242 SAMPLED-DATA CONTROL SYSTEMS 
When K(z,) represents a finite settling time there is little difference in 
labor involved in the two methods, but when K (z,) does not represent a 
finite settling time then (9.70) offers considerable advantage over the 
direct expansion method. The application of (9.70) is greatly simpli- 
fied by recognition of the fact that any function of z,—" may be factored 
out directly before evaluation of the integral. That is, as may be 
readily verified from an elementary consideration of Fig. 9.1 and (9.7), 
ZF (2n")G@(2n)] = F(2@)ZIG(2n)] (9.71) 
For the present example, substitution of (9.66) in (9.70) leads to the 
integral 
ZK (én)] = 

i 1 2," 1 = 22,2" gizn) + Geen Gee 
207 ie —z2,11—22,1 1 = 272s (erga) 
The application of the results given in (9.71) to the integral in (9.72) 
simplifies the problem to the form 
Fy yA bps ca) learn) (gi2n + go) den 
ASG) = 20] i, (Zn — 1) (én — 2)(1 — 2n"2 en (9.73) 

The path of integration T for the integral of (9.73) is a circle with 
radius greater than 2, and the evaluation of the integral is simply done 
by the method of residues. The final result is that 
DA 7 

Zee G —1)gi+ 0| 2! G32" sae (9.74) 
The application of the constraint equation (9.68) to (9.74), which 
requires the evaluation of (9.74) at z2 = 2”, can be simplified to the 
equation 
Qnvl 
291 + g2 = Ta (9.75) 
Finally, the two constants can now be evaluated from the solution of 
(9.75) and (9.69), with the result 
Qntl — 1 
n= eT 
Substitution of (9.76) in (9.66) and (9.74) permits one to calculate the 
two transfer functions necessary to get the controller pulse transfer 
function. 
g,) ae Denn 

— Zn” 1 ae 2°27 Qrtrl —- 1 
J eee = oa 
= Zi @ aa i 22 at) (9.77) 
It should be noted during the algebra leading to (9.77) that 1 — Z[K (2n)] 
