PROPT Linear Pendulum - Revision history

Per: /* Plot result */

2012-02-14T09:22:11Z

Plot result

← Older revision		Revision as of 04:22, 14 February 2012
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	[[File:linearPendulum_01.png]]		[[File:linearPendulum_01.png]]

			[[Category:PROPT Examples]]

Mbot at 12:09, 9 November 2011

2011-11-09T12:09:20Z

← Older revision		Revision as of 07:09, 9 November 2011
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	<math> J = x_1(t_f) </math>		<math> J = x_1(t_f) </math>


	subject to:		subject to:

	<math> \frac{dx_1}{dt} = x_2 </math>		<math> \frac{dx_1}{dt} = x_2 </math>

	<math> \frac{dx_2}{dt} = u-x_1 </math>		<math> \frac{dx_2}{dt} = u-x_1 </math>

	<math> x(t_0) = [0 \ 0] </math>		<math> x(t_0) = [0 \ 0] </math>

	<math> \|u\| <= 1 </math>		<math> \|u\| <= 1 </math>


	<source lang="matlab">		<source lang="matlab">

Mbot at 13:06, 8 November 2011

2011-11-08T13:06:19Z

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	<pre>		<pre>
	Problem type appears to be: lp		Problem type appears to be: lp
	Time for symbolic processing: 0.~~03003~~ seconds		Time for symbolic processing: 0.02984 seconds
	Starting numeric solver		Starting numeric solver
	===== * * * =================================================================== * * *		===== * * * =================================================================== * * *
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	FuncEv 206 Iter 206		FuncEv 206 Iter 206
	CPU time: 0.031200 sec. Elapsed time: 0.~~015000~~ sec.		CPU time: 0.031200 sec. Elapsed time: 0.013000 sec.

	</pre>		</pre>

Mbot at 18:24, 2 November 2011

2011-11-02T18:24:18Z

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	{{Part Of Manual\|title=the PROPT Manual\|link=[[PROPT \|PROPT Manual]]}}		{{Part Of Manual\|title=the PROPT Manual\|link=[[PROPT\|PROPT Manual]]}}

	Viscocity Solutions of Hamilton-Jacobi Equations and Optimal Control Problems. Alberto Bressan, S.I.S.S.A, Trieste, Italy.		Viscocity Solutions of Hamilton-Jacobi Equations and Optimal Control Problems. Alberto Bressan, S.I.S.S.A, Trieste, Italy.
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	title('Linear Pendulum known solution');		title('Linear Pendulum known solution');
	</source>		</source>

			[[File:linearPendulum_01.png]]

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2011-11-02T15:29:49Z

Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} Viscocity Solutions of Hamilton-Jacobi Equations and Optimal Control Problems. Alberto Bressan, S.I.S.S...."

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{{Part Of Manual|title=the PROPT Manual|link=[[PROPT |PROPT Manual]]}}

Viscocity Solutions of Hamilton-Jacobi Equations and Optimal Control Problems. Alberto Bressan, S.I.S.S.A, Trieste, Italy.

A linear pendulum problem controlled by an external force.

==Problem Description==

Find u over t in [0; 20 ] to maximize:

<math> J = x_1(t_f) </math>

subject to:

<math> \frac{dx_1}{dt} = x_2 </math>
<math> \frac{dx_2}{dt} = u-x_1 </math>
<math> x(t_0) = [0 \ 0] </math>
<math> |u| <= 1 </math>

<source lang="matlab">
% Copyright (c) 2007-2008 by Tomlab Optimization Inc.
</source>

==Problem setup==

<source lang="matlab">
toms t
t_f = 20;
p = tomPhase('p', t, 0, t_f, 60);
setPhase(p);

tomStates x1 x2
tomControls u

% Initial guess
x0 = {icollocate({x1 == 0; x2 == 0})
collocate(u == 0)};

% Box constraints and bounds
cb = {-1 <= collocate(u) <= 1
initial(x1 == 0)
initial(x2 == 0)};

% ODEs and path constraints
ceq = collocate({dot(x1) == x2
dot(x2) == u-x1});

% Objective
objective = -final(x1);
</source>

==Solve the problem==

<source lang="matlab">
options = struct;
options.name = 'Linear Pendulum';
solution = ezsolve(objective, {cb, ceq}, x0, options);
t = subs(collocate(t),solution);
x1 = subs(collocate(x1),solution);
x2 = subs(collocate(x2),solution);
u = subs(collocate(u),solution);
</source>

<pre>
Problem type appears to be: lp
Time for symbolic processing: 0.03003 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Linear Pendulum f_k -12.612222977985969000
sum(|constr|) 0.000000000002757663
f(x_k) + sum(|constr|) -12.612222977983212000
f(x_0) 0.000000000000000000

Solver: CPLEX. EXIT=0. INFORM=1.
CPLEX Dual Simplex LP solver
Optimal solution found

FuncEv 206 Iter 206
CPU time: 0.031200 sec. Elapsed time: 0.015000 sec.

</pre>

==Plot result==

<source lang="matlab">
subplot(3,1,1)
plot(t,x1,'*-',t,x2,'*-');
legend('x1','x2');
title('Linear Pendulum state variables');

subplot(3,1,2)
plot(t,u,'+-');
legend('u');
title('Linear Pendulum control');

subplot(3,1,3)
plot(t,sign(sin(t_f-t)),'*-');
legend('Known u');
title('Linear Pendulum known solution');
</source>