ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS
6.4 Further example
CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics
n’th-order linear time-invariant system.
Find u over t in [0; 1 ] to minimize
J = | ∫ |
| x′*x + u′*u dt + 10*x1(tF)2 |
subject to:
| = A*x + u |
A = [0 1 0 ... 0 0 0 1 ... 0 ... ... ... 0 0 0 ... 1 1 -2 3 ... (-1)^(n+1)*n]
The initial condition are:
x(0) = [ 1 2 ... n ] , |
−∞ <= u(1:n) <= ∞ . |
Reference: [25]
toms t n = 6; t_F = 1; p = tomPhase('p', t, 0, t_F, 25); setPhase(p); x = tomState('x', n, 1); u = tomState('u', n, 1); nvec = (1:n); A = [sparse(n-1,1), speye(n-1); ... sparse(nvec.*(-1).^(nvec+1))]; % Initial guess guess = icollocate(x == nvec'); % Initial conditions cinit = (initial(x) == nvec'); % ODEs and path constraints ceq = collocate(dot(x) == A*x+u); % Objective objective = 10*final(x(1))^2 + integrate(x'*x + u'*u);
options = struct; options.name = 'Nagurka Problem'; solution = ezsolve(objective, {ceq, cinit}, guess, 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: Nagurka Problem f_k 109.074347751904550000 sum(|constr|) 0.000000000002256926 f(x_k) + sum(|constr|) 109.074347751906810000 f(x_0) 0.000000000000000000 Solver: CPLEX. EXIT=0. INFORM=1. CPLEX Barrier QP solver Optimal solution found FuncEv 3 GradEv 3 ConstrEv 3 Iter 3 CPU time: 0.046875 sec. Elapsed time: 0.032000 sec.
subplot(2,1,1) x1 = x(:,1); plot(t,x1,'*-'); legend('x1'); title('Nagurka Problem - First state variable'); subplot(2,1,2) u1 = u(:,1); plot(t,u1,'+-'); legend('u1'); title('Nagurka Problem - First control variable'); figure(2) surf(t, 1:n, x') xlabel('t'); ylabel('i'); zlabel('x'); title('Nagurka Problem - All state variables'); figure(3) surf(t, 1:n, u') xlabel('t'); ylabel('i'); zlabel('u'); title('Nagurka Problem - All control variables');