PROPT Global Dynamic System

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

Deterministic Global Optimization of Nonlinear Dynamic Systems, Youdong Lin and Mark A. Stadtherr, Department of Chemical and Biomolecular Engineering, University of Notre Dame

Problem Description

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

$J=-x^{2}(t_{f})$

subject to:

${\frac {dx}{dt}}=-x^{2}+u$

$x(t_{0})=9$

$-5<=u<=5$

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

Problem setup

toms t
p = tomPhase('p', t, 0, 1, 20);
setPhase(p);

tomStates x
tomControls u

% Box constraints, bounds and odo
c = {-10 <= icollocate(x) <= 10
    -5  <= collocate(u)  <= 5
    initial(x == 9)
    collocate(dot(x) == -x.^2+u)};

Solve the problem

options = struct;
options.name = 'Global Dynamic System';
Prob = sym2prob('con',-final(x)^2, c, [], options);
Prob.xInit = 20;
Result = tomRun('multiMin', Prob, 1);
solution = getSolution(Result);
t = subs(collocate(t),solution);
x = subs(collocate(x),solution);
u = subs(collocate(u),solution);

===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Global Dynamic System - Trial 16  f_k      -8.232621699200782600
                                          sum(|constr|)      0.000000000019587887
                                 f(x_k) + sum(|constr|)     -8.232621699181194700

Solver: multiMin with local solver snopt.  EXIT=0.  INFORM=0.
Find local optima using multistart local search
Did 20 local tries. Found 1 global, 2 minima. TotFuncEv 822. TotConstrEv 782

FuncEv  822 GradEv  782 ConstrEv  782 ConJacEv   35 Iter  482 
CPU time: 0.452403 sec. Elapsed time: 0.451000 sec. 

Plot result

figure(1);
subplot(2,1,1);
plot(t,x,'*-');
legend('x');

subplot(2,1,2);
plot(t,u,'*-');
legend('u');