PROPT Time Delay 2: Difference between revisions
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<math> J = x_2(t_F) </math> | <math> J = x_2(t_F) </math> | ||
subject to: | subject to: | ||
<math> \frac{dx_1}{dt} = t*x_1 + x_1(t-tau) + u </math> | <math> \frac{dx_1}{dt} = t*x_1 + x_1(t-tau) + u </math> | ||
<math> \frac{dx_2}{dt} = x_1^2 + u^2 </math> | <math> \frac{dx_2}{dt} = x_1^2 + u^2 </math> | ||
<math> tau = 1 </math> | <math> tau = 1 </math> | ||
The initial condition are: | The initial condition are: | ||
<math> x(t<=0) = [1 \ 0] </math> | <math> x(t<=0) = [1 \ 0] </math> | ||
<math> -inf <= u <= inf </math> | <math> -inf <= u <= inf </math> | ||
<source lang="matlab"> | <source lang="matlab"> |
Revision as of 08:13, 9 November 2011
This page is part of the PROPT Manual. See PROPT Manual. |
ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS
8.3.2 Example 2
CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics
Linear time-delay system considered by Palanisamy et al.
Problem Formulation
Find u over t in [0; 2 ] 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, 2, 50);
setPhase(p1);
tomStates x1 x2
tomControls u
% Initial guess
x0 = {icollocate({x1 == 1; x2 == 0})
collocate(u == 0)};
% Boundary constraints
cbnd = initial({x1 == 1; x2 == 0});
% Expression for x1(t-tau)
tau = 1;
x1delayed = ifThenElse(t<tau, 1, subs(x1,t,t-tau));
% ODEs and path constraints
ceq = collocate({
dot(x1) == t.*x1 + x1delayed + u
dot(x2) == x1.^2 + u.^2});
% Objective
objective = final(x2);
Solve the problem
options = struct;
options.name = 'Time Delay 2';
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: 0.07238 seconds Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31 ===================================================================================== Problem: --- 1: Time Delay 2 f_k 4.796108536142885900 sum(|constr|) 0.000000305572513302 f(x_k) + sum(|constr|) 4.796108841715399000 f(x_0) 0.000000000000000000 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 32 ConJacEv 32 Iter 27 MinorIter 137 CPU time: 0.062400 sec. Elapsed time: 0.071000 sec.
Plot result
figure(1)
plot(t,u,'+-');
legend('u');
title('Time Delay 2 control');