PROPT Fuller Phenomenon: Difference between revisions
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Revision as of 14:23, 2 November 2011
This page is part of the PROPT Manual. See PROPT Manual. |
A Short Introduction to Optimal Control, Ugo Boscain, SISSA, Italy
3.6 Fuller Phenomenon.
Problem Description
Find u over t in [0; inf ] to minimize:
subject to:
% Copyright (c) 2007-2008 by Tomlab Optimization Inc.
Problem setup
toms t
toms t_f
p = tomPhase('p', t, 0, t_f, 60);
setPhase(p);
tomStates x1 x2
tomControls u
% Initial guess
x0 = {t_f == 10
icollocate(x1 == 10-10*t/t_f)
icollocate(x2 == 0)
collocate(u == -1+2*t/t_f)};
% Box constraints
cbox = {1 <= t_f <= 1e4
-1 <= collocate(u) <= 1};
% Boundary constraints
cbnd = {initial({x1 == 10; x2 == 0})
final({x1 == 0; x2 == 0})};
% ODEs and path constraints
ceq = collocate({dot(x1) == x2; dot(x2) == u});
% Objective
objective = integrate(x1.^2);
Solve the problem
options = struct;
options.name = 'Fuller Phenomenon';
options.solver = 'snopt';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
t = subs(collocate(t),solution);
x1 = subs(collocate(x1),solution);
x2 = subs(collocate(x2),solution);
u = subs(collocate(u),solution);
Problem type appears to be: con Time for symbolic processing: 0.10071 seconds Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31 ===================================================================================== Problem: --- 1: Fuller Phenomenon f_k 242.423532418144080000 sum(|constr|) 0.000000063716024856 f(x_k) + sum(|constr|) 242.423532481860090000 f(x_0) 333.333333333328370000 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 28 GradEv 26 ConstrEv 27 ConJacEv 26 Iter 14 MinorIter 248 CPU time: 0.062400 sec. Elapsed time: 0.079000 sec.
Plot result
subplot(2,1,1)
plot(x1,x2,'*-');
legend('x1 vs x2');
title('Fuller Phenomenon state variables');
subplot(2,1,2)
plot(t,u,'+-');
legend('u');
title('Fuller Phenomenon control');