DIGITAL COMPENSATION OF SAMPLED-DATA SYSTEMS AL7() 
‘sidered a primary disadvantage, then the slight increase in settling time_at 
sampling instants is a trivial factor. 
7.8 Equivalent Digital-controller Systems 
It is often possible to realize the over-all prototype pulse transfer func- 
tions described in previous sections without the use of an actual sampled- 
data processing unit or digital controller. Except for the sampler, all the 
elements in such systems are analogue or continuous and take the form of 
networks, tachometers, or active electronic elements. The general form 

Fic. 7.17. Early feedback from system state variables. 
of the system is shown in Fig. 7.17, where the error sampler is followed by 
a zero-order hold and a linear continuous plant. It is necessary that 
early feedback connections from each of the first-order cascaded elements 
of the plant transfer function G(s) be tapped either directly or by simula- 
tion. The quantities being tapped are the “‘state variables” of the sys- 
tem. It is seen by a little reflection that the quantities B,, Bo, ... , Bn 
being fed back are linear combinations of the output and its n derivatives. 
As will be shown, it is possible to control the plant into ripple-free finite- 
settling-time operation, having these quantities available for feedback to 
the error line. 
To facilitate the analysis of this system, a fictitious sampler will be 
inserted in each of the feedback lines, as shown in Fig. 7.18a. While 
these samplers do not exist in the actual system, their presence in the 
model used for analysis does not alter the system because they are seen to 
be redundant with the error sampler. If they operate synchronously 
with the error sampler, the additional sampling operations merely con- 
tribute a sequence of samples which are again sampled to form the error- 
sample sequence. Sampling twice synchronously is the same as sam- 
pling only once since any information between samples is not transmitted 
anyway. 
The advantage of adopting this model is evident when it is realized that 
the feedback sequences at Bi, Bs, . . . , B, can be related to the error 
sequence by a number of partial z transforms Gi(z), G2(z), etc., as shown 
in Fig. 7.18b. These partial z transforms need further explanation, but 
