Optimal Control CY3H2, Lecture notes by Victor M. Becerra, School of Systems Engineering, University of Reading
Heating a room using the least possible energy.
Find u over t in [0; 1] to minimize:
J = |
| ∫ |
| u2 dt |
subject to:
| = −2*x + u |
x(0) = 0, |
x(1) = 10 |
toms t p = tomPhase('p', t, 0, 1, 20); setPhase(p); tomStates x tomControls u % Initial guess x0 = {icollocate(x == 10*t) collocate(u == 1)}; % Box constraints cbox = collocate(0 <= u); % Boundary constraints cbnd = {initial(x == 0) final(x == 10)}; % ODEs and path constraints ceq = collocate(dot(x) == -2*x+u); % Objective objective = 0.5*integrate(u^2);
options = struct; options.name = 'Temperature Control'; solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options); t = subs(collocate(t),solution); x = subs(collocate(x),solution); u = subs(collocate(u),solution);
Problem type appears to be: qp Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2011-02-05 ===================================================================================== Problem: 1: Temperature Control f_k 203.731472072763980000 sum(|constr|) 0.000000000046672914 f(x_k) + sum(|constr|) 203.731472072810650000 f(x_0) 0.000000000000000000 Solver: CPLEX. EXIT=0. INFORM=1. CPLEX Barrier QP solver Optimal solution found FuncEv 9 GradEv 9 ConstrEv 9 Iter 9 CPU time: 0.093750 sec. Elapsed time: 0.063000 sec.
figure(1); subplot(2,1,1) plot(t,x,'*-'); legend('Temperature'); subplot(2,1,2) plot(t,u,'*-'); legend('Energy');