206 SAMPLED-DATA CONTROL SYSTEMS 
which, upon substitution of H(z”) from (8.14), becomes 
C(22) = D(eo?)[1 — K (22?)] R(22”)G(z2) (8.16) 
Inversion of C(z2) will give the mid-points of the output ripple of the 
system. It should be noted that if K(z) has been chosen to produce a 
finite-settling-time response, it does not necessarily follow that C(z2) will 
invert into a finite-settling-time sequence. Only every second sample 
will definitely exhibit this property since the mid-point samples are sensi- 
tive to ripple. On the other hand, if the over-all prototype response 
function has been designed to be ripple-free, inversion of C(z2) will pro- 
duce a ripple-free response at all sample times after the transient period 
has passed. 
8.3 Partial-fraction Expansion Technique 
A useful technique for obtaining the ripple in any chosen sampling 
interval involves the expansion of the plant continuous transfer function 

Fic. 8.5. Typical sampled-data system for which ripple is to be determined. 
into partial fractions.®° This technique takes advantage of the simple 
relationships obtainable with ordinary z transforms and combines them 
with the Laplace transform describing the plant. To illustrate the 
method, reference is made to Fig. 8.5, where it is seen that the continuous 
output of the plant is given by the inversion of C(s) in response to an 
input R(s). The continuous plant transfer function includes the data 
hold, although, as shown later, part of the data-hold transfer function is 
separated out and included in the relation describing the z transform of the 
error sequence. While the discussion which follows centers around the 
system illustrated in Fig. 8.5, it will be readily seen that only minor modi- 
fications of the method will be required to handle other configurations. 
Referring to Fig. 8.5, it is seen that the Laplace transform of the con- 
tinuous output is C(s) and is related to the error transform H*(s) by 
C(s) = G(s) E*(s) (8.17) 
where G(s) is G,(s)G,(s). It has been shown that the Laplace transform 
