PROPT Disturbance Control

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

Optimal On-Line Control and Classical Regulation Problem, Faina M. Kirillova, Institute of Mathematics National Academy of Sciences of Belarus.

Algorithm of Acting Optimal Controller

Problem Description

Find u over t in [0; 25 ] to minimize:

$J=0$

subject to:

${\frac {dx_{1}}{dt}}=x_{3}$

${\frac {dx_{2}}{dt}}=x_{4}$

${\frac {dx_{3}}{dt}}=-x_{1}+x_{2}+u$

${\frac {dx_{4}}{dt}}=0.1*x_{1}-1.02*x_{2}+0.3*sin(4*t)*(t<9.75)$

$x(t_{0})=[0\ 0\ 2\ 1]$

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

$0<=u<=1$

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

Problem setup

toms t
p = tomPhase('p', t, 0, 25, 80);
setPhase(p);

tomStates x1 x2 x3 x4
tomControls u

% Box constraints
cbox = {0 <= collocate(u) <= 1};

% Boundary constraints
cbnd = {initial({x1 == 0; x2 == 0
    x3 == 2; x4 == 1})
    final({x1 == 0; x2 == 0
    x3 == 0; x4 == 0})};

% ODEs and path constraints
ceq = collocate({
    dot(x1) == x3
    dot(x2) == x4
    dot(x3) == -x1+x2+u
    dot(x4) == 0.1*x1-1.02*x2+0.3*sin(4*t).*(t<9.75)});

% Objective
objective = 0;

Solve the problem

options = struct;
options.name = 'Disturbance Control';
solution = ezsolve(objective, {cbox, cbnd, ceq}, [], options);
t  = subs(collocate(t),solution);
x1 = subs(collocate(x1),solution);
x2 = subs(collocate(x2),solution);
x3 = subs(collocate(x3),solution);
x4 = subs(collocate(x4),solution);
u  = subs(collocate(u),solution);

Problem type appears to be: lp
Time for symbolic processing: 0.069168 seconds
Starting numeric solver
===== * * * =================================================================== * * *
TOMLAB - TOMLAB Development license  999007. Valid to 2011-12-31
=====================================================================================
Problem: ---  1: Disturbance Control            f_k       0.000000000000000000
                                       sum(|constr|)      0.000000000013822610
                              f(x_k) + sum(|constr|)      0.000000000013822610
                                              f(x_0)      0.000000000000000000

Solver: CPLEX.  EXIT=0.  INFORM=1.
CPLEX Dual Simplex LP solver
Optimal solution found

FuncEv  330 Iter  330 
CPU time: 0.124801 sec. Elapsed time: 0.080000 sec. 

Plot result

figure(1);
subplot(2,2,1)
plot(x1,x3,'-');
title('Disturbance control');
legend('x1 vs x3');
subplot(2,2,2)
plot(x2,x4,'-');
legend('x2 vs x4');

subplot(2,2,3)
plot(t,u,'-');
legend('u');

subplot(2,2,4)
plot(t,0.3*sin(4*t).*(t<9.75),'-');
legend('disturbance');