PROPT Quadruple Integral - Revision history

Per: /* Plot result */

2012-02-14T09:31:26Z

Plot result

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	[[File:quadrupleIntegral_01.png]]		[[File:quadrupleIntegral_01.png]]

			[[Category:PROPT Examples]]

Mbot at 12:11, 9 November 2011

2011-11-09T12:11:31Z

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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} = x_3 </math>		<math> \frac{dx_2}{dt} = x_3 </math>

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

	<math> \frac{dx_4}{dt} = u </math>		<math> \frac{dx_4}{dt} = u </math>


	The initial condition are:		The initial condition are:

	<math> x(0) = [0.1 \ 0.2 \ 0.3 \ 0] </math>		<math> x(0) = [0.1 \ 0.2 \ 0.3 \ 0] </math>

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

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


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

Mbot at 13:07, 8 November 2011

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	<pre>		<pre>
	Problem type appears to be: lpcon		Problem type appears to be: lpcon
	Time for symbolic processing: 0.~~12859~~ seconds		Time for symbolic processing: 0.12111 seconds
	Starting numeric solver		Starting numeric solver
	===== * * * =================================================================== * * *		===== * * * =================================================================== * * *
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	FuncEv 1 ConstrEv 58 ConJacEv 58 Iter 31 MinorIter 474		FuncEv 1 ConstrEv 58 ConJacEv 58 Iter 31 MinorIter 474
	CPU time: 0.109201 sec. Elapsed time: 0.~~096000~~ sec.		CPU time: 0.109201 sec. Elapsed time: 0.090000 sec.

	</pre>		</pre>

Mbot at 18:25, 2 November 2011

2011-11-02T18:25:58Z

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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('Quadruple Integral control');		title('Quadruple Integral control');
	</source>		</source>

			[[File:quadrupleIntegral_01.png]]

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

ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS

12.4.5 Example 5

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

==Problem Formulation==

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} = x_3 </math>
<math> \frac{dx_3}{dt} = x_4 </math>
<math> \frac{dx_4}{dt} = u </math>

The initial condition are:

<math> x(0) = [0.1 \ 0.2 \ 0.3 \ 0] </math>
<math> x(t_F) = [0 \ 0 \ 0 \ 0] </math>
<math> -1 <= 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, 30);
setPhase(p);

tomStates x1 x2 x3 x4
tomControls u

% Initial guess
x0 = {t_f == 5
icollocate({x1 == 0.1-0.1*t/t_f
x2 == 0.2-0.2*t/t_f; x3 == 0.3-0.3*t/t_f})
collocate(u == -1)};

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

% Boundary constraints
cbnd = {initial({x1 == 0.1; x2 == 0.2; x3 == 0.3; x4 == 0})
final({x1 == 0; x2 == 0; x3 == 0; x4 == 0})};

% ODEs and path constraints
ceq = collocate({
dot(x1) == x2; dot(x2) == x3
dot(x3) == x4; dot(x4) == u});

% Objective
objective = t_f;
</source>

==Solve the problem==

<source lang="matlab">
options = struct;
options.name = 'Quadruple Integral';
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);
</source>

<pre>
Problem type appears to be: lpcon
Time for symbolic processing: 0.12859 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Quadruple Integral f_k 4.849598187644263100
sum(|constr|) 0.000000078353877264
f(x_k) + sum(|constr|) 4.849598265998140300
f(x_0) 5.000000000000000000

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

FuncEv 1 ConstrEv 58 ConJacEv 58 Iter 31 MinorIter 474
CPU time: 0.109201 sec. Elapsed time: 0.096000 sec.

</pre>

==Plot result==

<source lang="matlab">
subplot(2,1,1)
plot(t,x1,'*-',t,x2,'*-',t,x3,'*-',t,x4,'*-');
legend('x1','x2','x3','x4');
title('Quadruple Integral state variables');

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