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27 Continuous State Constraint Problem
Problem 2: Miser3 manual
27.1 Problem description
Find u(t) over t in [0; 1 ] to minimize
J = | ∫ | | x1(t)2 + x2(t)2 + 0.005*u(t)2 dt |
subject to:
8*(t−0.5)2 − 0.5 − x2 >=0 |
Reference: [19]
27.2 Problem setup
toms t
p = tomPhase('p', t, 0, 1, 50);
setPhase(p);
tomStates x1 x2
tomControls u
% Initial guess
x0 = {icollocate({x1 == 0; x2 == -1})
collocate(u==0)};
% Box constraints
cbox = {-10 <= icollocate(x1) <= 10
-10 <= icollocate(x2) <= 10
-20 <= collocate(u) <= 20};
% Boundary constraints
cbnd = initial({x1 == 0; x2 == -1});
% ODEs and path constraints
ceq = collocate({
dot(x1) == x2
dot(x2) == -x2+u
8*(t-0.5).^2-0.5-x2 >= 0 % Path constr.
});
% Objective
objective = integrate(x1.^2 + x2.^2 + 0.005*u.^2);
27.3 Solve the problem
options = struct;
options.name = 'Cont State Constraint';
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: qp
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2011-02-05
=====================================================================================
Problem: 1: Cont State Constraint f_k 0.169824305998486440
sum(|constr|) 0.000000000079892583
f(x_k) + sum(|constr|) 0.169824306078379030
f(x_0) 0.000000000000000000
Solver: CPLEX. EXIT=0. INFORM=1.
CPLEX Barrier QP solver
Optimal solution found
FuncEv 10 GradEv 10 ConstrEv 10 Iter 10
CPU time: 0.531250 sec. Elapsed time: 0.532000 sec.
27.4 Plot result
subplot(3,1,1)
plot(t,x1,'*-',t,x2,'*-');
legend('x1','x2');
title('Cont State Constraint state variables');
subplot(3,1,2)
plot(t,u,'+-');
legend('u');
title('Cont State Constraint control');
subplot(3,1,3)
ieq = 8*(t-0.5).^2-0.5-x2;
plot(t,ieq,'+-');
axis([0 1 0 5]);
legend('path');
title('Cont State Constraint path constraint');
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