CHAPTER 5 
SAMPLED-DATA SYSTEMS 
In previous chapters, the individual components which are found in 
sampled-data systems were discussed and the mathematical relations 
describing their operation derived. When a number of such elements are 
interconnected, they constitute a system, and if one or more samplers are 
included, they area sampled-data system. The configuration may be either 
open-cycle or closed-cycle, with the latter form being the center of 
interest in control systems. The rules for combination of elements 
are somewhat complicated by the presence of samplers. For this reason 
a direct analogy with the rules of continuous systems cannot be found 
in all cases, and the tempting possibility that all that need be done 
is to substitute the z transform for the Laplace transform is not correct. 
In this chapter the behavior of systems at sampling instants only will be 
considered. The behavior between sampling instants will be treated later 
by extensions of the methods used in this chapter. 
5.1 Sampled Elements in Cascade 
If two linear elements are connected in cascade as shown in Fig. 5.1, 
it is desirable to be able to rclate the over-all input and output sequences 
in terms of the transfer functions. Figure 5.1 shows two linear elements 
whose individual pulse transfer functions are Gi(z) and G2(z). It is 
important to note that a synchronous sampler is located between the 
two elements. This distinction is most important because the relations 
which will be derived depend on its presence. The case where no sampler 
is included will be taken up later. The over-all output pulse sequence is 
given by C2(z), while the intermediate pulse sequence which constitutes 
the output of the first clement and the input of the second is C1(z). 
From the definition of the pulse transfer function given in Sec. 4.6, it 
follows immediately that 
The sequence C,(z) is the input to the second element whose pulse transfer 
function is G2(z), so that its output C2(z) is given by 
C2(z2) = G2(2)C1(z2) (5.2) 
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