90 SAMPLED-DATA CONTROL SYSTEMS 
The over-all pulse transfer function G(z) is given by 
G(z) = Gi(z)G2(z) 
which, upon substitution and simplification, becomes 
1 
(=) ee 
GG) — 
If the two elements are cascaded without a sampler between them, 
the over-all pulse transfer function is given by 
Gis(z) = Z[Gi(s)G2(s)] 
1 
Einyty fo Oey aR Nn 
(s + a)(s + b) 
From Appendix I, the z transform is 
1 (e8? — e-aT) 2-1 
é = bl => et2z) OS eran) 
It is clear that G1.(z) differs considerably from G(z), as expected. It 
could be shown that if the sampling period 7’ were made small, the two 
expressions would tend to the same limit. Another point is that the 
two expressions have the same poles but not the same zeros. 
G12(z) => 

5.2 Feedback Sampled-data Systems 
The over-all pulse transfer function for a feedback system is not 
arrived at as directly as that for a continuous system because of the 
various limitations imposed on the combination of cascaded elements out- 
we bes —-Oo 
| [ C(z) 


Ris) C(s) 
Fic. 5.4. Error-sampled feedback system. 
lined in the previous section. There is no unique form of over-all pulse 
transfer function for closed-loop systems but rather a number of forms 
dependent on the location of the samplers, as will be seen in this section. 
To illustrate the methods used to determine the over-all pulse transfer 
function, several typical forms will be used. 
The error-sampled system is shown in Fig. 5.4. In this system there is 
only one sampler, placed at the point in the system where the error signal 
