162 SAMPLED-DATA CONTROL SYSTEMS 
following expression after simplification: 
0.81 — 1.1062-1 + 0.4852~2 — 0.0692-* 
1 + 0.35221 — 1.1592? — 0.19823 

Dg) = 
This response can be physically implemented by using four number 
storages in the manner shown in Fig. 4.10. The number of storages 
required is a measure of the complexity of the digital controller. This 
factor will be referred to again when different over-all prototype 
response functions are used. 
The foregoing example has served to illustrate many of the pertinent 
points applying to the design of “minimal” prototype systems. As 
shown there, minimal prototype systems cannot always be realized 
because of the presence of poles and zeros of the plant pulse transfer 
function which lie outside of the unit circle. In such a case, only a 
‘““minimum”’ response function can be obtained, which produces longer 
settling times than the minimal function, depending on how many 
uncancellable poles and zeros there are. A critical examination of the 
results of the previous example will show that the digital controller sup- 
plies the additional integration in the feedforward line of the system 
which is required to respond without error to a unit ramp input. The 
integration process implemented by the digital controller is a numerical 
integration, but the results on the over-all response are the same. 
A precaution which must be observed relative to the design of digital 
controllers is that cognizance be taken as to their output under expected 
operating conditions. For instance, in the previous example, the digital 
controller implements a numerical integration process. In doing so it 
produces a system which responds to a unit ramp with zero system error. 
Physically, this means that the output of the digital controller in the 
steady state is a constant. On the other hand, if the plant contained no 
integration of its own, the digital-controller pulse transfer function would 
show a double integration process in order to produce a system which 
responds to a unit ramp input with zero system error. In this case, the 
output of the digital controller would have to rise without limit since it 
provides both integrations necessary to implement the condition. From 
the practical viewpoint, this is impossible, and the system would saturate 
very quickly upon application of a ramp input. The general rule which 
should be followed is that the output of the digital controller should 
never be expected to rise above some practical limit under operating 
conditions of the system. 
Arbitrary response functions can be realized by taking the expression 
for K(z) and including more than the minimum number of terms. For 
instance, in the illustrative example, only a; and ag were-required in the 
