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111 A Simple Terminal Constraint Problem
Problem 1: Miser3 manual
111.1 Problem Description
Find u(t) over t in [0; 1 ] to minimize
subject to:
111.2 Problem setup
toms t1
p1 = tomPhase('p1', t1, 0, 0.75, 20);
toms t2
p2 = tomPhase('p2', t2, 0.75, 0.25, 20);
setPhase(p1);
tomStates x1p1
tomControls up1
setPhase(p2);
tomStates x1p2
tomControls up2
setPhase(p1);
% Initial guess
x01 = {icollocate({x1p1 == 1-0.1*t1/0.75})
collocate(up1==0.9*t1/0.75)};
% Box constraints
cbox1 = {-10 <= icollocate(p1,x1p1) <= 10
-10 <= collocate(p1,up1) <= 10};
% Boundary constraints
cbnd1 = initial(x1p1 == 1);
% ODEs and path constraints
ceq1 = collocate(dot(x1p1) == up1);
% Objective
objective1 = integrate(x1p1.^2+up1.^2);
setPhase(p2);
% Initial guess
x02 = {icollocate({x1p2 == 1-0.1*t2})
collocate(up2==0.9+0.1*t2)};
% Box constraints
cbox2 = {-10 <= icollocate(p2,x1p2) <= 10
-10 <= collocate(p2,up2) <= 10};
% Boundary constraints
cbnd2 = {initial(x1p2 == 0.9)
final(x1p2 == 0.75)};
% ODEs and path constraints
ceq2 = collocate(dot(x1p2) == up2);
% Objective
objective2 = integrate(x1p2.^2+up2.^2);
% Objective
objective = objective1 + objective2;
% Link phase
link = {final(p1,x1p1) == initial(p2,x1p2)};
111.3 Solve the problem
options = struct;
options.name = 'Terminal Constraint 2';
constr = {cbox1, cbnd1, ceq1, cbox2, cbnd2, ceq2, link};
solution = ezsolve(objective, constr, {x01, x02}, options);
t = subs(collocate(p1,t1),solution);
t = [t;subs(collocate(p2,t2),solution)];
x1 = subs(collocate(p1,x1p1),solution);
x1 = [x1;subs(collocate(p2,x1p2),solution)];
u = subs(collocate(p1,up1),solution);
u = [u;subs(collocate(p2,up2),solution)];
Problem type appears to be: qp
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2011-02-05
=====================================================================================
Problem: 1: Terminal Constraint 2 f_k 0.920531441472477340
sum(|constr|) 0.000000000559899399
f(x_k) + sum(|constr|) 0.920531442032376690
f(x_0) 0.000000000000000000
Solver: CPLEX. EXIT=0. INFORM=1.
CPLEX Barrier QP solver
Optimal solution found
FuncEv 8 GradEv 8 ConstrEv 8 Iter 8
CPU time: 0.031250 sec. Elapsed time: 0.031000 sec.
111.4 Plot result
subplot(2,1,1)
plot(t,x1,'*-');
legend('x1');
title('Terminal Constraint state variable');
subplot(2,1,2)
plot(t,u,'+-');
legend('u');
title('Terminal Constraint control');
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