ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS
10.2.2 Example 2
CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics
Find u over t in [0; 5 ] to minimize
J = x3(tF) |
subject to:
| = x2 |
| = u |
| = x12 |
The initial condition are:
x(0) = [0 1 0] |
−1 <= u <= 1 |
Reference: [25]
toms t p = tomPhase('p', t, 0, 5, 50); setPhase(p); tomStates x1 x2 x3 tomControls u % Initial guess x0 = {icollocate({x1 == 0; x2 == 1; x3 == 0}) collocate(u == 0)}; % Box constraints cbox = {-1 <= collocate(u) <= 1}; % Boundary constraints cbnd = initial({x1 == 0; x2 == 1; x3 == 0}); % ODEs and path constraints ceq = collocate({dot(x1) == x2 dot(x2) == u; dot(x3) == x1.^2}); % Objective objective = final(x3);
options = struct; options.name = 'Singular Control 2'; solution = ezsolve(objective, {cbox, 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: Singular Control 2 f_k 0.268336059478540890 sum(|constr|) 0.000000004483254193 f(x_k) + sum(|constr|) 0.268336063961795100 f(x_0) 0.000000000000000000 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 107 ConJacEv 107 Iter 100 MinorIter 353 CPU time: 0.843750 sec. Elapsed time: 0.875000 sec.
figure(1) plot(t,u,'+-'); legend('u'); title('Singular Control 2 control');