174 SAMPLED-DATA CONTROL SYSTEMS 
their product should yield the correct over-all pulse transfer function of 
the plant, Gi.(z). In the system shown in Fig. 7.186, only two state 
variables C; and C are shown, indicating that the plant transfer function 
is second-order. The process to be described can readily be extended to 
higher-order systems. 
Gy2(e) maa) wee 
rp -0 

(a) (0) 
Fic. 7.18. Z-transform relationships for early feedback system. 
The over-all pulse transfer function of the plant in Fig. 7.18b is chosen 
as Gi.(z), and it includes the hold system as well. If the continuous 
transfer function is separated as shown in Fig. 7.17, then the output of the 
first element, say Gi(s), is the first state variable. The pulse transfer 
function corresponding to G(s), namely, Gi(z), is found. The second 
pulse transfer function G2(z) is then given by 
G12(z) 
Gi(z) 
It is noted that G.(z) will not be exactly equal to the z transform cor- 
responding to the element @2(s), as would have been the case if a sampler 
G.(z) = (7.49) 


Fic. 7.19. Feedback system employing early feedback from system states. 
had actually separated the two elements. The pulse transfer function 
G.(z) is fictitious and is introduced only the purposes of analysis. If 
there are more than two elements in the feedforward line, the process is 
repeated. The feedback pulse sequences B,(z) and B2(z) are given by the 
relation 
Biz) = Gi(z)E2(z) Ki (7.50) 
and B2(z) = G2(z)Ci(z) (7.51) 
Having these relations, and referring to Fig. 7.19, it is possible to derive 
