1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
|
clear
clc
//Example 22.2
disp('Example 22.2')
//Author: Dhruv Gupta....Aug 4, 2013
//<dgupta6@wisc.edu>
K=[0.2 0.58 0.35;0.25 1.10 1.3;0.3 0.7 1.2];
tau=[2 2 2;3 3 3;4 4 4];
s=%s;
G=K./(1+tau*s);
RGA=K.*inv(K');
disp(RGA,"RGA=")
//IMC based tuning
tauC=5;
Kc=diag(tau/tauC./K);
mprintf("\n\nThe tauI given in book are wrong\n...
refer to Table 11.1 for correct formula\n\n")
tauI=diag(tau)+1;
mprintf('\nWe still however use the ones given in book\n');
disp(Kc,"Kc=")
disp(tauI,"tauI=")
//Refer to Eqns 15-23 and 15-24
Gc=Kc.*(1+(1)./tauI/s);
//For the sake of brevity we write Gstar as G
//We will account for delays in the for loop that we will write
//Refer to Figure 15.9 Page 295 for details of Smith Predictor
//====Making step response models of the continuos transfer functions====//
Ts=0.1;//Sampling time ie delta_T
delay=3/Ts;
N=150/Ts;//Model Order
s=%s;
G=syslin('c',diag(matrix(G,1,9)));//Transfer function
t=0:Ts:N*Ts;
u_sim=ones(9,length(t));
//u_sim(:,1:4)=zeros(9,4); //input delay to account for 3 min delay in G
S=csim(u_sim,t,G)';//generating step response model for real plant
//plot(t,S);
S(1,:)=[];
//Now we have these step response models for each of the transfer functions
//[S1 S4 S7
//S2 S5 S8
//S3 S6 S9]
T=150+delay;//Simulation Run Time in minutes(we add delay because our for loop runs till n-delay)
n=T/Ts*2+1; //no. of discrete points in our domain of analysis
//Input initialization T is the Time for simulation
//========Set point as 10=============//
//p is the controller output
p=zeros(n,3);
delta_p=zeros(n,3);
ytilde=zeros(n,3); //Prediction of Smith Fig 15.9
e=zeros(n,3); //corrections
edash=zeros(n,3);
delta_edash=zeros(n,3);
ysp=zeros(n,3);
ysp((n-1)/2+1:n,1)=10*ones(n-((n-1)/2+1)+1,1);
t=-(n-1)/2*Ts:Ts:(n-1)/2*Ts;
y=zeros(n,3);
for k=(n-1)/2+1:n-delay
//Error e
e(k,:)=ysp(k-1,:)-y(k-1,:);
//Error edash
edash(k,:)=e(k-1,:)-ytilde(k-1,:)+ytilde(k-1-delay,:);
//Edash=E-(Y1-Y2)...where Y2 is delayed Y1
delta_edash(k,:)=edash(k,:)-edash(k-1,:);
//Controller calculation----Digital PID----Eqn 7-28 Pg 136 (Velocity form)
p(k,:)=p(k-1,:)+[delta_edash(k,:)+edash(k,:)*diag(Ts./tauI)]*diag(Kc);
//Limits on manipulated variables
p(k,:)=min([(345-180)*ones(1,3);p(k,:)],'r');
p(k,:)=max([(105-180)*ones(1,3);p(k,:)],'r');
delta_p(k,:)=p(k,:)-p(k-1,:);
//Prediction
ytilde(k,1)=[S(1:N-1,1);S(1:N-1,4);S(1:N-1,7)]'...
*[flipdim(delta_p(k-N+1:k-1,1),1);flipdim(delta_p(k-N+1:k-1,2),1);flipdim(delta_p(k-N+1:k-1,3),1)]...
+[S(N,1) S(N,4) S(N,7)]*[p(k-N,1);p(k-N,2);p(k-N,3)];
ytilde(k,2)=[S(1:N-1,2);S(1:N-1,5);S(1:N-1,8)]'...
*[flipdim(delta_p(k-N+1:k-1,1),1);flipdim(delta_p(k-N+1:k-1,2),1);flipdim(delta_p(k-N+1:k-1,3),1)]...
+[S(N,2) S(N,5) S(N,8)]*[p(k-N,1);p(k-N,2);p(k-N,3)];
ytilde(k,3)=[S(1:N-1,3);S(1:N-1,6);S(1:N-1,9)]'...
*[flipdim(delta_p(k-N+1:k-1,1),1);flipdim(delta_p(k-N+1:k-1,2),1);flipdim(delta_p(k-N+1:k-1,3),1)]...
+[S(N,3) S(N,6) S(N,9)]*[p(k-N,1);p(k-N,2);p(k-N,3)];
//Output
y(k+delay,1)=[S(1:N-1,1);S(1:N-1,4);S(1:N-1,7)]'...
*[flipdim(delta_p(k-N+1:k-1,1),1);flipdim(delta_p(k-N+1:k-1,2),1);flipdim(delta_p(k-N+1:k-1,3),1)]...
+[S(N,1) S(N,4) S(N,7)]*[p(k-N,1);p(k-N,2);p(k-N,3)];
y(k+delay,2)=[S(1:N-1,2);S(1:N-1,5);S(1:N-1,8)]'...
*[flipdim(delta_p(k-N+1:k-1,1),1);flipdim(delta_p(k-N+1:k-1,2),1);flipdim(delta_p(k-N+1:k-1,3),1)]...
+[S(N,2) S(N,5) S(N,8)]*[p(k-N,1);p(k-N,2);p(k-N,3)];
y(k+delay,3)=[S(1:N-1,3);S(1:N-1,6);S(1:N-1,9)]'...
*[flipdim(delta_p(k-N+1:k-1,1),1);flipdim(delta_p(k-N+1:k-1,2),1);flipdim(delta_p(k-N+1:k-1,3),1)]...
+[S(N,3) S(N,6) S(N,9)]*[p(k-N,1);p(k-N,2);p(k-N,3)];
end
subplot(2,2,1);
plot(t',y(:,1),'--',t',y(:,2),':',t',y(:,3),'-.',t',ysp(:,1),'-');
set(gca(),"data_bounds",[0 150 -4 12]); //putting bounds on display
l=legend("$y1$","$y2$","$y3$",position=1);
l.font_size=5;
xtitle("","Time(min)","$y$");
a=get("current_axes");
c=a.y_label;c.font_size=5;
subplot(2,2,2);
plot(t',p(:,1),'--',t',p(:,2),':',t',p(:,3),'-.');
set(gca(),"data_bounds",[-1 150 -40 100]); //putting bounds on display
l=legend("$p1$","$p2$","$p3$",position=1);
l.font_size=5;
xtitle("","Time(min)","$p$");
a=get("current_axes");
c=a.y_label;c.font_size=5;
mprintf("Note that there is no overshoot around time=25 mins \n...
which is in contrast to what is shown in book")
//========Now for set point as 50=============//
//p is the controller output
p=zeros(n,3);
delta_p=zeros(n,3);
ytilde=zeros(n,3); //Prediction of Smith Fig 15.9
e=zeros(n,3); //corrections
edash=zeros(n,3);
delta_edash=zeros(n,3);
ysp=zeros(n,3);
ysp((n-1)/2+1:n,1)=50*ones(n-((n-1)/2+1)+1,1);
t=-(n-1)/2*Ts:Ts:(n-1)/2*Ts;
y=zeros(n,3);
for k=(n-1)/2+1:n-delay
//Error e
e(k,:)=ysp(k-1,:)-y(k-1,:);
//Error edash
edash(k,:)=e(k-1,:)-ytilde(k-1,:)+ytilde(k-1-delay,:);
//Edash=E-(Y1-Y2)...where Y2 is delayed Y1
delta_edash(k,:)=edash(k,:)-edash(k-1,:);
//Controller calculation----Digital PID----Eqn 7-28 Pg 136 (Velocity form)
p(k,:)=p(k-1,:)+[delta_edash(k,:)+edash(k,:)*diag(Ts./tauI)]*diag(Kc);
//Limits on manipulated variables
p(k,:)=min([(345-180)*ones(1,3);p(k,:)],'r');
p(k,:)=max([(105-180)*ones(1,3);p(k,:)],'r');
delta_p(k,:)=p(k,:)-p(k-1,:);
//Prediction
ytilde(k,1)=[S(1:N-1,1);S(1:N-1,4);S(1:N-1,7)]'...
*[flipdim(delta_p(k-N+1:k-1,1),1);flipdim(delta_p(k-N+1:k-1,2),1);flipdim(delta_p(k-N+1:k-1,3),1)]...
+[S(N,1) S(N,4) S(N,7)]*[p(k-N,1);p(k-N,2);p(k-N,3)];
ytilde(k,2)=[S(1:N-1,2);S(1:N-1,5);S(1:N-1,8)]'...
*[flipdim(delta_p(k-N+1:k-1,1),1);flipdim(delta_p(k-N+1:k-1,2),1);flipdim(delta_p(k-N+1:k-1,3),1)]...
+[S(N,2) S(N,5) S(N,8)]*[p(k-N,1);p(k-N,2);p(k-N,3)];
ytilde(k,3)=[S(1:N-1,3);S(1:N-1,6);S(1:N-1,9)]'...
*[flipdim(delta_p(k-N+1:k-1,1),1);flipdim(delta_p(k-N+1:k-1,2),1);flipdim(delta_p(k-N+1:k-1,3),1)]...
+[S(N,3) S(N,6) S(N,9)]*[p(k-N,1);p(k-N,2);p(k-N,3)];
//Output
y(k+delay,1)=[S(1:N-1,1);S(1:N-1,4);S(1:N-1,7)]'...
*[flipdim(delta_p(k-N+1:k-1,1),1);flipdim(delta_p(k-N+1:k-1,2),1);flipdim(delta_p(k-N+1:k-1,3),1)]...
+[S(N,1) S(N,4) S(N,7)]*[p(k-N,1);p(k-N,2);p(k-N,3)];
y(k+delay,2)=[S(1:N-1,2);S(1:N-1,5);S(1:N-1,8)]'...
*[flipdim(delta_p(k-N+1:k-1,1),1);flipdim(delta_p(k-N+1:k-1,2),1);flipdim(delta_p(k-N+1:k-1,3),1)]...
+[S(N,2) S(N,5) S(N,8)]*[p(k-N,1);p(k-N,2);p(k-N,3)];
y(k+delay,3)=[S(1:N-1,3);S(1:N-1,6);S(1:N-1,9)]'...
*[flipdim(delta_p(k-N+1:k-1,1),1);flipdim(delta_p(k-N+1:k-1,2),1);flipdim(delta_p(k-N+1:k-1,3),1)]...
+[S(N,3) S(N,6) S(N,9)]*[p(k-N,1);p(k-N,2);p(k-N,3)];
end
subplot(2,2,3);
plot(t',y(:,1),'--',t',y(:,2),':',t',y(:,3),'-.',t',ysp(:,1),'-');
set(gca(),"data_bounds",[0 150 -10 60]); //putting bounds on display
l=legend("$y1$","$y2$","$y3$",position=1);
l.font_size=5;
xtitle("","Time(min)","$y$");
a=get("current_axes");
c=a.y_label;c.font_size=5;
subplot(2,2,4);
plot(t',p(:,1),'--',t',p(:,2),':',t',p(:,3),'-.');
set(gca(),"data_bounds",[-1 150 -100 200]); //putting bounds on display
l=legend("$p1$","$p2$","$p3$",position=1);
l.font_size=5;
xtitle("","Time(min)","$p$");
a=get("current_axes");
c=a.y_label;c.font_size=5;
|
