DIGITAL COMPENSATION OF SAMPLED-DATA SYSTEMS 155 
outside of the unit circle on the z plane. This means that the over-all 
sampled-data system is unstable. 
It follows, therefore, that in the compensation of an open-loop system 
by a digital compensator, the poles and zeros of the plant pulse transfer 
function which lie outside of the unit circle cannot be canceled by the 
digital unit. This means that if a plant is unstable, that is, if it has poles 
outside the unit circle on the z plane, it is impossible to stabilize the sys- 
tem by means of a cascaded digital unit. Fortunately, this is not the 
ease for a closed-loop implementation, as will be shown later. More 
serious than this limitation is the fact that if a stable open-loop plant has 
a pulse transfer function which has a zero outside the unit circle, it is not 
possible to cancel this zero by means of the digital compensator. To 
ensure this fact, (7.19) shows that the desired over-all pulse transfer func- 
tion must contain as one of its zeros those zeros of the plant pulse transfer 
function which lie outside the unit circle. Thus, there is no complete 
freedom of choice in K4(z), and often the minimal prototype cannot be 
achieved. 
7.5 Compensation of Closed-loop Systems 
Less obvious but equally important are the limitations imposed on 
the designer in the compensation of closed-loop sampled-data sys- 
tems.** In the error-sampled system shown in Fig. 7.7 the digital 

Fia. 7.7. Digital compensation of closed-loop system. 
controller has a pulse transfer function D(z) and the plant a pulse transfer 
function G(z). The digital controller has a direct effect of canceling the 
undesired poles and zeros of the plant and replacing them with the poles 
and zeros required to implement some over-all response K(z). To study 
the forbidden cancellations, let it be assumed that the plant pulse transfer 
function contains at least one pole and one zero which lie outside the unit 
circlein thez plane. For purposes of this development, it can be assumed 
that all the other singularities lie inside the unit circle. Thus, G(z) can 
be expressed by 
2 @& 
Ge) = Fe) (7.20) 

where a, and b, are the zero and pole, respectively, which lie outside the 
unit circle. 
If the specified over-all pulse transfer function for this system is K,(z), 
