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89 Quadruple Integral
ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS
12.4.5 Example 5
CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics
89.1 Problem Formulation
Find u over t in [0; t_F ] to minimize
subject to:
The initial condition are:
Reference: [25]
89.2 Problem setup
toms t
toms t_f
p = tomPhase('p', t, 0, t_f, 30);
setPhase(p);
tomStates x1 x2 x3 x4
tomControls u
% Initial guess
x0 = {t_f == 5
icollocate({x1 == 0.1-0.1*t/t_f
x2 == 0.2-0.2*t/t_f; x3 == 0.3-0.3*t/t_f})
collocate(u == -1)};
% Box constraints
cbox = {0.1 <= t_f <= 100
-1 <= collocate(u) <= 1};
% Boundary constraints
cbnd = {initial({x1 == 0.1; x2 == 0.2; x3 == 0.3; x4 == 0})
final({x1 == 0; x2 == 0; x3 == 0; x4 == 0})};
% ODEs and path constraints
ceq = collocate({
dot(x1) == x2; dot(x2) == x3
dot(x3) == x4; dot(x4) == u});
% Objective
objective = t_f;
89.3 Solve the problem
options = struct;
options.name = 'Quadruple Integral';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
t = subs(collocate(t),solution);
x1 = subs(collocate(x1),solution);
x2 = subs(collocate(x2),solution);
x3 = subs(collocate(x3),solution);
x4 = subs(collocate(x4),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: Quadruple Integral f_k 4.849598187644263100
sum(|constr|) 0.000000078353877264
f(x_k) + sum(|constr|) 4.849598265998140300
f(x_0) 5.000000000000000000
Solver: snopt. EXIT=0. INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied
FuncEv 1 ConstrEv 58 ConJacEv 58 Iter 31 MinorIter 474
CPU time: 0.250000 sec. Elapsed time: 0.282000 sec.
89.4 Plot result
subplot(2,1,1)
plot(t,x1,'*-',t,x2,'*-',t,x3,'*-',t,x4,'*-');
legend('x1','x2','x3','x4');
title('Quadruple Integral state variables');
subplot(2,1,2)
plot(t,u,'+-');
legend('u');
title('Quadruple Integral control');
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