Minimum Cost Optimal Control: An Application to Flight Level Tracking, John Lygeros, Department of Engineering, University of Cambridge, Cambridge, UK.
Find scalar w over t in [0; t_F ] to minimize:
J = | ∫ |
| (x32) dt |
subject to:
Equations in the code.
Reference: [26]
toms t p = tomPhase('p', t, 0, 100, 60); setPhase(p); tomStates x1s x2 x3s tomControls u1s u2 x1 = x1s*100; x3 = x3s*10; u1 = u1s*10e3; cr2d = pi/180; % Box constraints cbox = { 92 <= icollocate(x1) <= 170 -20*cr2d <= icollocate(x2) <= 25*cr2d -150 <= icollocate(x3) <= 150 60e3 <= collocate(u1) <= 120e3 -150 <= collocate(u2) <= 150}; % Boundary constraints cbnd = initial({x1 == 153.73 x2 == 0; x3 == 0}); L = 65.3; D = 3.18; m = 160e3; g = 9.81; c = 6; % ODEs and path constraints ceq = collocate({ dot(x1) == (-D/m*x1.^2-g*sin(x2)+u1/m) dot(x2) == L/m*x1.*(1-c*x2)-g*cos(x2)./x1+L*c/m*u2 dot(x3) == (x1.*sin(x2))}); % Objective objective = integrate(x3.^2);
options = struct; options.name = 'Flight Path Tracking'; solution = ezsolve(objective, {cbox, cbnd, ceq}, [], options); t = subs(collocate(t),solution); x1 = subs(collocate(x1),solution); x2 = subs(collocate(x2),solution); x3 = subs(collocate(x3),solution); u1 = subs(collocate(u1),solution); u2 = subs(collocate(u2),solution);
Problem type appears to be: qpcon Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2011-02-05 ===================================================================================== Problem: --- 1: Flight Path Tracking f_k 0.000000009789752895 sum(|constr|) 0.000000000014497456 f(x_k) + sum(|constr|) 0.000000009804250351 f(x_0) 0.000000000000000000 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 444 ConJacEv 444 Iter 421 MinorIter 720 CPU time: 13.125000 sec. Elapsed time: 13.344000 sec.
figure(1); subplot(2,3,1); plot(t,x3,'-'); title('Alt') subplot(2,3,2); plot(t,x1,'-'); title('Vel') subplot(2,3,3); plot(t,x2,'-'); title('Gamma') subplot(2,3,4); plot(t,u2,'-'); title('Angle') subplot(2,3,5); plot(t,u1,'-'); title('Thrust')