156 SAMPLED-DATA CONTROL SYSTEMS 
then the required digital-controller pulse transfer function to produce this 
is given by 
1 K,(z) 
DO -gaT= Ke (7.21) 

an expression equivalent to (7.5). Substituting G(z) from (7.29) in this 
expression, there results 
D(z) ce! (oe ba Kee) 
(z = aa) F a(2) 1 — AB) 

(7.22) 
where the constants bz and ag are made almost but not exactly equal to 
the plant constants b, and a,. Also, Fa(z) and F,(z) are made almost 
identical, although, as will develop later, slight mismatches are of small 
consequence because it is assumed that all their poles and zeros lie inside 
the unit circle on the z plane. 
To obtain the over-all pulse transfer function which actually results 
from inserting a digital-controller pulse transfer function given by (7.22) 
and a plant pulse transfer function given by (7.20), the actual over-all 
pulse transfer function K,(z) is obtained by substituting G(z) and D(z) 
from (7.20) and (7.22), respectively, into (7.4), resulting in 
(z a ba) (2 = a,)F’,(2)K.(2) 
(2 — aa)(z — b,)[1 — K.(z)]Falz) + (2 — ba) (2 — a,)K,(2)F (2) 
(7.23) 
In the ideal situation, the poles and the zeros of the digital controller 
match those of the plant exactly, so that 
ba = by 
a = Ag (7.24) 
Fa(z) = F,(2) 
K,(z) = 

Then the complicated expression in (7.23) reduces simply to 
KE @) eh) (7.25) 
For this condition, it is seen that the over-all pulse transfer function actu- 
ally obtained matches exactly the desired function. 
In the practical situation, however, exact cancellation of the plant 
poles and zeros by the controller poles and zeros cannot be realistically 
expected, so that bz and 6, or ag and a, are not exactly equal. This means 
that expression (7.23) cannot be reduced, and its poles and zeros may lie 
outside the unit circle, as will be shown. For this development, it will be 
assumed that F(z) and F(z) are identical and that the imperfect cancella- 
tions are confined to poles and zeros which lie outside the unit circle. 
To determine the effect of imperfect cancellation, it is assumed initially 
that the controller zero and pole, ag and bg, are identical to the plant zero 
