108 SAMPLED-DATA CONTROL SYSTEMS 
function G(z). In this case, G(z) contains one finite zero and one zero 
at infinity. The point of departure of the locus from the real axis is 
found to be at 0.648. It has been shown?! that for this system the root 
locus representing the complex closed-loop poles of the system is a 
circle whose center is at (—0.72,0). The radius of the circle is 1.368 
and, using these values, the root locus appears as shown in Fig. 5.16. 
It should be noted here that more complex systems do not have such a 
simple form for the root locus and that they must be plotted by identi- 
fying those points for which the total angle of G(z) is 180°. Upon 
reentry into the real axis, the root-locus branches, one branch going to 
infinity and the other toward the finite zero of the system at (—0.72,0). 
A scale drawing of the root locus will show that the intersection with 
the unit circle occurs at the points 0.24 + j0.97. The gain K required 
to place the poles at this point is found from the magnitude relationship 
0.264 + 0.3682 
- (2 — 1)(@ — 0.368) |z=0.244;0.97 oe 
Solving for K, the gain is fond to be 2.43. This result was obtained 
in the illustrative example of the previous section by consideration of 
the enclosure of the point (—1,0) by the pulse transfer locus. Thus, to 
produce a stable system, it is required to have the gain K be less than 
2.42. If the gain exceeds this figure, the poles will be located outside 
the unit circle, resulting in an unstable system. Inversion of the over- 
all response pulse transfer function of the system would show that for 
the case resulting in complex poles, the response to a step function 
would be stable, though oscillatory, if K is less than 2.43 and more 
than 0.196. 
An unusual condition obtains in sampled-data systems for which no 
comparable or analogous situation exists in continuous systems. There 
may be stable sampled-data systems which have poles in the over-all 
pulse transfer function K (z) that are both real and negatine. For instance, 
in the system giving rise to the pulse transfer function G(z) of the system 
used in the preceding example, this condition will be realized if the 
sampling interval Tis halved. If this happens, the root locus will appear 
as shown in Fig. 5.17, where it is evident that for some value of K two real 
negative roots will result. Such negative real roots will produce an 
oscillatory response, as can be seen by considering the transient produced 
by such a root. The transient contribution of a root of this type is 
c(kT) = A(—a)* (5.59) 
where A is an arbitrary constant and 2; is a positive number, the sign of 
the root being explicitly written. For negative real roots, the sequence 
