DIGITAL COMPENSATION OF SAMPLED-DATA SYSTEMS 189 
found by manipulating (7.79) to yield the equality 
1 — [AR(z)/R@)] 
Solving for the characteristic equation, 
1+ AH(z)D) = 1 — [ARG@)/R©)] — 0 (7.82) 
I Kriz) 
It is readily seen that in order that the characteristic equation not contain 
zeros stemming from 1 — [A R(z)/R(z)] whose magnitudes are greater 
than unity, it is necessary that Kr(z) be so specified that 1 — Kpr(z) con- 
tains such zeros. 
These rules of design may be summarized as follows: 
1. The specified over-all pulse transfer function Kr(z) must satisfy the 
conditions of physical realizability as expressed in (7.9). 
2. In order that the system respond to specified test inputs which are 
time functions of the form ¢” with no steady-state error, it is necessary 
that 1 — Kr(z) contain a factor (1 — z)”*1. 
3. The digital-controller pulse transfer function D(z) should not cancel 
those zeros of AH(z) which have magnitudes equal to or greater than 
unity. This condition is met by selecting Kr(z) such that the function 
Kr(z) — [A R(z)/R(z)] contains all those zeros as its own. 
4, The zeros of the function 1 — [A R(z)/R(z)] whose magnitudes are 
equal to or greater than unity must be contained in 1 — Kp(z). 
By meeting all of these conditions, a stable system using a bypass con- 
troller is assured. 
EXAMPLE 
To illustrate the design procedure, the system given in Fig. 7.22 is 
assumed that 
1 
G6) = s(s + 0.5) 
The hold system is to be of zero order, and H(s) is then 
1 —e-s? 
H(s) = 
where T is taken as 1.5 sec. It is desired that this system respond 
without steady-state error to a unit step input so that 
s 
s 
R(s) = 
