92 SAMPLED-DATA CONTROL SYSTEMS 
Another common and useful structure is the one shown in Fig. 5.5. 
In this system, the elements G and H are continuous feedforward and 
feedback components, whose transfer functions are G(s) and H(s), respec- 
tively. The element D is a digital controller; that is, it receives a 
sequence of pulses e7(t) at its input and delivers a processed sequence of 
pulses eX(¢) at its output. Internally, the element contains either a 
pulsed network or a numerical device such as a digital computer. The 
7 
7 
7 
Eo > ORE} ex 
—o 
! C(z) 

Fia. 5.5. Error-sampled feedback system with digital element. 
unit has been referred to at times as a sampled-data processing unit,’ 
although the term digital controller has the advantage of being more com- 
pact and more descriptive. It was shown in the previous chapter that 
such elements, if linear, are characterized by a pulse transfer function 
D(z) which relates the input and output pulse sequences. 
The relations between the various z transforms describing the variables 
and elements can be obtained directly from the figure. The control error 
z transform F(z) is given by 
E,(z) = R(z) — Be) (5.24) 
and B(z) = Gi (z) E2(z2) (5.25) 
The pulse sequence at the output of the digital controller is related to the 
control error sequence through the digital-controller pulse transfer func- 
tion D(z) as follows: 
E.(2) = D(z)Ex(2) (5.26) 
Solving (5.24), (5.25), and (5.26) for E2(z), 
iO = a) R(z) (5.27) 
1+ D(z)GH(z) 
The output z transform C(z) is given by 
C(z) = G(z)E2(z) (5.28) 
Substituting (5.27) in (5.28), there results the over-all relation 
6) SG, (5.29) 
1+ D@)GH() 
