PROPT Disturbance Control - Revision history

Per: /* Plot result */

2012-02-14T08:57:45Z

Plot result

← Older revision		Revision as of 03:57, 14 February 2012
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	[[File:disturbanceControl_01.png]]		[[File:disturbanceControl_01.png]]

			[[Category:PROPT Examples]]

Mbot at 12:07, 9 November 2011

2011-11-09T12:07:28Z

← Older revision		Revision as of 07:07, 9 November 2011
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	<math> J = 0 </math>		<math> J = 0 </math>


	subject to:		subject to:

	<math> \frac{dx_1}{dt} = x_3 </math>		<math> \frac{dx_1}{dt} = x_3 </math>

	<math> \frac{dx_2}{dt} = x_4 </math>		<math> \frac{dx_2}{dt} = x_4 </math>

	<math> \frac{dx_3}{dt} = -x_1+x_2+u </math>		<math> \frac{dx_3}{dt} = -x_1+x_2+u </math>

	<math> \frac{dx_4}{dt} = 0.1x_1-1.02x_2+0.3sin(4t)*(t<9.75) </math>		<math> \frac{dx_4}{dt} = 0.1x_1-1.02x_2+0.3sin(4t)*(t<9.75) </math>

	<math> x(t_0) = [0 \ 0 \ 2 \ 1] </math>		<math> x(t_0) = [0 \ 0 \ 2 \ 1] </math>

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

	<math> 0 <= u <= 1 </math>		<math> 0 <= u <= 1 </math>


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

Mbot at 13:04, 8 November 2011

2011-11-08T13:04:49Z

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	<pre>		<pre>
	Problem type appears to be: lp		Problem type appears to be: lp
	Time for symbolic processing: 0.~~10133~~ seconds		Time for symbolic processing: 0.069168 seconds
	Starting numeric solver		Starting numeric solver
	===== * * * =================================================================== * * *		===== * * * =================================================================== * * *
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	FuncEv 330 Iter 330		FuncEv 330 Iter 330
	CPU time: 0.~~109201~~ sec. Elapsed time: 0.~~091000~~ sec.		CPU time: 0.124801 sec. Elapsed time: 0.080000 sec.

	</pre>		</pre>

Mbot at 18:22, 2 November 2011

2011-11-02T18:22:56Z

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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]]}}

	Optimal On-Line Control and Classical Regulation Problem, Faina M. Kirillova, Institute of Mathematics National Academy of Sciences of Belarus.		Optimal On-Line Control and Classical Regulation Problem, Faina M. Kirillova, Institute of Mathematics National Academy of Sciences of Belarus.
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	legend('disturbance');		legend('disturbance');
	</source>		</source>

			[[File:disturbanceControl_01.png]]

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

Optimal On-Line Control and Classical Regulation Problem, Faina M. Kirillova, Institute of Mathematics National Academy of Sciences of Belarus.

Algorithm of Acting Optimal Controller

==Problem Description==

Find u over t in [0; 25 ] to minimize:

<math> J = 0 </math>

subject to:

<math> \frac{dx_1}{dt} = x_3 </math>
<math> \frac{dx_2}{dt} = x_4 </math>
<math> \frac{dx_3}{dt} = -x_1+x_2+u </math>
<math> \frac{dx_4}{dt} = 0.1*x_1-1.02*x_2+0.3*sin(4*t)*(t<9.75) </math>
<math> x(t_0) = [0 \ 0 \ 2 \ 1] </math>
<math> x(t_f) = [0 \ 0 \ 0 \ 0] </math>
<math> 0 <= u <= 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, 25, 80);
setPhase(p);

tomStates x1 x2 x3 x4
tomControls u

% Box constraints
cbox = {0 <= collocate(u) <= 1};

% Boundary constraints
cbnd = {initial({x1 == 0; x2 == 0
x3 == 2; x4 == 1})
final({x1 == 0; x2 == 0
x3 == 0; x4 == 0})};

% ODEs and path constraints
ceq = collocate({
dot(x1) == x3
dot(x2) == x4
dot(x3) == -x1+x2+u
dot(x4) == 0.1*x1-1.02*x2+0.3*sin(4*t).*(t<9.75)});

% Objective
objective = 0;
</source>

==Solve the problem==

<source lang="matlab">
options = struct;
options.name = 'Disturbance Control';
solution = ezsolve(objective, {cbox, cbnd, ceq}, [], 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: lp
Time for symbolic processing: 0.10133 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Disturbance Control f_k 0.000000000000000000
sum(|constr|) 0.000000000013822610
f(x_k) + sum(|constr|) 0.000000000013822610
f(x_0) 0.000000000000000000

Solver: CPLEX. EXIT=0. INFORM=1.
CPLEX Dual Simplex LP solver
Optimal solution found

FuncEv 330 Iter 330
CPU time: 0.109201 sec. Elapsed time: 0.091000 sec.

</pre>

==Plot result==

<source lang="matlab">
figure(1);
subplot(2,2,1)
plot(x1,x3,'-');
title('Disturbance control');
legend('x1 vs x3');
subplot(2,2,2)
plot(x2,x4,'-');
legend('x2 vs x4');

subplot(2,2,3)
plot(t,u,'-');
legend('u');

subplot(2,2,4)
plot(t,0.3*sin(4*t).*(t<9.75),'-');
legend('disturbance');
</source>