PROPT Grusins Metric

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

A Short Introduction to Optimal Control, Ugo Boscain, SISSA, Italy

3.4 A Detailed Application: the Grusin's Metric

Problem Description

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

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

subject to:

${\frac {dx_{1}}{dt}}=u_{1}$

${\frac {dx_{2}}{dt}}=u_{2}*x_{1}$

$x(t_{0})=[0\ 0]$

$x(t_{f})=[-0.001\ -1]$

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

Problem setup

toms t
p = tomPhase('p', t, 0, 1, 60);
setPhase(p);

tomStates x1 x2
tomControls u1 u2

% Boundary constraints
cbnd = {initial({x1 == 0; x2 == 0})
    final({x1 == -0.01; x2 == -1})};

% ODEs and path constraints
ceq = collocate({dot(x1) == u1
    dot(x2) == u2.*x1});

% Objective
objective = integrate(u1.^2+u2.^2);

Solve the problem

options = struct;
options.name = 'Grusins Metric';
solution = ezsolve(objective, {cbnd, ceq}, [], options);
t  = subs(collocate(t),solution);
x1 = subs(collocate(x1),solution);
x2 = subs(collocate(x2),solution);
u1 = subs(collocate(u1),solution);
u2 = subs(collocate(u2),solution);

Problem type appears to be: qpcon
Time for symbolic processing: 0.089288 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Grusins Metric                 f_k       6.233154129250785900
                                       sum(|constr|)      0.000000060262669206
                              f(x_k) + sum(|constr|)      6.233154189513455500
                                              f(x_0)      0.000000000000000000

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

FuncEv    1 ConstrEv   45 ConJacEv   45 Iter   38 MinorIter  269
CPU time: 0.218401 sec. Elapsed time: 0.210000 sec. 

Plot result

subplot(2,1,1)
plot(x1,x2,'*-');
legend('x1 vs x2');
title('Grusins Metric state variables');

subplot(2,1,2)
plot(t,u1,'+-',t,u2,'+-');
legend('u1','u2');
title('Grusins Metric control');