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.
Find u over t in [0; 2 ] to minimize
J = x2(tF) |
subject to:
| = t*x1 + x1(t−tau) + u |
| = x12 + u2 |
tau = 1 |
A Taylor series expansion gives:
| ≈ |
|
The initial condition are:
x(0) = [1 0] |
−inf <= u <= inf |
Reference: [25]
toms t p = tomPhase('p', t, 0, 2, 50); setPhase(p); tomStates x1 x2 tomControls u % Initial guess x0 = {icollocate({x1 == 1; x2 == 0}) collocate(u == 0)}; % Boundary constraints cbnd = initial({x1 == 1; x2 == 0}); % ODEs and path constraints tau = 1; ceq = collocate({ dot(x1) == ((t+1).*x1+u)/(1+tau) dot(x2) == x1.^2+u.^2}); % Objective objective = final(x2);
options = struct; options.name = 'Time Delay 2 Appr.'; solution = ezsolve(objective, {cbnd, ceq}, x0, options); t = subs(collocate(t),solution); u = subs(collocate(u),solution);
Problem type appears to be: lpcon Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2011-02-05 ===================================================================================== Problem: --- 1: Time Delay 2 Appr. f_k 5.340734691399960700 sum(|constr|) 0.000000236393505091 f(x_k) + sum(|constr|) 5.340734927793465500 f(x_0) 0.000000000000000000 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 31 ConJacEv 31 Iter 26 MinorIter 138 CPU time: 0.218750 sec. Elapsed time: 0.218000 sec.
figure(1) plot(t,u,'+-'); legend('u'); title('Time Delay 2 control');