SAMPLED-DATA SYSTEMS 109 
is alternating, being positive for even values of k and negative for odd 
values of k. So long as 2; has a magnitude less than unity, the sequence 
is stable; that is, it converges as k tends toward infinity. Thus, in 
sampled-data systems it is not necessary that the poles of the over-all 
transfer function be complex in order to obtain an oscillatory solution. 
If the poles of the over-all pulse transfer function are real and positive, 
then the system is nonoscillatory, as can be seen readily from (5.59). 
Re 

Fic. 5.17. Root locus for stable system containing poles on negative real axis. 
Generally, the proximity of the poles of the over-all pulse transfer func- 
tion to the unit circle is indicative of a pronounced transient response. 
For instance, if the poles are complex and close to the unit circle, the 
oscillatory transient which results when the system is excited will have 
significant values many sample times later since a magnitude slightly less 
than unity raised to a power will not be significantly less than unity unless 
the power is very high. Thus, considerable overshoot and oscillation can 
be expected from a system of this type. Similarly, if the roots are real 
and negative and lie close to the unit circle, a heavy oscillatory response 
can also be expected. If the roots are real and positive and lie close to 
the unit circle, the response is monotonic but the approach to steady state 
requires many sample times. In the design of sampled-data systems 
having moderate transient response and minimum overshoot, it is neces- 
sary that the poles of the system lie well within the unit circle. Quanti- 
tative criteria as to pole locations cannot be given any more than they 
can for continuous systems, except for specific cases. Dominant poles 
are characterized by their proximity to the unit circle, as contrasted to all 
the other poles of the system.”!_ For instance, if a pair of complex poles 
