SAMPLED-DATA SYSTEMS 97 
for values of z whose magnitude is greater than unity. The inequality 
relating the first term of (5.42) and the last term, infinity, satisfies the 
condition for instability, (5.37). Thus, an unstable system has a pulse 
transfer function G(z) which contains at least one pole which lies outside 
of the unit circle in the z plane. To recapitulate: 
A stable linear sampled-data system has a pulse transfer function G(z) 
which contains no poles or other singularities which lie outside of the unit 
circle of the 2 plane. 
An unstable linear sampled-data system has a pulse transfer function 
G(z) which contains one or more poles or other singularities outside the unit 
circle of the z plane. 
The pulse transfer functions of the elements generally found in sampled- 
data feedback control systems have only simple or multiple poles as 
singularities and are in the form of ratios of polynomials in z or z—1. The 
conditions for stability of such systems are readily apparent by expanding 
the pulse transfer function into partial fractions. Thus, if G(z) is of the 
form 
Ge) = Gy a5 GRE ap GE sb 9 9 2 46 Gee 
Il sb One" ap eB seo 8 9 ae Oye e 
where conditions of physical realizability require that the term 1 be 
present in the denominator. If the various roots of the denominator are 
designated as z;, then G(z) can be expanded into partial fractions as 
follows: 
Ga C. a Om 
G(z) = ao + 7 Th Geoeaict eee 
= Ze 1 1 — Zoe 

(5.43) 
(5.44) 
If the input to the system is an impulse, then the time-domain pulse 
sequence which results is simply 
g(kT) = ao + Cies)* + Co(ze)* + > > + + Ci(2,)* (5.45) 
If the roots of the denominator of (5.43) which are the poles of the pulse 
transfer function G(z) have magnitudes which are greater than unity, it is 
evident that the impulsive response g(k7’) of the sampled-data system will 
increase without bound as the various integral values of & increase with- 
out bound. The stability condition that G(z) not contain any poles out- 
side the unit circle of the z plane is clearly seen in this approach. 
As in the case of continuous systems, it is not convenient to determine 
the stability of a system by factoring the denominator polynomial of 
G(z) in order to locate the poles of the function. For simple systems of 
second or possibly third order this procedure is as direct as any, but for 
higher-order systems the labor required to factor the polynomials becomes 
