182 SAMPLED-DATA CONTROL SYSTEMS 
input condition. Referring to Fig. 7.21, this means that with the largest 
test input at R, the pulse sequence EH. should be limited to some upper 
bound E,,, the maximum command signal which the plant is capable of 
taking without saturating. The first condition required to bring this 
about is that the output of the digital controller be zero or a finite con- 
K(z) 
Dz) G(z) 
I 
Z Digital 
controller T 

Fic. 7.21. Typical system having plant with saturation. 
stant in the steady state. Assuming that this condition is met, there is a 
procedure’ for preventing the pulse sequence applied to the plant from 
exceeding its specified upper bound. 
This is done by noting that, by definition, 
C@ 
and that C(z) = E2(z)G(z) (7.64) 
from which there results the equality 
K(z)R(z) = E.(z)G(z) (7.65) 
The test input function pulse transform R(z) is specified, as is the plant 
pulse transfer function G(z). It remains to find the over-all pulse transfer 
function K(z) such that no pulse in E.2(z) has a magnitude exceeding the 
upper bound. 
Recalling the series form of the pulse transfer functions, (7.65) may be 
rewritten in the form 
[ro +rnezit-- Miiaeet + keen a) 5 kpz-?] 
= [eo tet +o + ee] [gme-™ + ging zoo" + + +] (7.66) 
. Multiplying this sequence through and collecting the coefficient of like 
orders of 2—!, there result a number of equalities obtained by equating the 
coefficients of like powers of z—!. These equalities are 
Tokm = €0Jm 
ToKm+1 + Tikm = €o9m+i + €19m 
TKm+2 + Pikm+s + r2km = CoYm+2 + €1Ym+1 + €29m 
etc. (7.67) 
Expressed in this form it is possible to obtain the various coefficients of 
the pulse sequence whose pulse transfer function is K(z). 
