In Section 20.6.1, MPC was applied to the Wood-Berry distillation column model. A MATLAB program for this example and constrained MPC are shown in Table E20.9.

The design parameters have the base case values (Case B in Fig.) 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 5

(b) Input weighting matrix, R = 0.1I and R = I

Consider plots of both inputs and outputs. Which choices of M and B provide the best control? Do any of these MPC controllers provide significantly better control than the controllers shown in Figs. 20.13 and 20.14? Justify your answer.

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Table E20.9 MATLAB Program. (Based on a program by Morari and Ricker (1994).) gl1-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; delt=1; % Model horizon, N % Sampling period % Number of outputs ny=2; model=tfd2step(tfinal,delt,ny,g11,g21,g12.g22) plant=model; % No plant/model mismatch dmodel=[] 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; % Default disturbance model %3D % 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=[]; tfilter=[ ]; [y1,ul]=cmpc(plant,model,ywt,uwt,M,P,tend,r, ulim,ylin, tfilter, dplant,dmodel,dstep); figure(1) subplot(211) plot(yl) legend('XD','XB') xlabel('Time (min)') subplot(212) stairs(ul) % Plot inputs as staircase functions legend('R','S') xlabel("Time (min)') % No y limits