158 SAMPLED-DATA CONTROL SYSTEMS 
its zeros all those poles of the plant pulse transfer function which lie outside or 
on the unit circle in the z plane. In applying these restrictions it is not 
always possible to realize a minimal over-all prototype, although if M(z) 
and N(z) are made polynomials in 2~! with a finite number of terms and 
containing no denominator polynomial, finite settling time can always be 
obtained. 
It is convenient to collect the various rules and restrictions which 
apply to the compensation of feedback sampled-data systems. They are: 
1. In order to satisfy conditions of physical realizability, the specified 
over-all pulse transfer function must take the form 


a Kime” + 6 oo oO nee 
TG) = i = fe “+ Soe em: 
where m is the lowest order in z~! in the pulse transfer function 
Gg a Die te + saen nie, Diem | 
() = Pee 
2. For a minimal prototype, K,(z) contains only the numerator poly- 
nomial of lowest order in 271. 
3. For the system to respond to an input of the form 
A(z) 
(1 — 21)* 
with zero steady-state error, it is necessary that K,(z) satisfy the relation 
1 — "Kee — (Ie) eee) 
where F(z) is an arbitrary polynomial or ratio of polynomials in 271. 
4. For those plants having pulse transfer functions containing poles or 
zeros which lie outside or on the unit circle of the z plane, the following 
must obtain 
R(z) = 
Ke) I] (1 — az")M(z) 
Kee I (1 — bye-)N(z) 
where the a; and b; are all zeros and poles, respectively, of the plant pulse 
transfer function which lie outside or on the unit circle of the z plane. 
The application of these rules of design is best understood by means of an 
illustrative example. 
EXAMPLE? 
The system which is to be compensated by means of a digital con- 
troller is shown in Fig. 7.8, where it is seen that the continuous plant 
