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104 Singular Control 4
ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS
10.2.3 Example 4
CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics
104.1 Problem Formulation
Find u over t in [0; 5 ] to minimize
subject to:
The initial condition are:
Reference: [25]
104.2 Problem setup
toms t
p = tomPhase('p', t, 0, 5, 100);
setPhase(p)
tomStates x1 x2 x3 x4
tomControls u
% Initial guess
x0 = {icollocate({x1 == 1; x2 == 0
x3 == 0; x4 == 0})
collocate(u == 0)};
% Box constraints
cbox = {-1 <= collocate(u) <= 1};
% Boundary constraints
cbnd = initial({x1 == 1; x2 == 0
x3 == 0; x4 == 0});
% ODEs and path constraints
ceq = collocate({dot(x1) == x2; dot(x2) == x3
dot(x3) == u; dot(x4) == x1.^2});
% Objective
objective = final(x4);
104.3 Solve the problem
options = struct;
options.name = 'Singular Control 4';
solution = ezsolve(objective, {cbox, cbnd, ceq}, x0, options);
t = subs(collocate(t),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: Singular Control 4 f_k 1.252389645383043400
sum(|constr|) 0.000000063932046493
f(x_k) + sum(|constr|) 1.252389709315090000
f(x_0) 0.000000000000000000
Solver: snopt. EXIT=0. INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied
FuncEv 1 ConstrEv 92 ConJacEv 92 Iter 89 MinorIter 652
CPU time: 5.484375 sec. Elapsed time: 5.672000 sec.
104.4 Plot result
figure(1)
plot(t,u,'+-');
legend('u');
title('Singular Control 4 control');
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