PROPT Time Delay 1
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This page is part of the PROPT Manual. See PROPT Manual. |
ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS
8.3.1 Example 1
CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics
Linear time-delay system used for optimal control studies by Chan and Perkins
Problem Formulation
Find u over t in [0; 5 ] to minimize
subject to:
The initial condition are:
% Copyright (c) 2007-2008 by Tomlab Optimization Inc.
Problem setup
toms t
p1 = tomPhase('p1', t, 0, 5, 50);
setPhase(p1);
tomStates x1 x2 x3
tomControls u
% Initial guess
x0 = {icollocate({x1 == 1
x2 == 1; x3 == 0})
collocate(u == 0)};
% Boundary constraints
cbnd = initial({x1 == 1; x2 == 1; x3 == 0});
% Expressions for x1(t-tau) and x2(t-tau)
tau = 0.25;
x1delayed = ifThenElse(t<tau, 1, subs(x1,t,t-tau));
x2delayed = ifThenElse(t<tau, 1, subs(x2,t,t-tau));
% ODEs and path constraints
ceq = collocate({dot(x1) == x2
dot(x2) == -10*x1 - 5*x2 - 2*x1delayed - x2delayed + u
dot(x3) == 0.5*(10*x1.^2+x2.^2+u.^2)});
% Objective
objective = final(x3);
Solve the problem
options = struct;
options.name = 'Time Delay 1';
solution = ezsolve(objective, {cbnd, ceq}, x0, options);
t = subs(collocate(t),solution);
u = subs(collocate(u),solution);
Problem type appears to be: lpcon Time for symbolic processing: 0.10007 seconds Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31 ===================================================================================== Problem: --- 1: Time Delay 1 f_k 2.525970860473680400 sum(|constr|) 0.000000011181939804 f(x_k) + sum(|constr|) 2.525970871655620200 f(x_0) 0.000000000000000000 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 70 ConJacEv 70 Iter 49 MinorIter 204 CPU time: 0.249602 sec. Elapsed time: 0.251000 sec.
Plot result
figure(1)
plot(t,u,'+-');
legend('u');
title('Time Delay 1 control');