248 SAMPLED-DATA CONTROL SYSTEMS 
two integrations it is necessary, to base the design on zero steady-state 
response to a ramp rather than a step input. In these cases, the usual 
single-rate design for K(z,) is used, but some care must be exercised in 
the choice of effective input R(z,) selected to be entered in the formula. 
The design based on a ramp response will result in a multirate controller 
which will drive the plant into a ramp upon receipt of a single error sample 
and will settle the output in the steady state on the presumed ramp 
before being able to check the validity of the assumption. The difficulty 
of the situation is as shown in Fig. 9.15, where several possible input 
ramps are shown which lead to the 
same first error sample of amplitude 
ro. By appropriate choice of R(2n) 
entered in (9.82) the designer can 
drive the plant output to a steady- 
S _ state correspondence with any of 
these or with the unlimited other 
class of inputs which give the same 
initial sample value. The choice of 
0 T oT which first-order polynomial to de- 
Time sign for must be made by the de- 
Fria. 9.15. Three ramp inputs which give signer of the system based on his 
the same initial sample value. knowledge or estimate of the best 
case for hissystem. The decision here is not unlike that of the first-order 
hold discussed in the early chapters, where one had a choice of designing 
for full or partial slope correction in a data-hold operation. 
A second alternative suggests itself for the design of multirate control 
systems upon inspection of the dilemma of Fig. 9.15. Since this con- 
templated design results in a new steady state of the plant being estab- 
lished in each period, there is no reason why the rule for determining this 
steady state cannot be modified from one period to the next. In other 
words, it should be possible to set up the controller with provision for 
selection between designs for zero error to a step and zero error to a ramp. 
The decision to select one or the other of these possible performance rules 
could be made by a device which operates on the back difference of EF. 
If this back difference is less than some threshold, the slower but smoother 
step-response design is used, and for rapidly changing inputs the ramp 
design is used. Further development of this idea is necessary to evaluate 
its effectiveness in reducing following errors in any particular case. 
9.5 Summary 
A multirate sampled data system is a system which contains two or 
more signals which are sampled at different rates. In case the ratios of- 
