Bryson-Denham Problem

Apps.BrysonDenhamProblem History

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July 05, 2020, at 03:10 AM by 12.217.227.229 -
Added lines 7-10:
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Deleted lines 27-30:

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July 05, 2020, at 03:09 AM by 173.166.196.185 -
Added lines 24-27:

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July 05, 2020, at 01:33 AM by 173.166.196.185 -
Changed line 34 from:
m = GEKKO()
to:
m = GEKKO(remote=False)
Changed lines 50-52 from:
m.Minimize(final*1e5*x**2)
m.Minimize(final*1e5*(v+1)**2)
to:
soft = True
if soft:
    # soft terminal constraint
   
m.Minimize(final*1e5*x**2)
   m.Minimize(final*1e5*(v+1)**2)
else:
    # hard terminal constraint
    xf = m.Param();    vf = m.Param()
    m.free(xf);        m.free(vf)
    m.fix_final(xf,0); m.fix_final(vf,-1)
    # connect endpoint parameters to x and v
    m.Equations([xf==x,vf==v]
)
Changed lines 71-72 from:
plt.figure(figsize=(10,6))
to:
# Create a figure
plt.figure(figsize=(10,4))
June 21, 2020, at 01:56 PM by 136.36.211.159 -
Changed line 40 from:
u = m.Var(value=-6,lb=-6)
to:
u = m.Var(value=-6)
June 21, 2020, at 01:55 PM by 136.36.211.159 -
Added lines 27-28:
%width=550px%Attach:bryson_denham_solution.png
Deleted lines 84-86:
Attach:download.png [[Attach:bryson_denham_solution.zip|Download Bryson-Denham Solution in Python]]

%width=550px%Attach:bryson_denham_solution.png
June 21, 2020, at 01:54 PM by 136.36.211.159 -
Changed lines 27-32 from:
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to:
(:source lang=python:)
import numpy as np
import matplotlib.pyplot as plt
from gekko import GEKKO

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

# Variables
x = m.Var(value=0,ub=1/9)
v = m.Var(value=1)
u = m.Var(value=-6,lb=-6)

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

# Equations
m.Equation(x.dt()==v)
m.Equation(v.dt()==u)

# Final conditions
m.Minimize(final*1e5*x**2)
m.Minimize(final*1e5*(v+1)**2)

# Objective Function
obj = m.Intermediate(0.5*m.integral(u**2))
m.Minimize(final*obj)

m.options.IMODE = 6
m.options.NODES = 2
m.solve()

plt.figure(figsize=(10,6))
plt.subplot(2,2,1)
plt.plot([0,1],[1/9,1/9],'r:',label=r'$x<\frac{1}{9}$')
plt.plot(m.time,x.value,'k-',lw=2,label=r'$x$')
plt.ylabel('Position')
plt.legend(loc='best')
plt.subplot(2,2,2)
plt.plot(m.time,v.value,'b--',lw=2,label=r'$v$')
plt.ylabel('Velocity')
plt.legend(loc='best')
plt.subplot(2,2,3)
plt.plot(m.time,u.value,'r--',lw=2,label=r'$u$')
plt.ylabel('Thrust')
plt.legend(loc='best')
plt.xlabel('Time')
plt.subplot(2,2,4)
plt.plot(m.time,obj.value,'g-',lw=2,label=r'$\frac{1}{2} \int u^2$')
plt.text(0.5,3.0,'Final Value = '+str(np.round(obj.value[-1],2)))
plt.ylabel('Objective')
plt.legend(loc='best')
plt.xlabel('Time')
plt.show()
(:sourceend:)

Changed line 85 from:
%width=350px%Attach:bryson_denham_solution.png
to:
%width=550px%Attach:bryson_denham_solution.png
June 21, 2020, at 05:04 AM by 136.36.211.159 -
Deleted lines 35-54:


----

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Changed line 9 from:
The parameter ''u'' is the acceleration can be adjusted over the time horizon from a starting time of zero to a final time of one. The variable ''x'' is the position and ''v'' is the velocity.
to:
The parameter ''u'' (acceleration) is adjusted over the time horizon from a starting time of zero to a final time of one. The variable ''x'' is the position and ''v'' is the velocity.
Added lines 24-36:

!!!! Solution

(:html:)
<!--
<iframe width="560" height="315" src="https://www.youtube.com/embed/pgJ0jbfFBUE" frameborder="0" allowfullscreen></iframe>
-->
(:htmlend:)

Attach:download.png [[Attach:bryson_denham_solution.zip|Download Bryson-Denham Solution in Python]]

%width=350px%Attach:bryson_denham_solution.png

Changed line 9 from:
The parameter ''u'' is the acceleration can be adjusted over the time horizon from a starting time of zero to a time of one. The variable ''x'' is the position and ''v'' is the velocity.
to:
The parameter ''u'' is the acceleration can be adjusted over the time horizon from a starting time of zero to a final time of one. The variable ''x'' is the position and ''v'' is the velocity.
Changed lines 15-23 from:
{$\quad \frac{dx(t)}{dt} = v $}

{$\quad \frac{dv(t)}{dt} = u$}

{$\quad x(0) \; = \; x(1) \; = \; 0$}

{$\quad v(0) \; = \; -v(1) \; = \; 1$}

{$\quad x(t) \le \ell, \ell=\frac{1}{9}$}
to:
{$\frac{dx(t)}{dt} = v(t) $}

{$\frac{dv(t)}{dt} = u(t) $}

{$x(0) \; = \; x(1) \; = \; 0$}

{$v(0) \; = \; -v(1) \; = \; 1$}

{$x(t) \le \ell, \; \ell=\frac{1}{9}$}
Added lines 1-42:
(:title Bryson-Denham Problem:)
(:keywords Benchmark, Python, nonlinear control, dynamic programming, optimal control:)
(:description Minimize the integral of the control input while meeting certain path and final time constraints.:)

The Bryson-Denham optimal control problem is a benchmark test problem for optimal control algorithms.

!!!! Problem Statement

The parameter ''u'' is the acceleration can be adjusted over the time horizon from a starting time of zero to a time of one. The variable ''x'' is the position and ''v'' is the velocity.

{$\min J = \frac{1}{2} \; \int_0^1 u^2(t) dt$}

{$\mathrm{subject\;to}$}

{$\quad \frac{dx(t)}{dt} = v $}

{$\quad \frac{dv(t)}{dt} = u$}

{$\quad x(0) \; = \; x(1) \; = \; 0$}

{$\quad v(0) \; = \; -v(1) \; = \; 1$}

{$\quad x(t) \le \ell, \ell=\frac{1}{9}$}

----

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