PROPT Euler Buckling Problem - Revision history

Per: /* Footnote */

2012-02-14T09:00:29Z

Footnote

← Older revision		Revision as of 04:00, 14 February 2012
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	Note, however, that because there was no constraint on u, and it was not included in the cost function, x3_t looks very strange.		Note, however, that because there was no constraint on u, and it was not included in the cost function, x3_t looks very strange.

			[[Category:PROPT Examples]]

Mbot at 12:07, 9 November 2011

2011-11-09T12:07:35Z

← Older revision		Revision as of 07:07, 9 November 2011
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	<math> J = -z_1 </math>		<math> J = -z_1 </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} = \frac{-z_1*x_1}{x_3^2} </math>		<math> \frac{dx_2}{dt} = \frac{-z_1*x_1}{x_3^2} </math>

	<math> \int_0^1 x_3(t) \mathrm{d}t - 1 = 0 </math>		<math> \int_0^1 x_3(t) \mathrm{d}t - 1 = 0 </math>

	<math> x_1(0) = 0 </math>		<math> x_1(0) = 0 </math>

	<math> x_1(1) = 0 </math>		<math> x_1(1) = 0 </math>

	<math> x_2(0) = 1 </math>		<math> x_2(0) = 1 </math>

	<math> x_3(0) >= 0.5 </math>		<math> x_3(0) >= 0.5 </math>

	<math> x_3(t) >= 0.5 </math>		<math> x_3(t) >= 0.5 </math>


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

Mbot at 13:05, 8 November 2011

2011-11-08T13:05:09Z

← Older revision		Revision as of 08:05, 8 November 2011
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	<pre>		<pre>
	Problem type appears to be: lpcon		Problem type appears to be: lpcon
	Time for symbolic processing: 0.~~12429~~ seconds		Time for symbolic processing: 0.11144 seconds
	Starting numeric solver		Starting numeric solver
	===== * * * =================================================================== * * *		===== * * * =================================================================== * * *

Mbot at 18:23, 2 November 2011

2011-11-02T18:23:05Z

← Older revision		Revision as of 13:23, 2 November 2011
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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 4: Miser3 manual		Problem 4: Miser3 manual
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	title('Euler Buckling state variables');		title('Euler Buckling state variables');
	</source>		</source>

			[[File:eulerBuckling_01.png]]

	==Footnote==		==Footnote==

Mbot: Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} Problem 4: Miser3 manual ==Problem description== Over t in [0; 1 ], minimize $J = -z_1$ ..."

2011-11-02T15:29:12Z

Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} Problem 4: Miser3 manual ==Problem description== Over t in [0; 1 ], minimize <math> J = -z_1 </math> ..."

New page

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

Problem 4: Miser3 manual

==Problem description==

Over t in [0; 1 ], minimize

<math> J = -z_1 </math>

subject to:

<math> \frac{dx_1}{dt} = x_2 </math>
<math> \frac{dx_2}{dt} = \frac{-z_1*x_1}{x_3^2} </math>
<math> \int_0^1 x_3(t) \mathrm{d}t - 1 = 0 </math>
<math> x_1(0) = 0 </math>
<math> x_1(1) = 0 </math>
<math> x_2(0) = 1 </math>
<math> x_3(0) >= 0.5 </math>
<math> x_3(t) >= 0.5 </math>

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

==Problem setup==

<source lang="matlab">
toms t
toms z1
p = tomPhase('p', t, 0, 1, 40);
setPhase(p);

% States
tomStates x1 x2 x3 x4
% We don't need to introduce any control variables.

% Initial guess
x0 = {icollocate({
x1 == 0; x2 == 1
x3 == 0.5; x4 == t/40})
z1 == 10};

% Box constraints
cbox = {
icollocate({-10 <= x1 <= 10
-10 <= x2 <= 10; 0.5 <= x3 <= 10})
0 <= z1 <= 500};

% Boundary constraints
cbnd = {initial({x3 >= 0.5; x1 == 0
x2 == 1; x4 == 0})
final({x1 == 0; x4 == 1})};

% ODEs and path constraints
ceq = {collocate({
dot(x1) == x2
dot(x2) == -z1*x1./x3.^2
x3 >= 0.5 % Path constr.
% Integral constr.
})
integrate(x3) == 1};

% Objective
objective = z1;
</source>

==Solve the problem==

<source lang="matlab">
options = struct;
options.name = 'Euler Buckling';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);

% Extract optimal states and controls from solution
t = subs(collocate(t),solution);
x1 = subs(collocate(x1),solution);
x2 = subs(collocate(x2),solution);
x3 = subs(collocate(x3),solution);
</source>

<pre>
Problem type appears to be: lpcon
Time for symbolic processing: 0.12429 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Euler Buckling f_k 9.881895688695372800
sum(|constr|) 0.000000016445692785
f(x_k) + sum(|constr|) 9.881895705141065600
f(x_0) 10.000000000000000000

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

FuncEv 1 ConstrEv 73 ConJacEv 72 Iter 33 MinorIter 240
CPU time: 0.062400 sec. Elapsed time: 0.068000 sec.

</pre>

==Plot result==

<source lang="matlab">
figure(1)
plot(t,x1,'*-',t,x2,'*-',t,x3,'*-');
legend('x1','x2','x3');
title('Euler Buckling state variables');
</source>

==Footnote==

In the original [Miser3] problem formulation, it is requested to compute "u", equal to x3_t. u is not included in the optimization problem, thereby speeding up the solution process. x3_t can be obtained by simple numeric differentiation of x3.

Note, however, that because there was no constraint on u, and it was not included in the cost function, x3_t looks very strange.

PROPT Euler Buckling Problem - Revision history

Per: /* Footnote */

Mbot at 12:07, 9 November 2011

Mbot at 13:05, 8 November 2011

Mbot at 18:23, 2 November 2011

Mbot: Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} Problem 4: Miser3 manual ==Problem description== Over t in [0; 1 ], minimize J = -z_1 ..."

Mbot: Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} Problem 4: Miser3 manual ==Problem description== Over t in [0; 1 ], minimize $J = -z_1$ ..."