PROPT Linear Tangent Steering Problem - Revision history

Per at 09:21, 14 February 2012

2012-02-14T09:21:55Z

← Older revision		Revision as of 04:21, 14 February 2012
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	[[File:linTanSte_01.png]]		[[File:linTanSte_01.png]]

			[[Category:PROPT Examples]]

Mbot at 12:09, 9 November 2011

2011-11-09T12:09:16Z

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


	subject to:		subject to:

	<math> \frac{d^{2}y_1}{dt^{2}} = a*cos(u) </math>		<math> \frac{d^{2}y_1}{dt^{2}} = a*cos(u) </math>

	<math> \frac{d^{2}y_2}{dt^{2}} = a*sin(u) </math>		<math> \frac{d^{2}y_2}{dt^{2}} = a*sin(u) </math>

	<math> \|u\| <= \frac{pi}{2} </math>		<math> \|u\| <= \frac{pi}{2} </math>


	<math> y_{1:2}(0) = 0 </math>		<math> y_{1:2}(0) = 0 </math>

	<math> \frac{dy_{1:2}}{dt} = 0 </math>		<math> \frac{dy_{1:2}}{dt} = 0 </math>

	<math> a = 1 </math>		<math> a = 1 </math>

	<math> y_{2}(f) = 5 </math>		<math> y_{2}(f) = 5 </math>

	<math> \frac{dy_{1:2}}{dt}(f) = [45 \ 0] </math>		<math> \frac{dy_{1:2}}{dt}(f) = [45 \ 0] </math>


	The following transformation gives a new formulation:		The following transformation gives a new formulation:

	<math> x_1 = y_1 </math>		<math> x_1 = y_1 </math>

	<math> x_2 = \frac{dy_1}{dt} </math>		<math> x_2 = \frac{dy_1}{dt} </math>

	<math> x_3 = y_2 </math>		<math> x_3 = y_2 </math>

	<math> x_4 = \frac{dy_2}{dt} </math>		<math> x_4 = \frac{dy_2}{dt} </math>


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

	<math> \frac{dx_2}{dt} = a*cos(u) </math>		<math> \frac{dx_2}{dt} = a*cos(u) </math>

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

	<math> \frac{dx_4}{dt} = a*sin(u) </math>		<math> \frac{dx_4}{dt} = a*sin(u) </math>


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

Mbot at 13:06, 8 November 2011

2011-11-08T13:06:14Z

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	<pre>		<pre>
	Problem type appears to be: lpcon		Problem type appears to be: lpcon
	Time for symbolic processing: 0.~~1312~~ seconds		Time for symbolic processing: 0.12675 seconds
	Starting numeric solver		Starting numeric solver
	===== * * * =================================================================== * * *		===== * * * =================================================================== * * *
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	FuncEv 14 GradEv 152 ConstrEv 13 ConJacEv 152 Iter 11 MinorIter 12		FuncEv 14 GradEv 152 ConstrEv 13 ConJacEv 152 Iter 11 MinorIter 12
	CPU time: 0.~~109201~~ sec. Elapsed time: 0.~~175000~~ sec.		CPU time: 0.093601 sec. Elapsed time: 0.106000 sec.

	</pre>		</pre>

Mbot at 18:24, 2 November 2011

2011-11-02T18:24:13Z

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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]]}}

	Benchmarking Optimization Software with COPS Elizabeth D. Dolan and Jorge J. More ARGONNE NATIONAL LABORATORY		Benchmarking Optimization Software with COPS Elizabeth D. Dolan and Jorge J. More ARGONNE NATIONAL LABORATORY
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	title('Linear Tangent Steering control');		title('Linear Tangent Steering control');
	</source>		</source>

			[[File:linTanSte_01.png]]

Mbot: Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} Benchmarking Optimization Software with COPS Elizabeth D. Dolan and Jorge J. More ARGONNE NATIONAL LABOR..."

2011-11-02T15:29:46Z

Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} Benchmarking Optimization Software with COPS Elizabeth D. Dolan and Jorge J. More ARGONNE NATIONAL LABOR..."

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

Benchmarking Optimization Software with COPS Elizabeth D. Dolan and Jorge J. More ARGONNE NATIONAL LABORATORY

==Problem Formulation==

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

<math> J = t_f </math>

subject to:

<math> \frac{d^{2}y_1}{dt^{2}} = a*cos(u) </math>
<math> \frac{d^{2}y_2}{dt^{2}} = a*sin(u) </math>
<math> |u| <= \frac{pi}{2} </math>

<math> y_{1:2}(0) = 0 </math>
<math> \frac{dy_{1:2}}{dt} = 0 </math>
<math> a = 1 </math>
<math> y_{2}(f) = 5 </math>
<math> \frac{dy_{1:2}}{dt}(f) = [45 \ 0] </math>

The following transformation gives a new formulation:

<math> x_1 = y_1 </math>
<math> x_2 = \frac{dy_1}{dt} </math>
<math> x_3 = y_2 </math>
<math> x_4 = \frac{dy_2}{dt} </math>

<math> \frac{dx_1}{dt} = x_2 </math>
<math> \frac{dx_2}{dt} = a*cos(u) </math>
<math> \frac{dx_3}{dt} = x_4 </math>
<math> \frac{dx_4}{dt} = a*sin(u) </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 == 1
icollocate({
x1 == 12*t/t_f
x2 == 45*t/t_f
x3 == 5*t/t_f
x4 == 0})};

% Box constraints
cbox = {sqrt(eps) <= t_f
-pi/2 <= collocate(u) <= pi/2};

% Boundary constraints
cbnd = {initial({x1 == 0; x2 == 0; x3 == 0; x4 == 0})
final({x2 == 45; x3 == 5; x4 == 0})};

% ODEs and path constraints
a = 100;
ceq = collocate({dot(x1) == x2
dot(x2) == a*cos(u)
dot(x3) == x4
dot(x4) == a*sin(u)});

% Objective
objective = t_f;
</source>

==Solve the problem==

<source lang="matlab">
options = struct;
options.name = 'Linear Tangent Steering';
options.solver = 'knitro';
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: 0.1312 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Linear Tangent Steering f_k 0.554570876848740510
sum(|constr|) 0.000053415505390106
f(x_k) + sum(|constr|) 0.554624292354130580
f(x_0) 1.000000000000000000

Solver: KNITRO. EXIT=0. INFORM=0.
Default NLP KNITRO
Locally optimal solution found

FuncEv 14 GradEv 152 ConstrEv 13 ConJacEv 152 Iter 11 MinorIter 12
CPU time: 0.109201 sec. Elapsed time: 0.175000 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('Linear Tangent Steering state variables');

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