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Chapter 20

Q. 20.8

In Section 20.6.1, MPC was applied to the Wood-Berry distillation column model. A MATLAB program for this example and constrained MPC is shown in Table E20.8. The design parameters have the base case values (Case B in Fig. 20.14) except for P=10 and M=5. The input constraints are the saturation limits for each input (-0.15 and +0.15 ). Evaluate the effects of control horizon M and input weighting matrix R by simulating the set-point change and the first disturbance of Section 20.6.1 for the following parameter values:

(a) Control horizon, M=2 and M=5

(b) Input weighting matrix, R =0.1 I and R = I Consider plots of both inputs and outputs. Which choices of M and R provide the best control? Do any of these MPC controllers provide significantly better control than the controllers shown in Figs. 20.14 and 20.15? Justify your answer.

Table E20.8 MATLAB Program (Based on a program by Morari and Ricker (1994))
g11=poly2tfd(12.8,[16.7 1],0.1); % model
g21=poly2tfd(6.6,[10.9 1],0,7);
g12=poly2tfd(−18.9,[21.0 1],0,3);
g22=poly2tfd(−19.4,[14.4 1],0,3);
gd1=poly2tfd(3.8,[14.9 1],0,8.1);
gd2=poly2tfd(4.9,[13.2 1],0,3.4);
tfinal=120; % Model horizon, N
delt=1; %Sampling perio
ny=2; %Number of outputs
model=tfd2step(tfinal,delt,ny,g11,g21,g12,g22)
plant=model; %No plant/model mismatch
dmodel=[ ] %Default disturbance model
dplant=tfd2step(tfinal,delt,ny,gd1,gd2)
P=10; M=5; % Horizons
ywt=[1 1]; uwt=[0.1 0.1];%Q and R
tend=120; % Final time for simulation
r=[0 1]; %Set-point change in XB
a=zeros([1,tend]);
for i=51:tend
a(i)=0.3∗2.45; %30 %step in F at t=50 min.
end
dstep=[a′];
ulim=[−.15 −.15 .15 .15 1000 1000];%u limits
ylim=[ ]; %No y limits
tfilter=[ ];
[y1,u1]=cmpc(plant,model,ywt,uwt,M,P,tend,r,
ulim,ylim, tfilter,dplant,dmodel,dstep);
figure(1)
subplot(211)
plot(y1)
legend(′XD′,′XB′)
xlabel(′Time (min)′)
subplot(212)
stairs(u1)% Plot inputs as staircase functions
legend(′R′,′S′)
xlabel(′Time (min)′)

Step-by-Step

Verified Solution

(Use MATLAB Model Predictive Control Toolbox)

a) M=5 vs. M=2

b) R =0.1 I . . vs \quad R = I

Notice that the larger control horizon M and the smaller input weighting R, the more control effort is needed.