246 SAMPLED-DATA CONTROL SYSTEMS 
and, if there is to be no transient after T sec, then the output given by 
(9.81) must have reached the steady state in n samples, and n is chosen 
accordingly. 
It is possible to express (9.81) completely in terms of the plant pulse 
transfer function if the design is based on zero-ripple step response. If 
Aen? 
Gz.) = {+—— (9.83) 
» Bien Bo #0 
j=0 
Then, from the principles of Chap. 7, K(z,) contains all the zeros of 
G(z,) and for proper steady-state behavior K(1) = 1. Therefore 
R¢,) = =——— (9.84) 
) a 
i=1 
The upper limit on the numerator sums in (9.83) and (9.84) is deliberately 
taken as n because the order of K(z,) determines the number of pulses 
necessary for the output to reach the steady state and this number equals 
the order of the numerator polynomial inG(z,). For example, if G(z,) has 
a second-order numerator, then a double-rate system is sufficient. Sub- 
stituting (9.84) and (9.83) in (9.82), with the z, transform of a step for 
R(zn), the controller pulse transfer function is found to be 
Dea eee es (9.85) 
If the plant has an integration, then the 1 — z,—! factor will cancel in 
(9.85). It is observed that the nwmber of pulses applied to the plant 
(which number is regulated by. the order of the controller transfer func- 
tion), is determined by the order of the difference equation describing the 
plant, or, in other words, by the number of poles in the plant pulse trans- 
fer function. The length of time necessary to reach the steady state, 
however, is determined by the order of the numerator polynomial of 
G(zn) or by the zeros of the plant pulse transfer function. Therefore, as 
mentioned earlier, it is the number of zeros of G(zn) which determine the 
speed-up rate n in the multirate system. 
