DIGITAL COMPENSATION OF SAMPLED-DATA SYSTEMS 157 
and pole a, and b,, respectively. When this is so, perfect cancellations in 
the numerator and denominator of (7.23) take place, and the actual over- 
all response function is identical to the desired over-all response function. 
Now, if the plant zero and pole drift slightly by an amount Aa, and Ab,, 
these cancellations can no longer be made since the numerator of (7.23) 
differs from the denominator. 
The denominator of (7.23) is a polynomial in z, and the locus of its roots 
as any parameter is varied is continuous. Starting then with Aa, and 
Ab, at zero, meaning that a, and b, are respectively equal to az and ba, it is 
seen that the denominator polynomial has zeros at ag and b, (or a, and 
ba), which lie outside the unit circle. In this case, they are canceled by 
the numerator roots, which are equal. However, as the slight shifts Aa, 
and Ab, are introduced, the denominator roots shift slightly but are still 
outside of the unit circle in the z plane, and they are not canceled by equal 
roots in the numerator since its roots shift in a different manner. 
The conclusion which is drawn from this discussion is that if a plant 
contained in a feedback sampled-data system has a pulse transfer function 
which contains zeros and poles which lie outside of the unit circle in the 
2 plane, or in the limit, on the unit circle, no attempt should be made to 
cancel such poles with a digital-controller pulse transfer function since 
instability would inevitably result. This does not mean that a feedback 
system containing such a plant cannot be stabilized or compensated but 
rather that complete freedom of choice of over-all response functions is 
not possible. 
By applying suitable restrictions on the form of the specified pulse 
transfer function K,(z), the cancellation of poles and zeros of the plant 
pulse transfer function by the digital-controller pulse transfer function 
can be prevented. ‘These restrictions obtain by substituting the plant 
pulse transfer function from (7.20) in (7.21). 
D(z) = C= aE Paced eat ACLNIES (7.26) 
If (7.26) is not to contain a pole a, and a zero b, in the pulse transfer 
function D(z), these terms are to be contained in K,(z) and 1 — K,(z), 
respectively. Thus, in specifying K,(z) which leads to a stable system, 
the following relations must be satisfied: 
K,(z) = ( — a,z-!)M() 
1— K,@) = @ — 6,2 YN) (e270) 
where M(z) and N(z) are unspecified ratios of polynomials in g—. In 
words, zt 1s necessary that the specified over-all pulse transfer function K.(z) 
contain as its zeros all those zeros of the plant pulse transfer function which 
lhe outside or on the unit circle in the z-plane and that 1 — K.(z) contain as 
