Per at 09:27, 14 February 2012

2012-02-14T09:27:18Z

← Older revision		Revision as of 04:27, 14 February 2012
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	[[File:terminalConstraint2_01.png]]		[[File:terminalConstraint2_01.png]]

			[[Category:PROPT Examples]]

Mbot at 12:13, 9 November 2011

2011-11-09T12:13:00Z

← Older revision		Revision as of 07:13, 9 November 2011
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	<math> J = \int_0^{1} (x^2 + u^2) \mathrm{d}t </math>		<math> J = \int_0^{1} (x^2 + u^2) \mathrm{d}t </math>


	subject to:		subject to:

	<math> \frac{dx}{dt} = u </math>		<math> \frac{dx}{dt} = u </math>

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

	<math> x(1) = 0.75 </math>		<math> x(1) = 0.75 </math>

	<math> x(0.75) = 0.9 </math>		<math> x(0.75) = 0.9 </math>


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

Mbot: Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} Problem 1: Miser3 manual ==Problem Description== Find u(t) over t in [0; 1 ] to minimize $J = \int..."$

2011-11-08T13:08:45Z

Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} Problem 1: Miser3 manual ==Problem Description== Find u(t) over t in [0; 1 ] to minimize <math> J = \int..."

New page

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

Problem 1: Miser3 manual

==Problem Description==

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

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

subject to:

<math> \frac{dx}{dt} = u </math>
<math> x(0) = 1 </math>
<math> x(1) = 0.75 </math>
<math> x(0.75) = 0.9 </math>

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

==Problem setup==

<source lang="matlab">
toms t1
p1 = tomPhase('p1', t1, 0, 0.75, 20);
toms t2
p2 = tomPhase('p2', t2, 0.75, 0.25, 20);

setPhase(p1);
tomStates x1p1
tomControls up1
setPhase(p2);
tomStates x1p2
tomControls up2
setPhase(p1);

% Initial guess
x01 = {icollocate({x1p1 == 1-0.1*t1/0.75})
collocate(up1==0.9*t1/0.75)};

% Box constraints
cbox1 = {-10 <= icollocate(p1,x1p1) <= 10
-10 <= collocate(p1,up1) <= 10};

% Boundary constraints
cbnd1 = initial(x1p1 == 1);

% ODEs and path constraints
ceq1 = collocate(dot(x1p1) == up1);

% Objective
objective1 = integrate(x1p1.^2+up1.^2);

setPhase(p2);
% Initial guess
x02 = {icollocate({x1p2 == 1-0.1*t2})
collocate(up2==0.9+0.1*t2)};

% Box constraints
cbox2 = {-10 <= icollocate(p2,x1p2) <= 10
-10 <= collocate(p2,up2) <= 10};

% Boundary constraints
cbnd2 = {initial(x1p2 == 0.9)
final(x1p2 == 0.75)};

% ODEs and path constraints
ceq2 = collocate(dot(x1p2) == up2);

% Objective
objective2 = integrate(x1p2.^2+up2.^2);

% Objective
objective = objective1 + objective2;

% Link phase
link = {final(p1,x1p1) == initial(p2,x1p2)};
</source>

==Solve the problem==

<source lang="matlab">
options = struct;
options.name = 'Terminal Constraint 2';
constr = {cbox1, cbnd1, ceq1, cbox2, cbnd2, ceq2, link};
solution = ezsolve(objective, constr, {x01, x02}, options);

t = subs(collocate(p1,t1),solution);
t = [t;subs(collocate(p2,t2),solution)];
x1 = subs(collocate(p1,x1p1),solution);
x1 = [x1;subs(collocate(p2,x1p2),solution)];
u = subs(collocate(p1,up1),solution);
u = [u;subs(collocate(p2,up2),solution)];
</source>

<pre>
Problem type appears to be: qp
Time for symbolic processing: 0.098918 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: 1: Terminal Constraint 2 f_k 0.920531441471499900
sum(|constr|) 0.000000000474625644
f(x_k) + sum(|constr|) 0.920531441946125570
f(x_0) 0.000000000000000000

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

FuncEv 8 GradEv 8 ConstrEv 8 Iter 8
CPU time: 0.015600 sec. Elapsed time: 0.004000 sec.

</pre>

==Plot result==

<source lang="matlab">
subplot(2,1,1)
plot(t,x1,'*-');
legend('x1');
title('Terminal Constraint state variable');

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

[[File:terminalConstraint2_01.png]]

PROPT A Simple Terminal Constraint Problem - 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}} Problem 1: Miser3 manual ==Problem Description== Find u(t) over t in [0; 1 ] to minimize J = \int..."

Mbot: Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} Problem 1: Miser3 manual ==Problem Description== Find u(t) over t in [0; 1 ] to minimize $J = \int..."$