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Integral Objective (Luus)

Main.IntegralObjective History

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(:toggle hide gekko button show="GEKKO (Python) Solution":)
(:div id=gekko:)
(:source lang=python:)
import numpy as np
import matplotlib.pyplot as plt
from gekko import GEKKO

m = GEKKO()

nt = 101
m.time = np.linspace(0,2,nt)

# Variables
x1 = m.Var(value=1)
x2 = m.Var(value=0)
u = m.Var(value=0,lb=-1,ub=1)

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

# Equations
m.Equation(x1.dt()==u)
m.Equation(x2.dt()==0.5*x1**2)

# Objective Function
m.Obj(x2*final)

m.options.IMODE = 6
m.solve()
# m.solve(remote=False)  # for local solve

plt.figure(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,u.value,'r--',LineWidth=2,label=r'$u$')
plt.legend(loc='best')
plt.xlabel('Time')
plt.ylabel('Value')
plt.show()
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{$\min_u \frac{1}{2} \int_0^2 x_1(t) \, dt$}
to:
{$\min_u \frac{1}{2} \int_0^2 x_1^2(t) \, dt$}
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{$x_2 = \frac{1}{2} \int_0^2 x_1(t) \, dt$}
to:
{$x_2 = \frac{1}{2} \int_0^2 x_1^2(t) \, dt$}
Changed line 24 from:
{$\frac{dx_2}{dt} = \frac{1}{2} x_1(t)$}
to:
{$\frac{dx_2}{dt} = \frac{1}{2} x_1^2(t)$}
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The initial condition of the integral starts at zero and becomes the integral in the time range of 0 to 2 at the final point for the optimal control problem.
to:
The initial condition of the integral starts at zero and becomes the integral in the time range of 0 to 2. The value that is minimized is at the final point in the time horizon of the optimal control problem.
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{$x(0) = 1$}
to:
{$x_1(0) = 1$}
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{$x(0) = 1$}
to:
{$x_1(0) = 1$}
{$x_2(0) = 0
$}
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The initial condition of the integral starts at zero and becomes the integral in the time range of 0 to 2 at the final point for the optimal control problem.
Changed line 17 from:
{$x_2 = \frac{1}{2} \int_0^2 x(t) \, dt$}
to:
{$x_2 = \frac{1}{2} \int_0^2 x_1(t) \, dt$}
Changed line 24 from:
{$\frac{dx_2}{dt} = \frac{1}{2} x_1(t)
to:
{$\frac{dx_2}{dt} = \frac{1}{2} x_1(t)$}
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(:title Integral Objective (Luus):)
(:keywords nonlinear control, Luus, optimal control, dynamic optimization, engineering optimization, MATLAB, Python, GEKKO, optimization suite:)
(:description Luus optimal control problem solved with MATLAB, Python, and GEKKO.:)

'''Objective:''' Set up and solve the Luus optimal control benchmark problem. Create a program to optimize and display the results. ''Estimated Time: 30 minutes''

The Luus optimal control problem has an integral objective function. Integrals are a natural expression of many systems where the accumulation of a quantity is maximized or minimized.

{$\min_u \frac{1}{2} \int_0^2 x_1(t) \, dt$}
{$\mathrm{subject \; to}$}
{$\frac{dx_1}{dt}=u$}
{$x(0) = 1$}
{$-1 \le u(t) \le 1$}

Integral expressions are reformulated in differential and algebraic form by defining a new variable.

{$x_2 = \frac{1}{2} \int_0^2 x(t) \, dt$}

The new variable and integral are differentiated and included as an additional equation. The problem then becomes a minimization of the new variable {`x_2`} at the final time.

{$\min_u x_2\left(t_f\right)$}
{$\mathrm{subject \; to}$}
{$\frac{dx_1}{dt}=u$}
{$\frac{dx_2}{dt} = \frac{1}{2} x_1(t)
{$x(0) = 1$}
{$t_f = 2$}
{$-1 \le u(t) \le 1$}

!!!! Solutions

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Attach:download.png [[Attach:dynamic_optimization_luus.zip|Luus Problem in MATLAB and Python]]

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!!!! References

# Beal, L., GEKKO for Dynamic Optimization, 2018, URL: http://gekko.readthedocs.io/en/latest/

# 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. [[http://dx.doi.org/10.1016/j.compchemeng.2014.04.013|Article]]