MULTIRATE SAMPLED SYSTEMS 243 
can always be factored because the design has included specification of 
the roots of this function. Substitution of (9.77) and (9.65) into the 
basic design equation (9.36) leads to the final design objective, the 
pulse transfer function of the multirate controller D(z,). 
Se ale Wve 2 , 
D(2n) = Te > ~ ee l= ax (9.78) 
The multirate design which has resulted in the specification of the con- 
troller pulse transfer function given in (9.78) may be evaluated by com- 
paring the response of the multirate system with that of a single-rate sys- 
tem operating on the same plant and with the same over-all specifications. 
1 15-1423! 
7 1-23} 


(i 


Fia. 9.12. (a) Triple-rate and (6) single-rate controllers for an unstable plant. 
To illustrate the point, a multirate controller having an output sample 
rate three times the input sample rate will be used. This means that 
n = 3 in evaluating (9.78). The triple-rate controller is shown in Fig. 
9.12a, along with the continuous plant transfer function leading to (9.65). 
The single-rate system is shown in Fig. 9.126. It is noted that the com- 
plexity of the controller in each case is the same, that is, both controllers 
require no more than two storage elements. The response of these sys- 
tems when subjected to a step input is shown in Fig. 9.13, where they 
may be compared. First of all, the overshoot in both cases is quite large 
and equal. This is to be expected with an unstable plant used in the 
example. Also, the triple-rate design does not show much improvement 
in settling time because the unstable plant can be stabilized only by feed- 
back and the first feedback sample is applied to the error system after 
1 sec in either case. A system which is open-loop stable can reach 
steady state in much less time, as shown in the first example. 
Each of the example problems worked out for multirate controller 
design have been of the minimum-finite-settling-time variety. The 
application of staleness factors or the design for zero ripple could also be 
