PROPT Singular Control 4: Difference between revisions
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<math> J = x_4(t_F) </math> | <math> J = x_4(t_F) </math> | ||
subject to: | subject to: | ||
<math> \frac{dx_1}{dt} = x_2 </math> | <math> \frac{dx_1}{dt} = x_2 </math> | ||
<math> \frac{dx_2}{dt} = x_3 </math> | <math> \frac{dx_2}{dt} = x_3 </math> | ||
<math> \frac{dx_3}{dt} = u </math> | <math> \frac{dx_3}{dt} = u </math> | ||
<math> \frac{dx_4}{dt} = x_1^2 </math> | <math> \frac{dx_4}{dt} = x_1^2 </math> | ||
The initial condition are: | The initial condition are: | ||
<math> x(0) = [1 \ 0 \ 0 \ 0] </math> | <math> x(0) = [1 \ 0 \ 0 \ 0] </math> | ||
<math> -1 <= u <= 1 </math> | <math> -1 <= u <= 1 </math> | ||
<source lang="matlab"> | <source lang="matlab"> |
Revision as of 08:12, 9 November 2011
This page is part of the PROPT Manual. See PROPT Manual. |
ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS
10.2.3 Example 4
CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics
Problem Formulation
Find u over t in [0; 5 ] to minimize
subject to:
The initial condition are:
% Copyright (c) 2007-2008 by Tomlab Optimization Inc.
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);
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 Time for symbolic processing: 0.089315 seconds Starting numeric solver ===== * * * =================================================================== * * * TOMLAB - TOMLAB Development license 999007. Valid to 2011-12-31 ===================================================================================== Problem: --- 1: Singular Control 4 f_k 1.252389645383044100 sum(|constr|) 0.000000063932037643 f(x_k) + sum(|constr|) 1.252389709315081800 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: 9.313260 sec. Elapsed time: 2.553000 sec.
Plot result
figure(1)
plot(t,u,'+-');
legend('u');
title('Singular Control 4 control');