164 SAMPLED-DATA CONTROL SYSTEMS 
that to obtain unit output for unit input, the constant a must be made 
equal to (1 — c). Thus, a unit steady-state response prototype function 
K(z) becomes 
‘ — c)z! 
— cet 
K@ = (7.30) 
For a unit step input, three response functions are computed and plotted 
in Fig. 7.11, one for c set equal to zero (the minimal prototype), one with 
c set equal to 0.5, and the third with c set equal to —0.5. It is seen that 
Output c(n7) 

3T 4T 5T 6T 
Time 
Fia. 7.11. Response of system with staleness factors of various amount to unit step 
Input. 
for the staleness factor c set equal to a positive number of damping effect 
very similar to that produced by a low-pass RC circuit in continuous 
systems is obtained. A negative staleness factor brings about an oscil- 
latory response, which is the usual result when a pulse transfer function 
has a pole which is negative real, in this case —0.5. In practice, staleness 
factors are taken usually as positive in systems having the form of (7.28) 
and the exponent JN is taken as unity or other small integer. 
EXAMPLE 
To illustrate the effect of a staleness factor on the performance of a 
system, the same problem which was considered in the example in the 
previous section will be considered here. The block diagram of the 
system is given in Fig. 7.8. The over-all pulse transfer function is the 
same as in the minimal prototype considered in this example, except 
for the addition of the staleness factor. Thus 
(+ 2.842"")(ae7! aoe +) 
LS U.ae- 
It is noted that K(z) contains the zero, 2.34, of G(z) which lies outside 
of the unit circle of the z plane. 
KG) — 
