MULTIRATE SAMPLED SYSTEMS 239 
The solution of the simultaneous equations given in (9.58) is given by 

a 
fr= 1 
and the final required form of the pulse transfer function is 
KC a I (" ile a i 2-1) (9.59) 
l=2,73 n 

To illustrate the design given above with a numerical example, con- 
sider the design of a double-rate controller for the control of a plant 
with the transfer function 

1—e? 4.6 
G(s) = z EG (9.60) 
with T = 1 sec. In this case, 
ah 0.925 
CD) a (9.61) 
and, from (9.59), 
1G) a) (lieu eo lsocm lame) (9.62) 
From (9.62), taking the terms in z2~°, 
K (22?) Ce (9.63) 
The substitution of (9.61), (9.62), and (9.63) in the fundamental design 
equation (9.36) yields 
l= Ota Laat 
Den) eet = eae oe 
A — Cie 0.5 = a) 
7 OSU = ae oe 


The block diagram of the multirate system is shown in Fig. 9.10a, and 
a single-rate system designed for the same system by the techniques of 
Chap. 7 is shown in Fig. 9.106. The responses of the two systems are 
shown in Fig. 9.11, where the two curves show clearly the advantages 
of the multirate controller in reducing the output ripple of a sampled- 
data system. The particular example chosen may seem slightly arti- 
ficial in that the basic sampling rate is taken to be quite slow compared 
to the time constant of the plant, but in fact this is the only situation 
in which ripple is a serious problem anyway. 
EXAMPLE 
Asa second example of the design of a multirate controller, the design 
of D(z,) for the finite-settling-time control of a plant with the pulse 
