SAMPLED-DATA SYSTEMS 115 
becomes very small and the frequency-response function K(w) becomes 
large. This peak in frequency response reflects the fact that a pole lies 
near the unit circle and that the transient response will be relatively oscil- 
latory. Thus, avoidance of the critical point (— 1,0) by the pulse transfer 
locus by a substantial margin is a means of obtaining an over-all response 
which is docile in the time domain. 
5.8 Summary 
Sampled-data systems are structures of interconnected linear elements 
characterized by the fact that samplers may be located at one or more 
points. The rules of combination of sampled systems are not entirely 
analogous to those relating to continuous linear systems but depend on 
the locations of the samplers in the structure. In the simple case of two 
cascaded elements which are separated by a sampler, the pulse transfer 
function of the combination is simply the product of the pulse transfer 
functions of the elements. On the other hand, in the case of the pulse 
transfer function of cascaded elements which are not separated by a 
sampler, the pulse transfer function of the combination is the z transform 
corresponding to the product of the continuous transfer functions of the 
two elements. The over-all pulse transfer function in this case is not 
equal to the product of the two individual pulse transfer functions. 
The over-all pulse transfer functions of closed-loop systems can be 
obtained by applying the rules developed for combining individual ele- 
ments. There is no one form for the over-all pulse transfer function of a 
closed-loop system since it depends on the location of the samplers in the 
system. Thus, for each configuration, an over-all pulse transfer function 
must be derived. For convenience, the forms for a number of common 
configurations are recorded in tables. Regardless of the detailed form, 
however, a characteristic equation in z which describes the condition of 
stability of the system can be found. This relation equates a general 
form 1 + F(z) to zero, where F(z) is the loop pulse transfer function of the 
system which is dependent on the exact distribution of continuous ele- 
ments and sampling switches. 
If the magnitude of the roots of the characteristic equation is greater 
than unity, a sampled system is unstable. More generally, if the summa- 
tion of the magnitudes of the samples obtained by sampling the impulsive 
response of the system is bounded, the system is stable. This necessary 
and sufficient condition for stability is met if the poles of the over-all pulse 
transfer function, or equivalently, the roots of the characteristic equation, 
lie within the unit circle of the z plane. A test for stability is one in which 
a modified Routh-Hurwitz criterion is applied. By using the bilinear 
transformation on the characteristic equation, the resultant transformed 
