PROPT Fuller Phenomenon - Revision history

Per: /* Plot result */

2012-02-14T09:03:05Z

Plot result

← Older revision		Revision as of 04:03, 14 February 2012
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	[[File:fullerPhenomenon_01.png]]		[[File:fullerPhenomenon_01.png]]

			[[Category:PROPT Examples]]

Mbot at 12:08, 9 November 2011

2011-11-09T12:08:12Z

← Older revision		Revision as of 07:08, 9 November 2011
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	<math> J = \int_0^{inf} x_1^2 \mathrm{d}t </math>		<math> J = \int_0^{inf} x_1^2 \mathrm{d}t </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 </math>		<math> \frac{dx_2}{dt} = u </math>

	<math> x(t_0) = [10 \ 0 ] </math>		<math> x(t_0) = [10 \ 0 ] </math>

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

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


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

Mbot at 13:05, 8 November 2011

2011-11-08T13:05:27Z

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	<pre>		<pre>
	Problem type appears to be: con		Problem type appears to be: con
	Time for symbolic processing: 0.~~10071~~ seconds		Time for symbolic processing: 0.095364 seconds
	Starting numeric solver		Starting numeric solver
	===== * * * =================================================================== * * *		===== * * * =================================================================== * * *
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	FuncEv 28 GradEv 26 ConstrEv 27 ConJacEv 26 Iter 14 MinorIter 248		FuncEv 28 GradEv 26 ConstrEv 27 ConJacEv 26 Iter 14 MinorIter 248
	CPU time: 0.062400 sec. Elapsed time: 0.~~079000~~ sec.		CPU time: 0.062400 sec. Elapsed time: 0.073000 sec.

	</pre>		</pre>

Mbot at 18:23, 2 November 2011

2011-11-02T18:23:20Z

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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]]}}

	A Short Introduction to Optimal Control, Ugo Boscain, SISSA, Italy		A Short Introduction to Optimal Control, Ugo Boscain, SISSA, Italy
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	title('Fuller Phenomenon control');		title('Fuller Phenomenon control');
	</source>		</source>

			[[File:fullerPhenomenon_01.png]]

Mbot: Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} A Short Introduction to Optimal Control, Ugo Boscain, SISSA, Italy 3.6 Fuller Phenomenon. ==Problem De..."

2011-11-02T15:29:19Z

Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} A Short Introduction to Optimal Control, Ugo Boscain, SISSA, Italy 3.6 Fuller Phenomenon. ==Problem De..."

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

A Short Introduction to Optimal Control, Ugo Boscain, SISSA, Italy

3.6 Fuller Phenomenon.

==Problem Description==

Find u over t in [0; inf ] to minimize:

<math> J = \int_0^{inf} x_1^2 \mathrm{d}t </math>

subject to:

<math> \frac{dx_1}{dt} = x_2 </math>
<math> \frac{dx_2}{dt} = u </math>
<math> x(t_0) = [10 \ 0 ] </math>
<math> x(t_f) = [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
toms t_f
p = tomPhase('p', t, 0, t_f, 60);
setPhase(p);

tomStates x1 x2
tomControls u

% Initial guess
x0 = {t_f == 10
icollocate(x1 == 10-10*t/t_f)
icollocate(x2 == 0)
collocate(u == -1+2*t/t_f)};

% Box constraints
cbox = {1 <= t_f <= 1e4
-1 <= collocate(u) <= 1};

% Boundary constraints
cbnd = {initial({x1 == 10; x2 == 0})
final({x1 == 0; x2 == 0})};

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

% Objective
objective = integrate(x1.^2);
</source>

==Solve the problem==

<source lang="matlab">
options = struct;
options.name = 'Fuller Phenomenon';
options.solver = 'snopt';
solution = ezsolve(objective, {cbox, cbnd, 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: con
Time for symbolic processing: 0.10071 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Fuller Phenomenon f_k 242.423532418144080000
sum(|constr|) 0.000000063716024856
f(x_k) + sum(|constr|) 242.423532481860090000
f(x_0) 333.333333333328370000

Solver: snopt. EXIT=0. INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv 28 GradEv 26 ConstrEv 27 ConJacEv 26 Iter 14 MinorIter 248
CPU time: 0.062400 sec. Elapsed time: 0.079000 sec.

</pre>

==Plot result==

<source lang="matlab">
subplot(2,1,1)
plot(x1,x2,'*-');
legend('x1 vs x2');
title('Fuller Phenomenon state variables');

subplot(2,1,2)
plot(t,u,'+-');
legend('u');
title('Fuller Phenomenon control');
</source>