234 SAMPLED-DATA CONTROL SYSTEMS 
The pulse transfer function of this system is defined as the ratio 
CG) 
ie) 
“ D(2n)G (én) 
Wie Zz D ence) 
Equation (9.32) is the fundamental equation of analysis for the multi- 
rate controller system. If the controller pulse transfer function D(zn) 
and the plant transfer function G(z,) are given, then (9.32) or (9.31) show 
the response of the system. In order to synthesize the controller transfer 
function, it is necessary to invert these expressions and solve for D(z,) in 
terms of K(z,) and G(z,). The first step toward this inversion is to take 
the z transform corresponding to (9.32). 
Tze) | — Kae) 
_ LD @n)Gn)] 
1 + Z[D(2n)G(en)] 
In writing (9.33), use has been made of the fact that 1 + Z[D(en)@(2n)] is 
a function of z and may therefore be factored out of the z transform opera- 
tion. Solving (9.33) for the term containing the unknown controller 
transfer function D(z,), 

K (én) = 
(9.32) 

(9.33) 
UD een] = ~~ (9.34) 
1+ 2D@)Ge)] = EG (9.34a) 
The solution for the controller pulse transfer function may be completed 
by substituting (9.34a) into (9.32): 
r(y\ — _D@n)G@n) 
1 K (én) 
therefore D(2n) (9.36) 

7G@) T2kG 
Once a desirable over-all transfer function K(z,) is prescribed, (9.36) 
gives the transfer function of the multirate controller necessary to produce 
the desired response. The design of the multirate controller, like that of 
the single-rate controller, is therefore reduced to the specification of a 
suitable over-all transfer function K (z,). 
The design objectives and limitations associated with the specification 
of a suitable over-all pulsed transfer function for use in a multirate con- 
trol system are essentially the same as those already outlined in Chap. 7 in 
connection with the design of single-rate systems. Any of the design 
methods discussed for single-rate systems applies to the design of multi- 
rate systems, with the sole exception that the formulas leading to the 
