MULTIRATE SAMPLED SYSTEMS 237 
where (9.44) is obtained in the manner defined earlier, which is to take the 
z transform of R(s) and replace z by z,”. The functions P(z,) and P(z,”) 
are polynomialsinz,!. If (9.43) and (9.44) are substituted in (9.42), one 
obtains the equation for the z, transform of the error 
eee) IP(@2 IK (Eq) 
The steady-state performance one would like to specify is that the final 
value of E;(z,) in response to the polynomial input be zero. The final- 
value theorem may be used to express this condition in terms of K(z,). 
Tf all poles of Hi(z,) are inside the unit circle, then 
€1(0) = me (1 = 2n7!) Ei (2n) ; (9.46) 
In order for Ei(zn) as given by (9.45) to meet the requirements for the 
application of (9.46), it is necessary, first of all, that K(z,) contain as 
zeros all the roots of (1 — z2,~”)* except the kth-order root at unity. As 
discussed in Sec. 9.2, these roots lie equally spaced around the unit circle 
and must be removed if the final-value theorem is to apply. Therefore, 
by inspection, it can be stated that for proper steady-state behavior, 
K (gn) must be of the form 
1 — 2,7") 
KG) = Sh Fe) 
= (1+ get + ag? ot apt) EF (2,,) (9.47) 
where the correctness of the polynomial in z,—! can be readily verified by a 
process of long division and where F'(z,) is as yet arbitrary. Substituting 
(9.47) in (9.45), 
IFAC) IZA) 
OD ee ek 
_ Ip) = PG EQOME») 
(Beam) 
If, now, the final value of £i(z,) is to be zero, then (9.48) must have no 
pole at z, = 1. This means that the numerator of (9.48) must have a 
kth-order zero at 2, = 1. That is, if Q(z,) is defined as 
(9.48) 
OG) FeCl EAE) (9.49) 
then the steady-state constraint may be satisfied by requiring that 
OC, y— 5 erm) ie) (9.50) 
Or, equivalently, 
EOE nc omen ol nee ee 1 (9.51) 
Ve Nae Zn=1 
