PROPT Bridge Crane System - Revision history

Per at 08:54, 14 February 2012

2012-02-14T08:54:41Z

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	[[File:bridgeCraneSystem_01.png]]		[[File:bridgeCraneSystem_01.png]]

			[[Category:PROPT Examples]]

Mbot at 12:06, 9 November 2011

2011-11-09T12:06:37Z

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	<math> J = t_F </math>		<math> J = 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 </math>		<math> \frac{dx_2}{dt} = u </math>

	<math> \frac{dx_3}{dt} = x_4 </math>		<math> \frac{dx_3}{dt} = x_4 </math>

	<math> \frac{dx_4}{dt} = -0.98x_3 + 0.1u </math>		<math> \frac{dx_4}{dt} = -0.98x_3 + 0.1u </math>


	The initial condition are:		The initial condition are:

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

	<math> x(t_F) = [15 \ 0 \ 0 \ 0] </math>		<math> x(t_F) = [15 \ 0 \ 0 \ 0] </math>

	<math> -1 <= u <= 1 </math>		<math> -1 <= u <= 1 </math>


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

Mbot at 13:03, 8 November 2011

2011-11-08T13:03:52Z

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	<pre>		<pre>
	Problem type appears to be: lpcon		Problem type appears to be: lpcon
	Time for symbolic processing: 0.~~14777~~ seconds		Time for symbolic processing: 0.13109 seconds
	Starting numeric solver		Starting numeric solver
	===== * * * =================================================================== * * *		===== * * * =================================================================== * * *
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	FuncEv 1 ConstrEv 37 ConJacEv 37 Iter 19 MinorIter 501		FuncEv 1 ConstrEv 37 ConJacEv 37 Iter 19 MinorIter 501
	CPU time: 0.171601 sec. Elapsed time: 0.~~176000~~ sec.		CPU time: 0.171601 sec. Elapsed time: 0.168000 sec.

	</pre>		</pre>

Mbot at 18:22, 2 November 2011

2011-11-02T18:22:18Z

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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]]}}

	ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS		ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS
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	title('Bridge Crane System control');		title('Bridge Crane System control');
	</source>		</source>

			[[File:bridgeCraneSystem_01.png]]

Mbot at 14:54, 2 November 2011

2011-11-02T14:54:23Z

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

	ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS		ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS
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	<math> -1 <= u <= 1 </math>		<math> -1 <= u <= 1 </math>

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

	==Problem setup==		==Problem setup==

	<~~syntaxhighlight~~ lang="matlab">		<source lang="matlab">
	toms t		toms t
	toms t_f		toms t_f
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	% Objective		% Objective
	objective = t_f;		objective = t_f;
	</~~syntaxhighlight~~>		</source>

	==Solve the problem==		==Solve the problem==

	<~~syntaxhighlight~~ lang="matlab">		<source lang="matlab">
	options = struct;		options = struct;
	options.name = 'Bridge Crane System';		options.name = 'Bridge Crane System';
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	x4 = subs(collocate(x4),solution);		x4 = subs(collocate(x4),solution);
	u = subs(collocate(u),solution);		u = subs(collocate(u),solution);
	</~~syntaxhighlight~~>		</source>

	<pre>		<pre>
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	==Plot result==		==Plot result==

	<~~syntaxhighlight~~ lang="matlab">		<source lang="matlab">
	subplot(2,1,1)		subplot(2,1,1)
	plot(t,x1,'-',t,x2,'-',t,x3,'-',t,x4,'-');		plot(t,x1,'-',t,x2,'-',t,x3,'-',t,x4,'-');
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	legend('u');		legend('u');
	title('Bridge Crane System control');		title('Bridge Crane System control');
	</~~syntaxhighlight~~>		</source>

Mbot: Created page with " ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS 12.4.1 Example 1: Bridge crane system CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics ==Problem description..."

2011-11-02T11:42:06Z

Created page with " ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS 12.4.1 Example 1: Bridge crane system CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics ==Problem description..."

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ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS

12.4.1 Example 1: Bridge crane system

CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics

==Problem description==

Find u over t in [0; t_F ] to minimize

<math> J = t_F </math>

subject to:

<math> \frac{dx_1}{dt} = x_2 </math>
<math> \frac{dx_2}{dt} = u </math>
<math> \frac{dx_3}{dt} = x_4 </math>
<math> \frac{dx_4}{dt} = -0.98*x_3 + 0.1*u </math>

The initial condition are:

<math> x(0) = [0 \ 0 \ 0 \ 0] </math>
<math> x(t_F) = [15 \ 0 \ 0 \ 0] </math>
<math> -1 <= u <= 1 </math>

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

==Problem setup==

<syntaxhighlight lang="matlab">
toms t
toms t_f

p = tomPhase('p', t, 0, t_f, 50);
setPhase(p);

tomStates x1 x2 x3 x4
tomControls u

% Initial guess
% Note: The guess for t_f must appear in the list before expression involving t.
x0 = {t_f == 8, ...
collocate(u==1-2*t/t_f)};

% Box constraints
cbox = {0.1 <= t_f <= 100
-1 <= collocate(u) <= 1};

% Boundary constraints
cbnd = {initial({x1 == 0; x2 == 0
x3 == 0; x4 == 0})
final({x1 == 15; x2 == 0
x3 == 0; x4 == 0})};

% ODEs and path constraints
ceq = collocate({
dot(x1) == x2
dot(x2) == u
dot(x3) == x4
dot(x4) == -0.98*x3+0.1*u});

% Objective
objective = t_f;
</syntaxhighlight>

==Solve the problem==

<syntaxhighlight lang="matlab">
options = struct;
options.name = 'Bridge Crane System';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
t = subs(collocate(t),solution);
x1 = subs(collocate(x1),solution);
x2 = subs(collocate(x2),solution);
x3 = subs(collocate(x3),solution);
x4 = subs(collocate(x4),solution);
u = subs(collocate(u),solution);
</syntaxhighlight>

<pre>
Problem type appears to be: lpcon
Time for symbolic processing: 0.14777 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Bridge Crane System f_k 8.578933610367172900
sum(|constr|) 0.000000187956526363
f(x_k) + sum(|constr|) 8.578933798323699700
f(x_0) 8.000000000000000000

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

FuncEv 1 ConstrEv 37 ConJacEv 37 Iter 19 MinorIter 501
CPU time: 0.171601 sec. Elapsed time: 0.176000 sec.

</pre>

==Plot result==

<syntaxhighlight lang="matlab">
subplot(2,1,1)
plot(t,x1,'*-',t,x2,'*-',t,x3,'*-',t,x4,'*-');
legend('x1','x2','x3','x4');
title('Bridge Crane System state variables');

subplot(2,1,2)
plot(t,u,'+-');
legend('u');
title('Bridge Crane System control');
</syntaxhighlight>