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2012-02-14T09:26:40Z

Plot result

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	[[File:timeDelay2_approximate_01.png]]		[[File:timeDelay2_approximate_01.png]]

			[[Category:PROPT Examples]]

Mbot at 12:13, 9 November 2011

2011-11-09T12:13:26Z

← Older revision		Revision as of 07:13, 9 November 2011
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	<math> J = x_2(t_F) </math>		<math> J = x_2(t_F) </math>


	subject to:		subject to:

	<math> \frac{dx_1}{dt} = t*x_1 + x_1(t-tau) + u </math>		<math> \frac{dx_1}{dt} = t*x_1 + x_1(t-tau) + u </math>

	<math> \frac{dx_2}{dt} = x_1^2 + u^2 </math>		<math> \frac{dx_2}{dt} = x_1^2 + u^2 </math>

	<math> tau = 1 </math>		<math> tau = 1 </math>


	A Taylor series expansion gives:		A Taylor series expansion gives:

	<math> \frac{dx_1}{dt} \approx \frac{(t+1)*x_1 + u }{1 + tau} </math>		<math> \frac{dx_1}{dt} \approx \frac{(t+1)*x_1 + u }{1 + tau} </math>


	The initial condition are:		The initial condition are:

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

8.3.2 Example 2

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

Linear time-delay system considered by Palanisamy et al.

==Problem Formulation==

Find u over t in [0; 2 ] to minimize

<math> J = x_2(t_F) </math>

subject to:

<math> \frac{dx_1}{dt} = t*x_1 + x_1(t-tau) + u </math>
<math> \frac{dx_2}{dt} = x_1^2 + u^2 </math>
<math> tau = 1 </math>

A Taylor series expansion gives:

<math> \frac{dx_1}{dt} \approx \frac{(t+1)*x_1 + u }{1 + tau} </math>

The initial condition are:

<math> x(0) = [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
p = tomPhase('p', t, 0, 2, 50);
setPhase(p);

tomStates x1 x2
tomControls u

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

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

% ODEs and path constraints
tau = 1;
ceq = collocate({
dot(x1) == ((t+1).*x1+u)/(1+tau)
dot(x2) == x1.^2+u.^2});

% Objective
objective = final(x2);
</source>

==Solve the problem==

<source lang="matlab">
options = struct;
options.name = 'Time Delay 2 Appr.';
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: 0.070509 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Time Delay 2 Appr. f_k 5.340734691399960700
sum(|constr|) 0.000000236393505091
f(x_k) + sum(|constr|) 5.340734927793465500
f(x_0) 0.000000000000000000

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

FuncEv 1 ConstrEv 31 ConJacEv 31 Iter 26 MinorIter 138
CPU time: 0.062400 sec. Elapsed time: 0.068000 sec.

</pre>

==Plot result==

<source lang="matlab">
figure(1)
plot(t,u,'+-');
legend('u');
title('Time Delay 2 control');
</source>

[[File:timeDelay2_approximate_01.png]]

PROPT Time Delay 2 (Approximate) - Revision history

Per: /* Plot result */

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 8.3.2 Example 2 CHAPMAN & HALL/CRC Monographs and Surveys in Pur..."