244 SAMPLED-DATA CONTROL SYSTEMS 
done with a multirate controller, of course. The techniques are the same 
as for the single-rate case discussed in Chap. 7, except for the change in 
details of constraint applications as discussed above. The comments 
regarding the relative merits of each of these designs as given in Chap. 7 
apply equally well to the multirate case and need not be repeated here. 
9.4 Design of Multirate Controllers with Single-rate Techniques 
For the usual problem in control-system design with plants to be con- 
trolled which are open-loop stable, the multirate controller may, under 
certain circumstances, be designed using single-rate techniques. This 
© Response of triple rate system 
X Response of single rate system 
_ 
PSO wo fF aA Dn N CO UO OO 
_ 

Time 
Fig. 9.13. Step responses of single- and triple-rate systems of Fig. 9.12. 
design procedure, which was first described by Bertram in a discussion of 
ref. 30, takes advantage of the fact that for finite-settling-time designs 
(which include the zero-ripple design) it is always possible to select the 
speed-up factor n in the multirate controller in such a fashion that all 
transients in the system at sampling instants are over in one period of 
length T. Although this technique applies to any finite-settling-time 
design, the emphasis here will be placed on the zero-ripple case, which, of 
the several possibilities, is the most attractive in terms of performance 
and among the most complicated in terms of design steps. For the zero- 
ripple design, the speed-up factor is so chosen that transients at all 
instants are made to terminate in one period of length T. To concentrate 
for the moment on the step response, the operation of a multirate system 
designed on the basis being proposed here can be described as follows, 
with reference to Fig. 9.9. Upon application of the step at R, a single 
