PROPT Genetic 2

This page is part of the PROPT Manual. See PROPT Manual.

PROCEEDINGS OF WORLD ACADEMY OF SCIENCE, ENGINEERING AND TECHNOLOGY VOLUME 21 JANUARY 2007 ISSN 1307-6884

Optimal Control Problem, Quasi-Assignment Problem and Genetic Algorithm Omid S. Fard and Akbar H. Borzabadi

See paper for failure of GA toolbox algorithm.

Example 2

Problem Formulation

Find u over t in [0; 1 ] to minimize

$J=\int _{0}^{1}u^{2}\mathrm {d} t$

subject to:

${\frac {dx}{dt}}={\frac {1}{2}}*x^{2}*sin(x)+u$

The initial condition are:

$x(0)=0$

$x(1)=0.5$

% Copyright (c) 2007-2008 by Tomlab Optimization Inc.

Problem setup

toms t
p = tomPhase('p', t, 0, 1, 50);
setPhase(p);

tomStates x
tomControls u

% Initial guess
x0 = {icollocate(x == 0.5*t)
    collocate(u == 0)};

% Boundary constraints
cbnd = {initial(x == 0)
    final(x == 0.5)};

% ODEs and path constraints
ceq = collocate(dot(x) == 1/2*x.^2.*sin(x)+u);

% Objective
objective = integrate(u.^2);

Solve the problem

options = struct;
options.name = 'Genetic 2';
solution = ezsolve(objective, {cbnd, ceq}, x0, options);
t = subs(collocate(t),solution);
x = subs(collocate(x),solution);
u = subs(collocate(u),solution);

Problem type appears to be: qpcon
Time for symbolic processing: 0.079692 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Genetic 2                      f_k       0.235327080033220280
                                       sum(|constr|)      0.000000001552022066
                              f(x_k) + sum(|constr|)      0.235327081585242350
                                              f(x_0)      0.000000000000000000

Solver: snopt.  EXIT=0.  INFORM=1.
SNOPT 7.2-5 NLP code
Optimality conditions satisfied

FuncEv    1 ConstrEv   23 ConJacEv   23 Iter   21 MinorIter   72
CPU time: 0.031200 sec. Elapsed time: 0.031000 sec. 

Plot result

subplot(2,1,1)
plot(t,x,'*-');
legend('x');
title('Genetic 2 state variables');

subplot(2,1,2)
plot(t,u,'+-');
legend('u');
title('Genetic 2 control');