228 SAMPLED-DATA CONTROL SYSTEMS 
is to be valid and the poles introduced by this factor must lie outside the 
contour of integration. The condition also implies that the region of the 
z plane over which the resultant C'(z) is valid lies outside the circle whose 
radius is z,”. Since I contains all the poles of C(z,) on the z, plane, the 
corresponding locus on the z plane contains all the poles of C(z), so that 
the inversion of C(z) by the usual inversion integral will be correct. The 
contour I which contains all the poles of C'(z,)/zn but which excludes the 
roots of (1 — z,"z~1) thus produces the correct expression for C(z). In 
evaluating (9.19), it is possible to do so either by obtaining the residues at 
the poles of C(z,)/2n which are contained inside I or by obtaining the 
residues at the n poles which are the roots of (1 — 2,2!) which lie outside 
IT. An example will illustrate the evaluation of (9.19). 
EXAMPLE 
As an example of the use of (9.19), consider again the situation shown 
in Fig. 9.4¢ with 
ik 
fi(s) = = 
it 
DOs rash 
and io 
no 
One can write down immediately that 
C(en) = R(2n)G(2n) 
: 1 
~ Gd —2 0 — 2,4) 
ae 
= = Ga (9.20) 

From (9.19) evaluated by obtaining the residues at the poles of 
C (én) /2n, 
1 Zn" iL dzn 
TO) ae | ee eye 
1 e-l/n 
a (tea) Ge) a (e" — 1) — e327) 


and, for n = 3, 
WS Ge) Sa =e) 
C@) = Gad 0 ee) 
oe 1 + (e-¥3 + Cae) 2m 
fan =e) Se 
which checks with the previous result obtained by switch decomposi- 
