

plt.subplot(4,1,2) plt.plot(ts,v.value,'b-',linewidth=2) plt.ylabel('Velocity') plt.legend(['v (Velocity)'])

plt.subplot(4,1,3) plt.plot(ts,mass.value,'k-',linewidth=2) plt.ylabel('Mass') plt.legend(['m (Mass)'])

plt.subplot(4,1,4) plt.plot(ts,u.value,'g-',linewidth=2) plt.ylabel('Force') plt.legend(['u (Force)'])

plt.xlabel('Time') plt.show() (:sourceend:)

Problem Statement

 minimize tf

 subject to
   ds/dt = v
   dv/dt = (u-0.2*v^2)/m
   dm/dt = -0.01 * u^2

 path constraints
   0.0 <= v(t) <= 1.7
   -1.1 <= u(t) <= 1.1

 initial boundary conditions
   s(0) = 0
   v(0) = 0
   m(0) = 1

 final boundary conditions
   s(tf) = 10.0
   v(tf) = 0.0

Solution

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(:title Rocket Launch: Classic Optimal Control:) (:keywords Python, MATLAB, nonlinear control, Rocket, Goddard, model predictive control, dynamic programming:) (:description Minimize final time for rocket launch by manipulating the force exerted by the thruster. This is a classic dynamic optimization problem benchmark used in many research papers to test the application of new algorithms.:)

A rocket burn trajectory is desired to minimize a travel time between a starting point and a final point, 10 units of distance away. The thrust can be between an upper limit of 1.1 and a lower limit of -1.1. The initial and final velocity must be zero and the maximum velocity can never exceed 1.7. It is also desirable to minimize the use of fuel to perform the maneuver. There is a drag resistance the is proportional to the square of the velocity and mass is lost as the fuel is burned during thrust operations.

Download Rocket Launch Planning Solution in MATLAB and Python

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