Paper: Solving Tough Optimal Control Problems by Network Enabled Optimization Server (NEOS)
Jinsong Liang, YangQuan Chen, Max Q.-H. Meng, Rees Fullmer Utah State University and Chinese University of Hong Kong (Meng)
EXAMPLE-1: A TEXTBOOK BANG-BANG OPTIMAL CONTROL PROBLEM
Find u over t in [0; t_F ] to minimize
J = tF |
subject to:
| = x2 |
| = u |
x1(0) = 0 |
x1(tF) = 300 |
x2(0) = 0 |
x2(tF) = 0 |
−2 <= u <= 1 |
Reference: [23]
toms t toms t_f p = tomPhase('p', t, 0, t_f, 30); setPhase(p); tomStates x1 x2 tomControls u % Initial guess % Note: The guess for t_f must appear in the list before expression involving t. x0 = {t_f == 20 icollocate({x1 == 300*t/t_f; x2 == 0}) collocate(u==1-2*t/t_f)}; % Box constraints cbox = {10 <= t_f <= 40 -2 <= collocate(u) <= 1}; % Boundary constraints cbnd = {initial({x1 == 0; x2 == 0}) final({x1 == 300; x2 == 0})}; % ODEs and path constraints ceq = collocate({dot(x1) == x2; dot(x2) == u}); % Objective objective = t_f;
options = struct; options.name = 'Bang-Bang Free Time'; options.prilev = 1; solution = ezsolve(objective, {cbox, cbnd, 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: lpcon Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2011-02-05 ===================================================================================== Problem: --- 1: Bang-Bang Free Time f_k 30.019823270451944000 sum(|constr|) 0.000028878808460443 f(x_k) + sum(|constr|) 30.019852149260405000 f(x_0) 20.000000000000000000 Solver: snopt. EXIT=0. INFORM=1. SNOPT 7.2-5 NLP code Optimality conditions satisfied FuncEv 1 ConstrEv 526 ConJacEv 526 Iter 136 MinorIter 185 CPU time: 0.750000 sec. Elapsed time: 0.766000 sec.
subplot(2,1,1) plot(t,x1,'*-',t,x2,'*-'); legend('x1','x2'); title('Bang-Bang Free Time state variables'); subplot(2,1,2) plot(t,u,'+-'); legend('u'); title('Bang-Bang Free Time control');