Arthur Bryson - Dynamic Optimization
Example about landing an object.
Reference: [9]
toms t toms t_f p = tomPhase('p', t, 0, t_f, 60); setPhase(p); tomStates altitude speed mass tomControls thrust % Initial guess x0 = {t_f == 1.5 icollocate({ altitude == 1-t/t_f speed == -0.783+0.783*t/t_f mass == 1-0.99*t/t_f }) collocate(thrust == 0)}; % Box constraints cbox = { 0 <= t_f <= 1000 -20 <= icollocate(altitude) <= 20 -20 <= icollocate(speed) <= 20 0.01 <= icollocate(mass) <= 1 0 <= collocate(thrust) <= 1.227}; % Boundary constraints cbnd = {initial({altitude == 1; speed == -0.783; mass == 1}) final({altitude == 0; speed == 0})}; % ODEs and path constraints exhaustvelocity = 2.349; gravity = 1; ceq = collocate({ dot(altitude) == speed dot(speed) == -gravity + thrust./mass dot(mass) == -thrust./exhaustvelocity}); % Objective objective = integrate(thrust);
options = struct; options.name = 'Moon Lander'; solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options); t = subs(collocate(t),solution); altitude = subs(collocate(altitude),solution); speed = subs(collocate(speed),solution); mass = subs(collocate(mass),solution); thrust = subs(collocate(thrust),solution);
Problem type appears to be: qpcon Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2011-02-05 ===================================================================================== Problem: --- 1: Moon Lander f_k 1.420346223923628400 sum(|constr|) 0.000000000118734605 f(x_k) + sum(|constr|) 1.420346224042363000 f(x_0) 0.000000000000000000 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 70 ConJacEv 70 Iter 21 MinorIter 1434 CPU time: 0.937500 sec. Elapsed time: 0.984000 sec.
subplot(2,1,1) plot(t,altitude,'*-',t,speed,'*-',t,mass,'*-'); legend('altitude','speed','mass'); title('Moon Lander state variables'); subplot(2,1,2) plot(t,thrust,'+-'); legend('thrust'); title('Moon Lander control');