DIGITAL COMPENSATION OF SAMPLED-DATA SYSTEMS 17/7 
which, upon substitution of the expressions for K(z) and 1 — K(e), 
becomes 
OFS Sige EO yals zm) 
G.(z) = @ — 250 + 0.4182) 

The selection of the gain constants Ky) and K, is made by causing the 
actual feedforward pulse transfer function of the system G,(z) to be 
equal to the required equivalent function G.(z). To obtain the expres- 
sion for the actual feedforward pulse transfer function, Fig. 7.20 is seen 
to consist of two elements Gi(z) and G2(z) such that their product is 
Gi2(z), which is to be altered by means of the early feedback to G,(z). 
The first step is to note that Gi(z) is given by 
1 
<= — gil 
Gi(z) (1-2 Gee D 
which is, after simplification, 
= 
Be = 0.6322 
I — OCs 

The second pulse transfer function G2(z) is given by 
Gi2(z2) 
Gi(z2) 
Substituting the expressions for Gi2(z) and G,(z), there results 
0.582(1 + 0.718271) 
1—2! 
Go(z) = 


G2(z) = 
_ Now from Fig. 7.20, the actual feedforward pulse transfer function 
G.(z) which relates the output C(z) and the command £;(z) is seen to be 
KGi(z) 
if + KoKiGi(z) 
Substituting Gi(z) and G2(z) previously obtained, G.(z) becomes 
K ((0.632) (0.582) (1 + 0.7182—1)z7} 
[1 + (0.632K 0K, — 0.368)z](1 — 272) 
If the system being designed is to be identical with the required feed- 
forward pulse transfer function, then 
Ga(z) = G.(z) 
Equating these two functions and evaluating the constants Ky and Ki 
by identity, the following relations are obtained: 
0.581 = (0.632) (Ky) (0.582) 
and 0.418 = 0.632K.K, — 0.368 
Ga(z) = G2(z) 
GA@)— 

