214 SAMPLED-DATA CONTROL SYSTEMS 
Inverting C(z,A) and keeping A as a constant parameter, the output 
pulse c(mT’,A) is, by evaluation of (4.20), the following: 
— p=olp—maLr 
AGE Nn = (: ES ) eer 
1 — e@? 

Thus, if the ripple in any sampling interval following the mth sampling 
instant is desired, the appropriate value of m is substituted in 
this expression and A is permitted to vary from zero to unity. It is 
seen that for this simple example, the ripple at any sampling instant 
is an exponentially decaying term whose initial value is that in the 
parenthesis. 
While the example is a very simple one, it does illustrate the basic 
technique. Its application to more complex systems involves merely 
more labor but does not add to the theoretical complexity. The same 
technique can be applied directly to feedback systems with only minor 
modifications. Referring to Fig. 8.12, the block diagram shown is that 
G(z,A) 

Fig. 8.12. Feedback system using modified z transform to determine output ripple. 
of an error-sampled system having a feedforward transfer function G(s). 
The technique is to insert a fictitious time advance, just as in the case of 
open-cycle systems. In so doing, however, the loop pulse transfer func- 
tion is affected, thus resulting in an erroneous over-all response function. 
To restore the system so that it is the same as the actual system except 
for the time advance in the feedforward element, it is necessary to insert 
an equal time delay in the feedback line. In this manner, the over-all 
pulse transfer function is unaffected except for the advance in the output 
pulse transfer function. By varying the number A, it is possible to deter- 
mine the variations of the output between chosen sampling instants. 
Referring once again to Fig. 8.12, it is seen that the sampled output is 
given by 
G(z,A) 
1+G@) 
Inversion of C(z,A) gives the output samples as a function of A, which is 
varied as a parameter from zero to unity. The application of the method 
is best illustrated by means of an example. 
C(z,A) = Rs) (8.31) 
