232 SAMPLED-DATA CONTROL SYSTEMS 
The algebra involved in obtaining (9.26) from (9.25) is fairly com- 
plicated and, in particular, requires that several common factors in the 
transforms be divided out. In carrying out such an analysis it is help- 
ful to keep the order of the characteristic equation in mind and to 
continually look for common factors to reduce the expressions. The 
output of the system is readily calculated from one of the equations in 
(9.24). 
9.3 Synthesis of a Multirate Digital Controller 
Methods for the design of digital controllers for control of single-rate 
sampled-data systems were given in Chap. 7. In that chapter were given 
several design criteria, such as minimum finite settling time, staleness- 
factor design, and ripple-free design. All these methods, as developed in 
that chapter, were applied to systems which are single-rate. The 
manipulated variable which came from the controllers operated at the 
same basic rate as the error information fed to the controllers. 
In many cases of practical importance the design can be improved by 
adopting a multirate controller. The block diagram of a system which 
employs a multirate controller is shown in Fig. 9.9. In this system the 
error signal, i, is sampled at some basic period 7 which is presumed to 
be fixed beyond the control of the system designer. The digital con- 
troller is designed to operate at a reduced period, T/n, so that the samples 
to the plant G are supplied more frequently than the input is sampled. 
The properties of such a design are briefly these: 
1. Both the output ripple and the response time, or ‘‘rise time,” are 
less than in the comparable single-rate system because of the shorter 
period of the samples at the input to the plant. 
2. The input signals to the plant from the multirate controller are 
larger in amplitude than comparable signals from the single-rate system, 
with the result that saturation effects are more serious in the multirate 
system. 
3. The multirate controller has the same memory or storage require- 
ments as the corresponding single-rate design, but must operate at a 
higher speed. 
The designer must decide whether or not the improved speed and the 
ripple performance are worth the price of increased speed of computa- 
tion of the controller and greater saturation limits in the plant. If the 
decision is favorable to a multirate controller, then the design may be 
readily carried out using the procedures to be described here. 
The design problem for a multirate controller to be used in the system 
