Per: /* Plot result */

2012-02-14T09:26:31Z

Plot result

← Older revision		Revision as of 05:26, 14 February 2012
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	[[File:transferMinSwing_01.png]]		[[File:transferMinSwing_01.png]]

			[[Category:PROPT Examples]]

Mbot at 12:13, 9 November 2011

2011-11-09T12:13:29Z

Mbot: Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} Example 7.1: K.L. Teo, K. K. Leong, G.J. Goh ==Problem Formulation== Find u over t in [0; 1 ] to minimize..."

2011-11-08T13:09:01Z

Created page with "{{Part Of Manual|title=the PROPT Manual|link=PROPT Manual}} Example 7.1: K.L. Teo, K. K. Leong, G.J. Goh ==Problem Formulation== Find u over t in [0; 1 ] to minimize..."

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

Example 7.1: K.L. Teo, K. K. Leong, G.J. Goh

==Problem Formulation==

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

<math> J = \int_0^{1} 4.5 (x_3^2 + x_6^2) \mathrm{d}t </math>

subject to:

<math> \frac{dx_1}{dt} = 9*x_4 </math>
<math> \frac{dx_2}{dt} = 9*x_5 </math>
<math> \frac{dx_3}{dt} = 9*x_6 </math>
<math> \frac{dx_4}{dt} = 9*(x_7+17.2656*x_3) </math>
<math> \frac{dx_5}{dt} = 9*x_8 </math>
<math> \frac{dx_6}{dt} = -9*\frac{x_7+27.0756*x_3+2*x_5*x_6}{x_2} </math>
<math> \frac{dx_7}{dt} = 9*u_1 </math>
<math> \frac{dx_8}{dt} = 9*u_2 </math>

<math> x(0) = [0 \ 22 \ 0 \ 0 \ -1 \ 0 \ NaN \ NaN] </math>
<math> x(1) = [10 \ 14 \ 0 \ 2.5 \ 0 \ 0 \ NaN \ NaN] </math>
<math> |x_4| <= 2.5 </math>
<math> |x_5| <= 1.0 </math>
<math> |x_7| <= 2.83374 </math>
<math> -0.80865 <= x_8 <= 0.71265 </math>
<math> |u| <= 10 </math>

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

==Problem setup==

<source lang="matlab">
toms t phi1 phi2

% Starting guess
speed = 5;
xopt = 1.2*t;
yopt = 1.6*t;
thetaopt = pi/4;

phi1opt = 1;
phi2opt = 1;
x1opt = 10*t;
x2opt = 22-8*t;
x3opt = 0;
x4opt = 2.5*t;
x5opt = -1+t;
x6opt = 0;
x7opt = 0;
x8opt = 0;
u1opt = 0;
u2opt = 0;
</source>

==Solve the problem, using a successively larger number collocation points==

<source lang="matlab">
for n=[20 40]
</source>

<source lang="matlab">
% Create a new phase and states, using n collocation points
p = tomPhase('p', t, 0, 1, n);
setPhase(p);
tomStates x1 x2 x3 x4 x5 x6 x7 x8
tomControls u1 u2

% Initial guess
x0 = {phi1 == phi1opt; phi2 == phi2opt
icollocate({
x1 == x1opt; x2 == x2opt
x3 == x3opt; x4 == x4opt
x5 == x5opt; x6 == x6opt
x7 == x7opt; x8 == x8opt})
collocate({
u1 == u1opt; u2 == u2opt})};

% Box constraints
cbox = {-10 <= phi1 <= 10
-10 <= phi2 <= 10
-2.5 <= icollocate(x4) <= 2.5
-1 <= icollocate(x5) <= 1
-2.83374 <= icollocate(x7) <= 2.83374
-0.80865 <= icollocate(x8) <= 0.71265
-10 <= collocate(u1) <= 10
-10 <= collocate(u2) <= 10};

% Boundary constraints
cbnd = {initial({x1 == 0
x2 == 22; x3 == 0
x4 == 0; x5 == -1
x6 == 0; x7 == phi1
x8 == phi2
})
final({x1 == 10
x2 == 14; x3 == 0
x4 == 2.5; x5 == 0
x6 == 0})};

% ODEs and path constraints
ceq = collocate({dot(x1) == 9*x4
dot(x2) == 9*x5; dot(x3) == 9*x6
dot(x4) == 9*(x7+17.2656*x3)
dot(x5) == 9*x8
dot(x6) == -9*(x7+27.0756*x3+2*x5.*x6)./x2
dot(x7) == 9*u1; dot(x8) == 9*u2});

% Objective
objective = integrate(4.5*(x3.^2 + x6.^2));
</source>

==Solve the problem==

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

% Optimal x and u to use as starting guess in the next iteration
x1opt = subs(x1, solution);
x2opt = subs(x2, solution);
x3opt = subs(x3, solution);
x4opt = subs(x4, solution);
x5opt = subs(x5, solution);
x6opt = subs(x6, solution);
x7opt = subs(x7, solution);
x8opt = subs(x8, solution);
u1opt = subs(u1, solution);
u2opt = subs(u2, solution);
phi1opt = subs(phi1, solution);
phi2opt = subs(phi2, solution);
</source>

<pre>
Problem type appears to be: qpcon
Time for symbolic processing: 0.25702 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Transfer Min Swing f_k 0.005155677076381541
sum(|constr|) 0.000000000220310318
f(x_k) + sum(|constr|) 0.005155677296691859
f(x_0) 0.000000000000000000

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

FuncEv 1 ConstrEv 107 ConJacEv 107 Iter 100 MinorIter 374
CPU time: 0.187201 sec. Elapsed time: 0.186000 sec.

</pre>

<pre>
Problem type appears to be: qpcon
Time for symbolic processing: 0.25108 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31
=====================================================================================
Problem: --- 1: Transfer Min Swing f_k 0.005157874701172190
sum(|constr|) 0.000000001277942393
f(x_k) + sum(|constr|) 0.005157875979114583
f(x_0) 0.005155729096774491

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

FuncEv 1 ConstrEv 126 ConJacEv 126 Iter 123 MinorIter 594
CPU time: 0.951606 sec. Elapsed time: 0.965000 sec.

</pre>

<source lang="matlab">
end

t = subs(collocate(t),solution);
x1 = collocate(x1opt);
x2 = collocate(x2opt);
x3 = collocate(x3opt);
x7 = collocate(x7opt);
x8 = collocate(x8opt);
</source>

==Plot result==

<source lang="matlab">
subplot(2,1,1)
plot(t,x1,'*-',t,x2,'*-',t,x3,'*-');
legend('x1','x2','x3');
title('Transfer Min Swing state variables');

subplot(2,1,2)
plot(t,x7,'+-',t,x8,'+-');
legend('x7','x8');
title('Transfer Min Swing control');
</source>

[[File:transferMinSwing_01.png]]

← Older revision		Revision as of 08:13, 9 November 2011
Line 8:		Line 8:

	<math> J = \int_0^{1} 4.5 (x_3^2 + x_6^2) \mathrm{d}t </math>		<math> J = \int_0^{1} 4.5 (x_3^2 + x_6^2) \mathrm{d}t </math>


	subject to:		subject to:

	<math> \frac{dx_1}{dt} = 9*x_4 </math>		<math> \frac{dx_1}{dt} = 9*x_4 </math>

	<math> \frac{dx_2}{dt} = 9*x_5 </math>		<math> \frac{dx_2}{dt} = 9*x_5 </math>

	<math> \frac{dx_3}{dt} = 9*x_6 </math>		<math> \frac{dx_3}{dt} = 9*x_6 </math>

	<math> \frac{dx_4}{dt} = 9(x_7+17.2656x_3) </math>		<math> \frac{dx_4}{dt} = 9(x_7+17.2656x_3) </math>

	<math> \frac{dx_5}{dt} = 9*x_8 </math>		<math> \frac{dx_5}{dt} = 9*x_8 </math>

	<math> \frac{dx_6}{dt} = -9\frac{x_7+27.0756x_3+2x_5x_6}{x_2} </math>		<math> \frac{dx_6}{dt} = -9\frac{x_7+27.0756x_3+2x_5x_6}{x_2} </math>

	<math> \frac{dx_7}{dt} = 9*u_1 </math>		<math> \frac{dx_7}{dt} = 9*u_1 </math>

	<math> \frac{dx_8}{dt} = 9*u_2 </math>		<math> \frac{dx_8}{dt} = 9*u_2 </math>


	<math> x(0) = [0 \ 22 \ 0 \ 0 \ -1 \ 0 \ NaN \ NaN] </math>		<math> x(0) = [0 \ 22 \ 0 \ 0 \ -1 \ 0 \ NaN \ NaN] </math>

	<math> x(1) = [10 \ 14 \ 0 \ 2.5 \ 0 \ 0 \ NaN \ NaN] </math>		<math> x(1) = [10 \ 14 \ 0 \ 2.5 \ 0 \ 0 \ NaN \ NaN] </math>

	<math> \|x_4\| <= 2.5 </math>		<math> \|x_4\| <= 2.5 </math>

	<math> \|x_5\| <= 1.0 </math>		<math> \|x_5\| <= 1.0 </math>

	<math> \|x_7\| <= 2.83374 </math>		<math> \|x_7\| <= 2.83374 </math>

	<math> -0.80865 <= x_8 <= 0.71265 </math>		<math> -0.80865 <= x_8 <= 0.71265 </math>

	<math> \|u\| <= 10 </math>		<math> \|u\| <= 10 </math>


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

PROPT Transfer Min Swing - 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}} Example 7.1: K.L. Teo, K. K. Leong, G.J. Goh ==Problem Formulation== Find u over t in [0; 1 ] to minimize..."