MULTIRATE SAMPLED SYSTEMS 249 
the different rates are the ratios of small numbers, it is possible to analyze 
a multirate system by reducing it to an equivalent multiloop single-rate 
system. This reduction of the multirate problem to a single-rate problem 
involves the decomposition of a sampling switch which operates with a 
period 7'/n into n parallel switches (with delay and advance operators) 
which operate with the period T. The method, which is called switch 
decomposition, is practical for pencil-and-paper analyses only if the total 
number of switches in the equivalent system is relatively small. For 
multirate systems where an exact equivalent single-rate system would 
involve a large number of samplers, it is frequently possible to obtain an 
approximate system which should provide sufficient accuracy to answer 
the most important questions. For example, switches with sampling 
rates in the ratio 22:7 could probably be considered to be in the ratio 3:1 
without a great loss of accuracy. 
The theory of multirate control systems is important to the synthesis of 
new systems as well as to the analysis of existing systems. A multirate 
controller can be designed which accepts error data at intervals of T sec 
and supplies data at intervals of T’/n sec. This effective speed-up in the 
data rate into the plant or process means that, in general, the response 
time of the output will be reduced and the ripple between samples will 
be lessened. The multirate controller can be designed for the standard 
prototype response functions, such as minimum finite settling time, stale- 
ness factor, or zero ripple, and will in each case effectively speed up the 
time scale of the response. There is a limitation on the response-time 
improvement in case the plant is open-loop unstable because such systems 
can only be stabilized by feedback and the multirate controller does not 
increase the speed of feedback sampling. 
For finite-settling-time designs, it is possible to obtain the pulse transfer 
function of the multirate controller by using single-rate techniques. 
With this combined design method, the speed-up factor in the multirate 
system is included as a design parameter and not left to arbitrary choice. 
There is some indication that the method can be extended to provide a 
basis for a nonlinear adaptive control-system design. 
