134 SAMPLED-DATA CONTROL SYSTEMS 
the designer may then specify an open-loop continuous transfer function 
which has the desired characteristics and solve for the necessary com- 
pensating network. The difficulty with the method lies in the initial 
selection of a K(z) such that the resulting continuous network is realizable 
and practical. There does not seem to be a simple method for introduc- 
ing the necessary constraints on K(z) for this purpose. It should be 
mentioned, however, that this particular problem has not been seriously 
studied because of the development of the more attractive time-domain 
design methods for digital controllers which are described in Chap. 7. 
The nature of the difficulty in the application of method 4 is best 
illustrated by example. Consider the basic system shown in Fig. 6.2 and 
assume a zero-order-hold circuit. A typical specification on the over-all 
transfer function K(z) might be that the system follow a first-order poly- 
nomial in time (ramp) with zero steady-state error. That is, the steady- 
state value of the error is to be zero. For the system being considered, 
the error transform is 
E(z) = R(z) — Cle) 
= R(z)[l — K@)] 
sete taf olay (6.17) 
cm) } 
From the final-value theorem and (6.17) 
lim e(nT) = lim (_ — 27) Ges a [1 — K(z)] (6.18) 
n— 2 
Inspection of (6.18) shows that, if the final value of e(nT) is to be zero, 
then 1 — K(z) must contain factor (1 — z~!)?. The simplest possible 
transfer function which satisfies this condition is 
Gj = LOS Se (6.19) 
For the control system shown in Fig. 6.2, direct analysis shows that if 
H(s) is a zero-order hold, 
JaeeG! 4) Ke) 
Meee ee he) 
= a (6.20) 
Therefore, the transfer function of the compensating network must 
satisfy the relation 
NGaeen eae 
[sf a=) 
z(2z — 1) 
ee (6.21) 
Z 
