DIGITAL COMPENSATION OF SAMPLED-DATA SYSTEMS 149 
powers of 2—! as follows: 
G(z) = Azm+ Be-MD +1... (7.7) 
where A, B, etc., are constants which are functions of the various p’s and 
q’s. Since G(z) defines the impulsive response of the plant, it follows 
that the presence of 2~” as the first term assures the fact that the output 
of the plant does not precede the input. In the limiting condition, the 
exponent m can be zero, although in practical plants it is never less than 
unity. 
The digital controller whose pulse transfer function is D(z) is also sub- 
ject to this restriction, and its form must be as given in (7.2). Inci- 
dentally, in this form, the gain constant is ao since it can be factored out 
as a multiplicative term for the whole expression. The impulsive 
response of a digital controller of this form is seen to have a first output 
pulse whose amplitude is @. Asa consequence of the restrictions placed 
on the forms of the pulse transfer functions G(z) and D(z), the over-all 
prototype response function K(z) as given in (7.4) will likewise have 
restrictions. These are found by substituting the required forms of 
G(z) and D(z) in (7.4) as follows: 





(Qe seo 8 2 fe (Go ar GE sp ee ae 2s 2 Ge) 
Re) = (Gosr GE ar 2 8 8 Ge ap los Se pe SP le 2 2 le) 
ane Oe -+ dere Dn”) (ao + aye + oz? + 660 Cem) 
~ o + get + + ss qe )(1 + biz! + baz? + +: - Bye) 
(7.8) 
Simplifying this expression and collecting terms, the following results 
ee ae 2 99 Tiere 
I) = i SR ae Pe ee eae (7-9) 
where the various k’s and l’s are combinations of the various p’s, q’s, a’s, 
and b’s. 
It is concluded from this simple development that in order to accom- 
modate physically realizable elements in the closed loop, the numerator 
of the over-all pulse transfer function K(z) must contain z—! to a power 
equal to and possibly greater than the lowest power m in the plant pulse 
transfer function G(z). Also, it is necessary that the term J) appear in 
the denominator of K(z), as shown in (7.9). By observing these simple 
rules it is assured that the specified prototype does not require physically 
unrealizable components in its implementation. 
The second requirement for the minimal prototype response function is 
that it respond to a specified test function with zero systematic steady- 
state error. It is implied in stating the condition that this need only 
apply at sampling instants. From a practical viewpoint, however, it 
