BEHAVIOR OF SYSTEMS BETWEEN SAMPLING INSTANTS 205 
By contrast, if the normal z transform at the basic rate, C’(z), is taken, 
its inversion will yield 
c*() =14+ tee ae — T) + A +e? +e") 6(t— 2T) + + - 
It is seen that every second term of the double-rate sampling sequence 
corresponds to the terms of the basic rate sequence. The extra point 
obtained from C(z2) gives the mid-point value of the ripple. 
By using higher than double-rate sampling at the output, more points 
describing the ripple can be obtained within a given sampling interval. 
The procedure is very much the same as that for double-rate sampling 
and will be discussed in the next chapter. Though the resulting expres- 
sions are more complex than those for double-rate sampling, the cost in 
labor is not prohibitive. 
T E,(e) | Da T alo | 
Controller 



—o 
P/2 Cleo) 


C(s) 
Fic. 8.4. Digitally compensated system with double-rate output sampling. 
Feedback sampled-data systems are analyzed for the ripple in very much 
the same manner as for open-cyclesystems. Forinstance, atypical system 
is one given in Fig. 8.4, where a digitally compensated control system is 
shown. The double-rate sampled output of interest here is expressed by 
C(z2), which is the double-rate z transform of the output obtained by 
means of a fictitious sampler. The continuous plant transfer function, 
including the data hold, is G(s), and it is the output from this component, 
when subjected to the pulse sequence /2(z), that describes the ripple. It 
has been shown that the z transform F(z) of the control error is given by 
Ey(z) = [1 — K@)]R@) (8.12) 
where K(z) is the over-all pulse transfer function. The output of the 
digital controller, which is the input to the plant including the data hold, 
is then given by 
E.(z) = D(z){1 — K(z)|R@) (8.13) 
As shown previously, the double-rate z transform corresponding to E2(z) 
is simply E2(z22), where z2? replaces z. Thus, from (8.13), 
E2(22”) = D(22?){1 —d K (22?)] R (22?) (8.14) 
The double-rate z transform of the output C(z2) is given by 
C(z2) = E(22?)G(z2) (8.15) 
