MULTIRATE SAMPLED SYSTEMS 245 
pulse of amplitude rp is supplied at ZH, to the multirate controller. The 
controller then applies a sequence of pulses to the plant at intervals of 
T/n sec, which pulses are so designed in amplitude that the plant output 
is brought to the new required level or position with no further error 
detected at H;. The parameter n must be chosen sufficiently large that 
the required change in output level can be accomplished in the allotted 
time. The minimum value of n necessary to do this can be determined 
by inspection of the design equations. 
The reason that single-rate techniques can be used to design the multi- 
rate controller pulse transfer function for the operation described above is 
that the entire design is essentially based on open-loop equations. The 
controller receives a single pulse from H#; and proceeds to change the out- 
put level of the plant sufficiently to reduce the detected error to zero with 
no feedback at all. Of course, if for any reason the output level did not 
reach the desired position, then a second error pulse would supply the 
feedback information. In other words, although the system operates on 
feedback principles, it can be designed on an open-loop basis. 
In terms of design equations, the procedure for realizing the pulse 
transfer function of the multirate controller using these principles is quite 
simple. The design objective may be realized by requiring the multirate 
system of Fig. 9.9 to produce an output in response to a step input which 
is identical to the output of a single-rate system designed for the same 
plant and a sampling period T/n. The only difference between them is 
that the multirate controller gets only a single pulse input instead of a 
sequence of pulses. 
To be more specific, the transform of the output of the system of Fig. 
9.9 is 
C(én) = Ey(2n") D(en)G (Zn) (9.79) 
and, if n is so chosen that only one pulse appears at Hj, then (9.79) reduces 
to 
Cen) = roD(en)G (en) (9.80) 
where 7o is the value of the first error sample supplied to the controller. 
If the input is a unit step, then rois unity. A single-rate system designed 
for zero-ripple response with the plant G would have an over-all pulse 
transfer function K(z,). The output of the single-rate system would be 
C(en) = R(zn) BK (en) (9.81) 
If these two system outputs are to be identical, then the multirate con- 
troller pulse transfer function must be 
R(en) K(2n) 
LNGa) = ro Gen) 
(9.82) 
