Disturbance Control

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

Contents

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
===== * * * =================================================================== * * *
TOMLAB - Tomlab Optimization Inc. Development license  999001. Valid to 2010-02-05
=====================================================================================
Problem: ---  1: Disturbance Control            f_k       0.000000000000000000
                                       sum(|constr|)      0.000000000046160833
                              f(x_k) + sum(|constr|)      0.000000000046160833
                                              f(x_0)      0.000000000000000000

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

FuncEv  336 Iter  336 
CPU time: 0.171875 sec. Elapsed time: 0.172000 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');