Dynamic Matrix Control Tutorial

Dynamic Matrix Control (DMC) was the first Model Predictive Control (MPC) algorithm introduced in early 1980s. Nowadays, DMC is available in almost all commercial industrial distributed control systems and process simulation software packages. This tutorial intends to explain the features of DMC using the dmc function developed by the author.

Contents

• Example: A Water Heater
• The Step Response Model
• DMC without Setpoint Prediction
• DMC with Setpoint Prediction
• DMC with Different Control Weight
• Reference

Example: A Water Heater

Consider a water heater as shown in the following figure, where the cold water is heated by means of a gas burner. The aim of DMC is by manipulating valve to control the gas flow so that the outlet temperature is at desired level.

The Step Response Model

The DMC algorithm works with a step response model. The first step to design a DMC controller is to perform a step test on the plant to generate a step response model. The step response of the water heater is obtained through such a test and given as follows:

p.sr = [0;0;0.271;0.498;0.687;0.845;0.977;1.087;1.179;1.256;...
        1.320;1.374;1.419;1.456;1.487;1.513;1.535;1.553;1.565;1.581;...
        1.592;1.600;1.608;1.614;1.619;1.632;1.627;1.630;1.633;1.635];

% The response is converged after 30 steps.

DMC without Setpoint Prediction

Setup the DMC

N=120;  % Total simulation length (samples)
p.p=10; % Prediction horizon
p.m=5;  % Moving horizon
p.la=1; % Control weight
% Reference (setpoint)
R=[ones(30,1);zeros(30,1);ones(30,1);zeros(30,1)];
p.y=0;  % Initial output
p.v=[]; % empty past input to indicate initialization
% buffer of input to cope with time delay
u=zeros(3,1);
% Initialization of variables for results
Y=zeros(N,1);
U=zeros(N,1);
% DMC Simulation
for k=1:120
    p.a=0;
    p.r=R(k);   % DMC only knows current setpoint
    if k>60     % change smoothing factor for second half simulation
        p.a=0.7;
    end
    p=dmc(p);
    Y(k)=p.y;
    U(k)=p.u;
    u=[u(2:3);p.u];
    p.y=0.8351*p.y+0.2713*u(1); % actual plant output
end
% DMC results
subplot(211)
plot(1:N,Y,'b-',1:N,R,'r--',[60 60],[-0.5 1.5],':','linewidth',2)
title('solid: output, dashed: reference')
text(35,1,'\alpha=0')
text(95,1,'\alpha=0.7')
axis([0 120 -0.5 1.5])
subplot(212)
[xx,yy]=stairs(1:N,U);
plot(xx,yy,'-',[60 60],[-0.5 1.5],':','linewidth',2)
axis([0 120 -0.5 1.5])
title('input, \lambda=1')
xlabel('time, min')

DMC with Setpoint Prediction

Setup the DMC

N=120;  % Total simulation length (samples)
p.p=10; % Prediction horizon
p.m=5;  % Moving horizon
p.la=1; % Control weight
% Reference (setpoint)
R=[ones(30,1);zeros(30,1);ones(30,1);zeros(30,1)];
p.y=0;  % Initial output
p.v=[]; % empty past input to indicate initialization
% buffer of input to cope with time delay
u=zeros(3,1);
% Initialization of variables for results
Y=zeros(N,1);
U=zeros(N,1);
% DMC Simulation
for k=1:120
    p.a=0;
    p.r=R(k:min(N,k+p.p)); % DMC knows future setpoint
    if k>60     % change smoothing factor for second half simulation
        p.a=0.7;
    end
    p=dmc(p);
    Y(k)=p.y;
    U(k)=p.u;
    u=[u(2:3);p.u];
    p.y=0.8351*p.y+0.2713*u(1); % actual plant output
end
% DMC results
subplot(211)
plot(1:N,Y,'b-',1:N,R,'r--',[60 60],[-0.5 1.5],':','linewidth',2)
title('solid: output, dashed: reference')
text(35,1,'\alpha=0')
text(95,1,'\alpha=0.7')
axis([0 120 -0.5 1.5])
subplot(212)
[xx,yy]=stairs(1:N,U);
plot(xx,yy,'-',[60 60],[-0.5 1.5],':','linewidth',2)
axis([0 120 -0.5 1.5])
title('input, \lambda=1')
xlabel('time, min')

DMC with Different Control Weight

Setup the DMC

N=120;  % Total simulation length (samples)
p.p=10; % Prediction horizon
p.m=5;  % Moving horizon
p.la=0.1; % Control weight
% Reference (setpoint)
R=[ones(30,1);zeros(30,1);ones(30,1);zeros(30,1)];
p.y=0;  % Initial output
p.v=[]; % empty past input to indicate initialization
% buffer of input to cope with time delay
u=zeros(3,1);
% Initialization of variables for results
Y=zeros(N,1);
U=zeros(N,1);
% DMC Simulation
for k=1:120
    p.a=0;
    p.r=R(k:min(N,k+p.p)); % DMC knows future setpoint
    if k>60     % change smoothing factor for second half simulation
        p.a=0.7;
    end
    p=dmc(p);
    Y(k)=p.y;
    U(k)=p.u;
    u=[u(2:3);p.u];
    p.y=0.8351*p.y+0.2713*u(1); % actual plant output
end
% DMC results
subplot(211)
plot(1:N,Y,'b-',1:N,R,'r--',[60 60],[-0.5 1.5],':','linewidth',2)
title('solid: output, dashed: reference')
text(35,1,'\alpha=0')
text(95,1,'\alpha=0.7')
axis([0 120 -0.5 1.5])
subplot(212)
[xx,yy]=stairs(1:N,U);
plot(xx,yy,'-',[60 60],[-0.5 1.5],':','linewidth',2)
axis([0 120 -1 2])
title('input, \lambda=0.1')
xlabel('time, min')

Reference

Camacho, E.F. and Bordons, C., Model Predictive Control, Springer-Verlag, 1999.

Published with MATLAB® 7.6