PROPT Zermelos problem (version 2): Difference between revisions
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Revision as of 09:09, 8 November 2011
This page is part of the PROPT Manual. See PROPT Manual. |
Problem description
Time-optimal crossing by boat of a river with a position dependent current stream.
Applied Optimal Control, Bryson & Ho, 1975. Example 1 on page 77.
Programmers: Gerard Van Willigenburg (Wageningen University) Willem De Koning (retired from Delft University of Technology)
% Copyright (c) 2009-2009 by Tomlab Optimization Inc.
Problem setup
% Array with consecutive number of collocation points
narr = [20 40];
toms t t_f % Free final time
for n=narr
p = tomPhase('p', t, 0, t_f, n);
setPhase(p)
tomStates x1 x2
tomControls u1
% Initial & terminal states
xi = [0; 0];
xf = [31; 0];
% Initial guess
if n==narr(1)
x0 = {t_f == 2; icollocate({x1 == xi(1); x2 == xi(2)})
collocate({u1 == 0})};
else
x0 = {t_f == tfopt; icollocate({x1 == xopt1; x2 == xopt2})
collocate({u1 == uopt1})};
end
% Box constraints
cbox = {1 <= t_f <= 10};
% Boundary constraints
cbnd = {initial({x1 == xi(1); x2 == xi(2)});
final({x1 == xf(1); x2 == xf(2)})};
% ODEs and path constraints
v = 9;
% No water motion in x1 direction
dx1 = v*cos(u1);
% Water motion in x2 direction: 5*sin(pi*x1/31)
dx2 = v*sin(u1)+5*sin(pi*x1/31);
ceq = collocate({
dot(x1) == dx1
dot(x2) == dx2});
% Objective
objective = t_f;
Solve the problem
options = struct;
options.name = 'Ferry trajectory optimization';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
tfopt = subs(t_f,solution);
xopt1 = subs(x1,solution);
xopt2 = subs(x2,solution);
uopt1 = subs(u1,solution);
Problem type appears to be: lpcon Time for symbolic processing: 0.10473 seconds Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31 ===================================================================================== Problem: --- 1: Ferry trajectory optimization f_k 3.681324334091920800 sum(|constr|) 0.000000334380569889 f(x_k) + sum(|constr|) 3.681324668472490700 f(x_0) 2.000000000000000000 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 84 ConJacEv 84 Iter 51 MinorIter 98 CPU time: 0.062400 sec. Elapsed time: 0.054000 sec.
Problem type appears to be: lpcon Time for symbolic processing: 0.10344 seconds Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31 ===================================================================================== Problem: --- 1: Ferry trajectory optimization f_k 3.681324335373934000 sum(|constr|) 0.000002472598427159 f(x_k) + sum(|constr|) 3.681326807972361300 f(x_0) 3.681324334091920800 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 40 ConJacEv 40 Iter 32 MinorIter 115 CPU time: 0.062400 sec. Elapsed time: 0.058000 sec.
end
% Get solution
t = subs(collocate(t),solution);
x1 = subs(collocate(x1),solution);
x2 = subs(collocate(x2),solution);
u1 = subs(collocate(u1),solution);
%Bound u1 to [0,2pi]
u1 = rem(u1,2*pi); u1 = (u1<0)*2*pi+u1;
% Plot final solution
figure(1)
subplot(2,1,1)
plot(t,x1,'*-',t,x2,'*-');
legend('x1','x2');
title('Ferry states');
subplot(2,1,2)
plot(t,u1,'+-');
legend('u1');
title('Ferry control');