MULTIRATE SAMPLED SYSTEMS 241 
satisfy the two equations 
I= py? Il == Bhp © 
K(@n) = Eeaiiqpestos = (HA oP RS) (9.66) 
eae) — (ls 2t2n (Dien | Ooeamc) (9.67) 
The two constants gi and gz in (9.66) are to be determined from the 
steady-state requirement and from the fact expressed by (9.67) that 
1 — Z[K(z,)] must contain the factor 1 — 2"2z-!. The constants b; and 
be in (9.67) are only functions of the g’s and will automatically fall out 
of the solution. The essence of the constraint expressed by (9.67) can 
also be stated in the form 
Z[K (zn)] = 1 (9.68) 
Z=2n 

The steady-state requirement given by (9.51) requires the formation 
of the Q(z,) function from (9.49). In the present example the design 
requirement is for zero steady-state error to a step, so that one can 
write immediately 
1 
R(s) = a 
1 
Me Digi 
1 
gs 
Caan 
Je) = Il 
and, from (9.49) and (9.66), 
Iie Bg oe 
Q(én) = 1 — TT So (Gis > AP Gee 9) 
The steady-state requirement of (9.51) reduces to 
Q(zn) eid = 0 
1 
for a step input for which k = 1. Therefore, the two constants g; and 
g2 must satisfy the equation 
1 
27 — 1 
The second equation satisfied by gi and gz is obtained from (9.68) and 
requires the calculation of Z[K(z,)]. This calculation can be made 
either by expansion of K(z,) and selection of the z,—” terms or else by 
use of the formula developed earlier that 
RC eee (9.70) 
20j T (1 — ecm Cn 
gi + g2 = (9.69) 
