154 SAMPLED-DATA CONTROL SYSTEMS 
pensation of open-loop systems first. In Fig. 7.6, a plant whose pulse 
transfer function is G(z) is to be compensated by a digital device whose 
pulse transfer function is D(z) to implement an over-all pulse transfer 
function K,(z). For purposes of discussion it may be assumed that a 
desired over-all response function K4(z) may be, though it does not neces- 
sarily have to be, a minimal prototype function described in the previous 
section. In this case, the desired response would be related to the plant 
and compensator by the following relation: 
K.(z) = D(z)G() (7.18) 
where D(z) is so chosen that it cancels the undesirable poles and zeros of 
G(z) and replaces them by the poles and zeros of the desired over-all 
response function. 
pa 
—~ 
R(z) C(z) 
Fia. 7.6. Digital compensation of open-cycle system. 
In implementing a technique like this one, certain limitations present 
themselves. For instance, if the plant pulse transfer function G(z) con- 
tains a pole b, and a zero a, which lie outside the unit circle, difficulties 
due to inaccurate cancellation are severe. The presence of a pole in the 
plant outside the unit circle implies the instability of the plant, an 
unlikely circumstance. On the other hand, the presence of a zero outside 
the unit circle is perfectly possible with stable plants found in practice. 
The digital compensator has a pulse transfer function D(z) which cancels 
these poles and zeros and replaces them with desired poles and zeros. 
However, since the magnitudes of the actual plant poles and zeros are only 
approximately known and, furthermore, since the implementation of the 
digital compensator D(z) cannot be perfectly implemented, it follows 
that poles and zeros of the plant can never be perfectly canceled. This 
results in serious difficulties for those poles and zeros which lie outside 
the unit circle, as can be seen from the expression 
OSU R ep ae, 
K.(2) re (2 Si b,) (z =o aa) Ka(e) (7.19) 
where K,,(z) is the actual over-all pulse transfer function and K,4(z) is the 
desired over-all pulse transfer function. Ideally, the poles and zeros 
introduced by the plant, b, and a,, are canceled by the poles and zeros 
introduced by the digital compensator, bg and ag. Actually, perfect 
cancellation is not achieved, and it is noted that the compensated over-all 
response function K,(z) then contains a pair of poles b, and ag which lie 

