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	[[File:timeDelay1_01.png]]		[[File:timeDelay1_01.png]]

			[[Category:PROPT Examples]]

Mbot at 12:13, 9 November 2011

2011-11-09T12:13:08Z

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	<math> J = x_3(t_F) </math>		<math> J = x_3(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} = -10x_1-5x_2-2*x_1(t-tau)-x_2(t-tau)+u </math>		<math> \frac{dx_2}{dt} = -10x_1-5x_2-2*x_1(t-tau)-x_2(t-tau)+u </math>

	<math> \frac{dx_3}{dt} = 0.5(10x_1^2+x_2^2+u^2) </math>		<math> \frac{dx_3}{dt} = 0.5(10x_1^2+x_2^2+u^2) </math>

	<math> tau = 0.25 </math>		<math> tau = 0.25 </math>


	The initial condition are:		The initial condition are:

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

	<math> -inf <= u <= inf </math>		<math> -inf <= u <= inf </math>


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

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

ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS

8.3.1 Example 1

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

Linear time-delay system used for optimal control studies by Chan and Perkins

==Problem Formulation==

Find u over t in [0; 5 ] to minimize

<math> J = x_3(t_F) </math>

subject to:

<math> \frac{dx_1}{dt} = x_2 </math>
<math> \frac{dx_2}{dt} = -10*x_1-5*x_2-2*x_1(t-tau)-x_2(t-tau)+u </math>
<math> \frac{dx_3}{dt} = 0.5*(10*x_1^2+x_2^2+u^2) </math>
<math> tau = 0.25 </math>

The initial condition are:

<math> x(t<=0) = [1 \ 1 \ 0] </math>
<math> -inf <= u <= inf </math>

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

==Problem setup==

<source lang="matlab">
toms t
p1 = tomPhase('p1', t, 0, 5, 50);
setPhase(p1);

tomStates x1 x2 x3
tomControls u

% Initial guess
x0 = {icollocate({x1 == 1
x2 == 1; x3 == 0})
collocate(u == 0)};

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

% Expressions for x1(t-tau) and x2(t-tau)
tau = 0.25;
x1delayed = ifThenElse(t<tau, 1, subs(x1,t,t-tau));
x2delayed = ifThenElse(t<tau, 1, subs(x2,t,t-tau));

% ODEs and path constraints
ceq = collocate({dot(x1) == x2
dot(x2) == -10*x1 - 5*x2 - 2*x1delayed - x2delayed + u
dot(x3) == 0.5*(10*x1.^2+x2.^2+u.^2)});

% Objective
objective = final(x3);
</source>

==Solve the problem==

<source lang="matlab">
options = struct;
options.name = 'Time Delay 1';
solution = ezsolve(objective, {cbnd, ceq}, x0, options);
t = subs(collocate(t),solution);
u = subs(collocate(u),solution);
</source>

<pre>
Problem type appears to be: lpcon
Time for symbolic processing: 0.10007 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Time Delay 1 f_k 2.525970860473680400
sum(|constr|) 0.000000011181939804
f(x_k) + sum(|constr|) 2.525970871655620200
f(x_0) 0.000000000000000000

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

FuncEv 1 ConstrEv 70 ConJacEv 70 Iter 49 MinorIter 204
CPU time: 0.249602 sec. Elapsed time: 0.251000 sec.

</pre>

==Plot result==

<source lang="matlab">
figure(1)
plot(t,u,'+-');
legend('u');
title('Time Delay 1 control');
</source>

[[File:timeDelay1_01.png]]

PROPT Time Delay 1 - Revision history

Per at 09:27, 14 February 2012

Mbot at 12:13, 9 November 2011

Mbot: Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS 8.3.1 Example 1 CHAPMAN & HALL/CRC Monographs and Surveys in Pur..."