SAMPLED-DATA SYSTEMS 107 
and if outside, an unstable system. Also, in view of (5.55), the closer the 
poles approach the magnitude unity, the longer will the transient persist 
in a system subjected to a sudden input. Generally, proximity to the 
unit circle has the same effect as proximity to the imaginary axis of poles 
ina continuous system. The best way of showing these effects is to use an 
example. 
EXAMPLE 
To illustrate the rules governing the plotting of root loci for sampled- 
data systems, the same system used in the illustrative example of the 
Im 
0.24 +70.97 
(K=2.43) 





Root locus 
unit circle 
Yj UY K=0.196 
Yj 
yyy 
Vif, 
is) 
Ly 
ASN, 0 
Z, , 
‘Vithpyy A, 
bf 
by 
1,0 Re 
Fig. 5.16. Root locus for system used in example. 
previous section will be used. The system considered there has a unity 
feedback and a feedforward pulse transfer function G(z) given by 
0.264 + 0.3682 
Ne) = i (2 — 1)(e — 0.368) 
The system uses error sampling, resulting in a characteristic equation 
of the form 
1+Ge) =0 
whose roots must be determined. The open-loop pulse transfer func- 
tion G(z) has poles at (1,0) and (0.368,0) and has a zero at (—0.72,0). 
These poles and zeros are marked on Fig. 5.16. 
Starting with the gain constant K at zero, the root loci initiate at the 
poles of the open-loop transfer function G(z). When the gain K goes 
_to infinity, the loci terminate at the zeros of the open-loop transfer 
