138 SAMPLED-DATA CONTROL SYSTEMS 
EXAMPLE Method 5 
The design of a pulsed network to compensate the system shown in 
Fig. 6.6 may be easily done from root-locus or pole-zero considerations. 
The pulse transfer function of the plant and hold combination is 
z+ 0.719 
HG(z) = 0.368 (2 — 1)(z — 0.368) 

(6.25) 
which may be represented by the pole-zero constellation and cor- 
responding root locus shown in Fig. 6.19. The relative damping of 
Unit circle 
Re 

Fria. 6.19. Gain root locus for system of Fig. 6.6. 
the transient terms corresponding to the poles of the closed-loop trans- 
fer function depends upon the distance of these poles from the unit 
circle. Poles on the unit circle correspond to transients which do not 
decay with time, and poles at the origin of the z plane correspond to 
transient terms which vanish completely in a finite time. In general 
terms, the objective of the design is to bring the closed-loop roots close 
to or exactly to the origin. Any of the well-known techniques of root- 
locus modification may be used toward this end, using either the simple 
networks shown in Fig. 6.17 or more complicated structures. In the 
present example, a very satisfactory result can be obtained by use of 
the “lead” structure of Fig. 6.17b to cancel the zero and the pole of 
the GH(z) transfer function at 0.368. To accomplish this end, it is 
necessary to select the gain K and the time constant a of the com- 
pensating network in such a fashion that 
z2—eor 
Cee — eqs ican” 
z — 0.368 
y Sacra (6.26) 
N(z) = 
