MULTIRATE SAMPLED SYSTEMS 235 
desired results differ in the two cases. It is possible to design a multirate 
system for minimum finite settling time with zero steady-state error to a 
polynomial input, or to design for a ‘“‘staleness-factor”’ type of response, 
or to design for a ripple-free response. Since each of these designs has 
been discussed in detail in Chap. 7, only the essential parts of the designs 
will be worked out here to illustrate the special problems introduced by 
the multirate character of the system. 
Three basic requirements on the design of any control system are 
physical realizability, stability, and steady-state behavior. Each of 
these requirements will be discussed in turn for the system of Fig. 9.9. 
The most fundamental of these requirements is that of physical realizabil- 
ity. Inits most elementary terms, this limitation of physical equipment 
is expressed by the statement that to be physical, a device must not 
respond prior to the application of the excitation. In terms of the multi- 
rate control system, the controller pulse transfer function described by 
(9.36) will be physically realizable if K(z,) is made to have a zero at 
infinity of at least the order of the zero of G(z,) at infinity. That is, if 
the expansion of G(z,) about infinity (in powers of z,—!) starts with the 
Zn” term, then the similar expansion of K(z,) must not start until the 
Bo? WEA, 
The requirement that the system be stable is met by having all the poles 
of K(z,) lie inside the unit circle and by avoiding the cancellation of poles 
or zeros of G(z,) which are outside the unit circle by the controller pulse 
transfer function D(z,). This restriction of the poles and zeros of the 
controller transfer function is equivalent to the similar situation dis- 
cussed in Chap. 7 in connection with the design of unit-rate systems. 
For the multirate controller described by (9.36), stability requires that all 
the zeros of G(z,) which are on or outside the unit circle be contained as 
factors of K(z,) and all poles of G(z,) which are on or outside the unit 
circle be contained in 1 — A(z,"). In other words, if 
GG) = == ne) (9.37) 
yaa anes 
where F'(z,) is a ratio of polynomials in z, containing all its zeros and poles 
inside the unit circle of the z, plane and where |a|, |b) > 1. Then, for 
stability, it is necessary that 
KG(@*))e— ei (lee) AN (2) (9.38) 
and LS 1K(@ee) = Cl = Was 2 BeX) (9.39) 
A peculiarity of the multirate character of the problem is indicated by 
(9.39). In this equation, because of the nature of K(z,”), it has been 
necessary to introduce n roots in the function 1 — K(z,”) in order to 
remove one root from G(z,). As mentioned in Sec. 9.2, all these roots of 
