PROPT LQR Problem - Revision history

Per: /* Plot result */

2012-02-14T09:22:26Z

Plot result

← Older revision		Revision as of 05:22, 14 February 2012
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	[[File:lqrProblem_01.png]]		[[File:lqrProblem_01.png]]

			[[Category:PROPT Examples]]

Mbot at 12:09, 9 November 2011

2011-11-09T12:09:25Z

← Older revision		Revision as of 08:09, 9 November 2011
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	<math> J = \int_0^{1} (0.625x^2 + 0.5xu +0.5u^2) \mathrm{d}t </math>		<math> J = \int_0^{1} (0.625x^2 + 0.5xu +0.5u^2) \mathrm{d}t </math>


	subject to:		subject to:

	<math> \frac{dx}{dt} = \frac{1}{2}*x + u </math>		<math> \frac{dx}{dt} = \frac{1}{2}*x + u </math>

	<math> x(0) = 1 </math>		<math> x(0) = 1 </math>


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

Mbot at 13:06, 8 November 2011

2011-11-08T13:06:24Z

← Older revision		Revision as of 09:06, 8 November 2011
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	<pre>		<pre>
	Problem type appears to be: qp		Problem type appears to be: qp
	Time for symbolic processing: 0.~~055439~~ seconds		Time for symbolic processing: 0.053812 seconds
	Starting numeric solver		Starting numeric solver
	===== * * * =================================================================== * * *		===== * * * =================================================================== * * *

Mbot at 18:24, 2 November 2011

2011-11-02T18:24:24Z

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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]]}}

	Problem: LQR: RIOTS 95 Manual		Problem: LQR: RIOTS 95 Manual
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	title('LQR Problem control');		title('LQR Problem control');
	</source>		</source>

			[[File:lqrProblem_01.png]]

Mbot: Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} Problem: LQR: RIOTS 95 Manual ==Problem Description== Find u(t) over t in [0; 1 ] to minimize $..."$

2011-11-02T15:29:51Z

Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} Problem: LQR: RIOTS 95 Manual ==Problem Description== Find u(t) over t in [0; 1 ] to minimize <math> ..."

New page

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

Problem: LQR: RIOTS 95 Manual

==Problem Description==

Find u(t) over t in [0; 1 ] to minimize

<math> J = \int_0^{1} (0.625*x^2 + 0.5*x*u +0.5*u^2) \mathrm{d}t </math>

subject to:

<math> \frac{dx}{dt} = \frac{1}{2}*x + u </math>
<math> x(0) = 1 </math>

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

==Problem setup==

<source lang="matlab">
toms t
p = tomPhase('p', t, 0, 1, 20);
setPhase(p);

tomStates x
tomControls u

% Initial guess
x0 = icollocate(x == 1-t);

% ODEs and constraints
ceq = {collocate(dot(x) == 0.5*x+u)
initial(x == 1)};

% Objective
objective = integrate(0.625*x.^2+0.5*x.*u+0.5*u.^2);
</source>

==Solve the problem==

<source lang="matlab">
options = struct;
options.name = 'LQR Problem';
solution = ezsolve(objective, ceq, x0, options);
t = subs(collocate(t),solution);
x = subs(collocate(x),solution);
u = subs(collocate(u),solution);
</source>

<pre>
Problem type appears to be: qp
Time for symbolic processing: 0.055439 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: 1: LQR Problem f_k 0.380797077977355130
sum(|constr|) 0.000000000050276185
f(x_k) + sum(|constr|) 0.380797078027631300
f(x_0) 0.000000000000000000

Solver: CPLEX. EXIT=0. INFORM=1.
CPLEX Barrier QP solver
Optimal solution found

FuncEv 3 GradEv 3 ConstrEv 3 Iter 3
Elapsed time: 0.003000 sec.

</pre>

==Plot result==

<source lang="matlab">
subplot(2,1,1)
plot(t,x,'*-');
legend('x');
title('LQR Problem state variable');

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

PROPT LQR Problem - Revision history

Per: /* Plot result */

Mbot at 12:09, 9 November 2011

Mbot at 13:06, 8 November 2011

Mbot at 18:24, 2 November 2011

Mbot: Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} Problem: LQR: RIOTS 95 Manual ==Problem Description== Find u(t) over t in [0; 1 ] to minimize ..."

Mbot: Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} Problem: LQR: RIOTS 95 Manual ==Problem Description== Find u(t) over t in [0; 1 ] to minimize $..."$