94 SAMPLED-DATA CONTROL SYSTEMS 
analysis it is necessary that the period of the hidden oscillation match 
exactly the sampling interval or a fraction thereof. 
Discarding the possibility outlined above, the definition of stability of 
a linear sampled-data system can be based on the pulse-to-pulse con- 
vergence or divergence. Thus, 
A linear sampled-data system is stable if the pulse sequence at its output 
in response to a bounded sequence at its input 1s bounded. 
Conversely, a sampled-data system is unstable if the pulse sequence at its 
output in response to a bounded sequence at its input 1s unbounded. 
The problem is to find working criteria by which a system can be tested as 
} to its condition of stability. To- 
A mae _ ©!) ward this end, the system shown 
Ris) T Re) TC) in Fig. 5.7 is used as a means of ob- 
Fig. 5.7, System used to derive stability taining analytic forms for this deter- 
criterion. : : 
mination. 
If the pulse transfer function of the system shown in Fig. 5.7 is G(2), 
then, by definition, 
Gz) = » g(kT)2-* (5.30) 
k=0 
where g(kT’) is the value of the impulsive response of the system at the 
kth sampling instant. The output pulses of the system are obtained 
from the convolution summation 
co 
c(mT) = > g(kT)r[(m — k)T] (5.31) 
k=0 
Now, the input r*(t) is bounded so that it satisfies the relation 
Max |r[(m — k)T]| = M < w (5.32) 
where JM is some positive number less than infinity. For the system to be 
stable in accordance with the definition given previously, the output 
c(mT) must be bounded for all integral values of m. Thus, for stability, 
|e(mT)| < (5.33) 
for all integral values of m. The magnitude of c(mT’) is given by the 
magnitude of its equivalent from (5.31), 
le(mT)| = | a9 (kT)r[(m — k)T] (5.34) 
