184 SAMPLED-DATA CONTROL SYSTEMS 
The system is to respond to a step function with zero steady-state 
error, so that a requirement on the over-all pulse transfer function is 
that 
KG) — (ee bien, Ose ee 
where the various b’s are arbitrary or dependent on conditions brought 
about by saturation limitations. It is assumed that the maximum 
command signal applied to the plant not exceed 1.5 units. This limit 
is the saturation limit of the plant, and any higher signal at the input 
would produce no additional output. 
Starting with the first equation of (7.67), substitution of ro and gi: 
produces the relationship 
ky = (€o) (0.368) 
If there were no other terms in the over-all pulse transfer function K (2), 
all b’s in the expression 1 — K(z) would be zero and K(z) would be 
z1. Thus, if k; were taken at the minimal prototype value of unity, 
€o would be given by 
1 
~ 0.368 
SUT 
€0 
If the minimal prototype were desired, the first pulse applied to the 
plant is 2.72, which is higher than allowable. The maximum value of 
é) can be only 1.5, and hence the first term of the over-all pulse transfer 
function must be, from (7.67), 
ke = (1-5) (0-368) 
from which 
Po = (OSPR 
Thus, the first term of the series representation of the over-all pulse 
transfer function K(z) has been ascertained. 
To find the second term of K(z), the next higher-order prototype for 
K(z) is used. To respond to a unit step input without error, it is 
necessary that 
1— K(z) = (1 — 2) 4+ die) 
which, upon multiplying out, becomes 
1 — K(z) = 14 (b1 — 1)z7! — dye? 
from which K(z) is 
K(z) = — (by = zee + by27? 
Returning now to the second expression in (7.67) and substituting 
To, T1, k1, 0, 91, and ge, there results a relation between the second com- 
