176 SAMPLED-DATA CONTROL SYSTEMS 
consisting of an integrator and simple time delay. The latter is a 
typical transfer function found in small servomotors. It is assumed 
that the intermediate output C, is available for instrumentation, 
although in fact the equivalent effect could be obtained by using a 
tachometer at the output shaft of the servomechanism. 
The design objective is to produce a system which is ripple-free, 
which has a finite settling time, and which is capable of responding to a 
step input without steady-state error. The desired over-all pulse 
transfer function K(z) is specified by these requirements. As outlined 
in the preceding section, K(z) must contain all the zeros of the plant 
pulse transfer function, which is given by 
Is (ame 
s*(s + 1) 
If it is assumed that the sampling interval JT is unity, then G12(z) 
becomes 
Gi2(z) =Z 
O868251(L -- Or 8ee1) 
Girlz) = @. = 290 = 03682) 
The zeros of Gi2.(z) must be contained in the desired over-all pulse 
prototype function K(z), thus, 
K(z) = 271 + 0.7182—!)ao 
Only dp is included in this function since a minimum settling time is 
sought. 
In order to respond without error to a unit step input without steady- 
state error, the following relation holds: 
1 — K(z) = 1 — 2) 4+ bie“) 
Solving these two relations containing K(z) for the constants do and bi, 
there result the numerical values 
a = 0.581 
b; = 0.418 
Substituting these values back in the expressions for K(z) and 1 — K(z), 
K(z) = 0.581z-4(1 + 0.71827) 
1 KZ) =e. = 2-7) lea, OA Lace 
In order to achieve an over-all pulse transfer function K(z) as specified, 
it is necessary that the equivalent feedforward pulse transfer function 
G.(z) be given by the usual expression 
K(e) 
G.(z) = Pee) 
