218 SAMPLED-DATA CONTROL SYSTEMS 
among others, partial fractions of the form 
E@uGr fs SAE in aa (8.31) 
where wo is 2r/T and kis aninteger. If the z transform corresponding to 
G(s)H(s) is taken, it results in the following expression for GH(z): 
A As 
_ gvikeoTy—1 aly 1 — etikeoTz-1 ae 

GH(z) = i (8.32) 
From the definition of wo, it is seen that kwoT is simply 2rk and that 
e~tkwoT ig unity. Thus, the loop pulse transfer function GH(z) becomes 
Ai Ao 
SiN ee Gare 
It is seen from (8.33) that the oscillatory effects which are present in 
the impulsive response resulting from the inversion of G(s)H(s) are 
totally hidden in the sampled sequence resulting from the inversion of 
GH(z). This condition is the basic theoretical source of the hidden oscil- 
lation. However, unless the imaginary component of the pole of G(s) H(s) 
is precisely a multiple of the sampling frequency this condition will not 
occur. 
In the practical situation, if it is desired to observe the effect of hidden 
oscillations of this type by means of ordinary z transforms, all that need 
be done is to alter the loop transfer function slightly so that it is not 
“tuned” to the sampling frequency. This will ensure that the inversion 
of the z transform will contain such oscillatory terms. This simple pro- 
cedure forms the basis of the statement that hidden oscillations can 
readily be coped with and that the subject tends to be of academic 
interest only. 
see (8.33) 
8.6 Summary 
The application of the ordinary z transform to the analysis and syn- 
thesis of sampled-data systems gives information which can be used to 
evaluate the variables at sampling instants only. In a complete study, 
however, the behavior of the system between sampling instants is of con- 
siderable importance. For instance, finite-settling-time-systems proto- 
types may have considerable overshoots between sampling instants dur- 
ing the transient period, even though the response at sampling instants 
has settled completely. The oscillatory behavior in the time domain 
between sampling instants is called ‘“‘ripple.”’ 
Unfortunately, the direct inversion of the Laplace transform which 
gives the continuous output of sampled systems is very complex, and its 
use does not lead to results which are readily assessed or corrected should 
