Time Delay 2 (Approximate)

ITERATIVE DYNAMIC PROGRAMMING, REIN LUUS

8.3.2 Example 2

CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics

Linear time-delay system considered by Palanisamy et al.

Contents

Problem Formulation

Find u over t in [0; 2 ] to minimize

$$ J = x_2(t_F) $$

subject to:

$$ \frac{dx_1}{dt} = t*x_1 + x_1(t-tau) + u $$

$$ \frac{dx_2}{dt} = x_1^2 + u^2 $$

$$ tau = 1 $$

A Taylor series expansion gives:

$$ \frac{dx_1}{dt} \approx \frac{(t+1)*x_1 + u }{1 + tau} $$

The initial condition are:

$$ x(0) = [1 \ 0] $$

$$ -inf <= u <= inf $$

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

Problem setup

toms t
p = tomPhase('p', t, 0, 2, 50);
setPhase(p);

tomStates x1 x2
tomControls u

% Initial guess
x0 = {icollocate({x1 == 1; x2 == 0})
    collocate(u == 0)};

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

% ODEs and path constraints
tau = 1;
ceq = collocate({
    dot(x1) == ((t+1).*x1+u)/(1+tau)
    dot(x2) == x1.^2+u.^2});

% Objective
objective = final(x2);

Solve the problem

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

Problem type appears to be: lpcon
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Time Delay 2 Appr.             f_k       5.340734691399960700
                                       sum(|constr|)      0.000000236393505091
                              f(x_k) + sum(|constr|)      5.340734927793465500
                                              f(x_0)      0.000000000000000000

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

FuncEv    1 ConstrEv   31 ConJacEv   31 Iter   26 MinorIter  138
CPU time: 0.187500 sec. Elapsed time: 0.172000 sec.

Plot result

figure(1)
plot(t,u,'+-');
legend('u');
title('Time Delay 2 control');