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105 Singular Control 5
ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS
10.3 Yeo’s singular control problem
CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics
105.1 Problem Formulation
Find u over t in [0; 1 ] to minimize:
subject to:
| = x12 + x22 + 0.0005*(x2+16*x5−8−0.1*x3*u2)2 |
The initial condition are:
x(0) = [0 −1 −sqrt(5) 0 0] |
The state x4 is implemented as a cost directly. x4 in the implementation is x5. u has a low limit of 9 in the code.
Reference: [25]
105.2 Problem setup
toms t
p = tomPhase('p', t, 0, 1, 80);
setPhase(p)
tomStates x1 x2 x3 x4
tomControls u
% Initial guess
x0 = {icollocate({x1 == 0; x2 == -1
x3 == -sqrt(5); x4 == 0})
collocate(u == 3)};
% Box constraints
cbox = {0 <= collocate(u) <= 10};
% Boundary constraints
cbnd = initial({x1 == 0; x2 == -1
x3 == -sqrt(5); x4 == 0});
% ODEs and path constraints
ceq = collocate({dot(x1) == x2
dot(x2) == -x3.*u + 16*x4 - 8
dot(x3) == u; dot(x4) == 1});
% Objective
objective = integrate(x1.^2 + x2.^2 + ...
0.0005*(x2+16*x4-8-0.1*x3.*u.^2).^2);
105.3 Solve the problem
options = struct;
options.name = 'Singular Control 5';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
t = subs(collocate(t),solution);
u = subs(collocate(u),solution);
Problem type appears to be: con
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license 999001. Valid to 2011-02-05
=====================================================================================
Problem: --- 1: Singular Control 5 f_k 0.119318612949793070
sum(|constr|) 0.000000070783551966
f(x_k) + sum(|constr|) 0.119318683733345030
f(x_0) 1.024412849382205600
Solver: snopt. EXIT=0. INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied
FuncEv 372 GradEv 370 ConstrEv 370 ConJacEv 370 Iter 346 MinorIter 939
CPU time: 10.265625 sec. Elapsed time: 10.609000 sec.
105.4 Plot result
figure(1)
plot(t,u,'+-');
legend('u');
title('Singular Control 5 control');
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