208 SAMPLED-DATA CONTROL SYSTEMS 
its z transform, C,(z), given by 
acme ee) 
OE) eaten) 
where G,(z) is the pulse transfer function of the qth path and the term 
(1 — 2!) enters in consequence of the difference operation between E,(z) 
and E(z). 
Having the expression for C,(z) makes it possible to obtain the contribu- 
tion to the ripple by this elementary path during any desired sampling 
interval. The reason for this is that by having expanded the plant trans- 
G,(z) (8.22) 
* 
c(t 
figee Ae 
! 
| 
cx(t) 

Fia. 8.7. Expanded feedforward transfer function. 
fer function into partial fractions of the first order, the time function in 
any interval is fully specified by the initial value of the time function at 
the beginning of that interval. The latter is obtainable directly by 
inversion of (8.22) for the sampling instant in question. For instance, the 
time function c,(t) in the ath path is given by 
BAG Se ere (m+ 1)T >t > mT (8.23) 
for the time interval specified. It is seen that the time function depends 
on a knowledge of c.(mT), which is the mth sample obtained from the 
inversion of C,(z) in Fig. 8.7. 
For those elementary paths which are dependent on simple and multiple 
poles at the origin of the s plane, the contributions to the continuous out- 
put during the interval extending from mT to (m + 1)T are given by 
e(f) =L+M(t—mT)+N¢— mT)?+°--- (8.24) 
