PROPT Hyper Sensitive Optimal Control

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

Eigenvector approximate dichotomic basis method for solving hyper-sensitive optimal control problems 2000, Anil V. Rao and Kenneth D. Mease

3.1. Motivating example, a hyper-sensitive HBVP

Problem Formulation

Find u(t) over t in [0; t_f ] to minimize

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J = \int_0^{t_f} (x^2 + u^2) \mathrm{d}t }

subject to:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{dx}{dt} = -x^3+u } Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0 = 1 } Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{t_f} = 1.5 } Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t_f = 10 }

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

Problem setup

toms t
p = tomPhase('p', t, 0, 10, 50);
setPhase(p);

tomStates x
tomControls u

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

% bounds and ODEs
bceq = {collocate(dot(x) == -x.^3+u)
    initial(x) == 1; final(x) == 1.5};

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

Solve the problem

options = struct;
options.name = 'Hyper Sensitive';
solution = ezsolve(objective, bceq, 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: 0.080469 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Hyper Sensitive                f_k       6.723925391388339900
                                       sum(|constr|)      0.000000002440693436
                              f(x_k) + sum(|constr|)      6.723925393829033800
                                              f(x_0)      0.000000000000000000

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

FuncEv    1 ConstrEv   26 ConJacEv   26 Iter   21 MinorIter   70
CPU time: 0.031200 sec. Elapsed time: 0.031000 sec. 

Plot result

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

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