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
Find u over t in [0; 5 ] to minimize
J = x3(tF) |
subject to:
| = x2 |
| = −10*x1−5*x2−2*x1(t−tau)−x2(t−tau)+u |
| = 0.5*(10*x12+x22+u2) |
tau = 0.25 |
A Taylor series expansion gives:
| ≈ (−12*x1+(2*tau−6)*x2 + u)/(1−tau) |
The initial condition are:
x(0) = [1 1 0] |
−inf <= u <= inf |
Reference: [25]
toms t p = tomPhase('p', t, 0, 5, 50); setPhase(p); 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}); % ODEs and path constraints tau = 0.25; ceq = collocate({dot(x1) == x2 dot(x2) == (-12*x1+(2*tau-6)*x2 + u)/(1-tau) dot(x3) == 0.5*(10*x1.^2+x2.^2+u.^2)}); % Objective objective = final(x3);
options = struct; options.name = 'Time Delay 1 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 1 Appr. f_k 2.387051416916649200 sum(|constr|) 0.000000035059442522 f(x_k) + sum(|constr|) 2.387051451976091700 f(x_0) 0.000000000000000000 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 67 ConJacEv 67 Iter 48 MinorIter 214 CPU time: 0.671875 sec. Elapsed time: 0.734000 sec.
figure(1) plot(t,u,'+-'); legend('u'); title('Time Delay 1 control');