188 S\MPLED-DATA CONTROL SYSTEMS 
rewritten as 
C(s) = A(s)R(s) + A(s)H(s) D*(s)R*(s) — A(s)H(s)D*(s)C*(s) (7.76) 
Taking the z transform of both sides and recalling that the z transform of 
a pulse transfer function such as D*(s) is the pulse transfer function itself 
and replacing D*(s) by D(z), (7.76) becomes 
C(z) = AR(z) + AH(z) D(z) R(z) — AH(z) D(z) C(z) (7.77) 
Solving for C(z), 
AR(z) + AH(z) D(z) R(z) 
C@) = 1+ AH(2)D(2) 
(7.78) 
It is seen from (7.78) that an over-all pulse transfer function for the 
system is not readily obtained since R(z) is not separable from the expres- 
sion. It is possible, however, to define a quasi-over-all pulse transfer 
function as follows 
Kel) < £@ — (AR@/R@) + ANODE 
R(z) 1+ AA(z) D(z) 
It is seen that Kr(z) is dependent on the input R(z). On the other hand, 
if a design procedure is adopted where a desired response is sought for a 
given test input, it is acceptable to consider Kr(z) as an over-all pulse 
transfer function and to design a digital-controller pulse transfer function 
which will implement this desired result. Solving (7.79) for D(z), there 
results 
(7.79) 
1 Karz) — [AR()/RG@)] 
AH(z) 1 — Kr(z) 
As in the case of the other digital-controller designs studied in this 
chapter, there will be certain restrictions placed on the selection of Kr(z). 
The conditions of physical realizability are the same as those given in 
(7.9), where K(z) must have a term of zero order in the denominator. 
Secondly, if the system is to respond to a test input which is a power of 
time such as a step, ramp, or constant acceleration, the restriction on 
1 — K(z) given in (7.15) must be satisfied. Less obvious are the restric- 
tions placed on Kp(z) required to maintain a stable system. 
In order to maintain a stable system, the digital-controller pulse trans- 
fer function D(z) cannot be allowed to cancel poles or zeros of the system 
indiscriminately. It is seen from (7.80) that a situation similar to that 
discussed in Sec. 7.5 exists. The only difference is seen by contrasting 
(7.80) and (7.21). The numerator of (7.80), which must contain as its 
zeros all those zeros of AH(z) whose magnitudes are equal to or greater 
than unity, is Kr(z) — [AR(z)/R(z)], rather than the simpler form that 
would be found in (7.21), namely, K,(z). 
The condition relative to the form of 1 — Kar(z) is less obvious and is 

De) — (7.80) 
