DIGITAL COMPENSATION OF SAMPLED-DATA SYSTEMS 147 
desirable. More detailed discussions of implementation will be covered 
in later sections. 
7.2 Basic Principle of Linear Digital Compensation 
If it is assumed that the digital controller is linear, it follows that the 
relation between the input number sequence and the output number 
sequence is linear. Furthermore, a desirable system will require the 
storage of only a finite number of input and output samples. A general 
linear relation subject to these conditions will relate the input and output 
sequences, e;(t) and e}(t), respectively, in the following manner: 
e2(nT) + bye.[(n — 1)T] + bee[ (n —= 2) {ui + Sek Rte bpeo[(n > k)T| 
=aei(nT) + aei(m —1)T)+-:-:-: gael — u)T)] (7.1) 
where the various a’s and b’s are constants. This equation is the same as 
that given in (4.53), leading to the pulse transfer function (4.55). The 
pulse transfer function which generates this relationship is given by 
a Giza Qazm 5 06 (ha 
D(z) = 0 = 1 + 2 = u 


1 + byez! + bog? +: - > Dye (22) 
where D(z) is defined by 
be E.(2) 
De) = nee (7.3) 
If the digital controller is a pulsed network whose transfer function is 
G(s), then the digital controller pulse transfer function isG(z). If, on the 
other hand, the digital controller is an active element, then the various 
a’s and 6’s can be set without restriction. The only restriction is the 
usual one of physical realizability, which requires that in the form of 
(7.2), the denominator must contain the term ‘1.” 
The basic concept of digital compensation is that the various constants 
in the pulse transfer function D(z) are adjusted so that some desired over- 
all pulse transfer function K(z) is realized. Depending on the structure 
of the sampled-data feedback system, that is, on the relative location of 
the various samplers, the D(z) appears in various ways. For instance, in 
the system shown in Fig. 7.2, the over-all pulse transfer function K(¢) is 
_D@G® __ 
1+ DG) 
For a given G(z), D(z) must be so chosen as to produce a desired K(z). 
Supposing, for the moment, that the desired over-all pulse transfer 
function K (z) has been chosen, then the D(z) which is required is given by 
1 K(z) 
EC atee) ae) 9) 
K@) = (7.4) 

