Viscocity Solutions of Hamilton-Jacobi Equations and Optimal Control Problems. Alberto Bressan, S.I.S.S.A, Trieste, Italy.
A linear pendulum problem controlled by an external force.
Find u over t in [0; 20 ] to maximize:
J = x1(tf) |
subject to:
| = x2 |
| = u−x1 |
x(t0) = [0 0] |
|u| <= 1 |
Reference: [8]
toms t t_f = 20; p = tomPhase('p', t, 0, t_f, 60); setPhase(p); tomStates x1 x2 tomControls u % Initial guess x0 = {icollocate({x1 == 0; x2 == 0}) collocate(u == 0)}; % Box constraints and bounds cb = {-1 <= collocate(u) <= 1 initial(x1 == 0) initial(x2 == 0)}; % ODEs and path constraints ceq = collocate({dot(x1) == x2 dot(x2) == u-x1}); % Objective objective = -final(x1);
options = struct; options.name = 'Linear Pendulum'; solution = ezsolve(objective, {cb, ceq}, x0, options); t = subs(collocate(t),solution); x1 = subs(collocate(x1),solution); x2 = subs(collocate(x2),solution); u = subs(collocate(u),solution);
Problem type appears to be: lp Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2011-02-05 ===================================================================================== Problem: --- 1: Linear Pendulum f_k -12.612222977985938000 sum(|constr|) 0.000000000003687556 f(x_k) + sum(|constr|) -12.612222977982251000 f(x_0) 0.000000000000000000 Solver: CPLEX. EXIT=0. INFORM=1. CPLEX Dual Simplex LP solver Optimal solution found FuncEv 206 Iter 206 CPU time: 0.031250 sec. Elapsed time: 0.031000 sec.
subplot(3,1,1) plot(t,x1,'*-',t,x2,'*-'); legend('x1','x2'); title('Linear Pendulum state variables'); subplot(3,1,2) plot(t,u,'+-'); legend('u'); title('Linear Pendulum control'); subplot(3,1,3) plot(t,sign(sin(t_f-t)),'*-'); legend('Known u'); title('Linear Pendulum known solution');