238 SAMPLED-DATA CONTROL SYSTEMS 
Equations (9.47) and either (9.50) or (9.51) express the necessary and 
sufficient conditions on A(z,) such that the final value of the error will be 
zero in response to an input with the transform 1/s*. These same condi- 
tions take care of the requirement that the steady-state error be zero in 
response to any (k — 1)th-order polynomial input. 
EXAMPLE 
As an example of the application of the steady-state constraints 
expressed by (9.47) and (9.51), consider the design of a minimum finite- 
settling-time pulse transfer function which must have zero steady-state 
error in response toaramp input. The plant has no poles or zeros out- 
side the unit circle except for a simple zero at infinity. In this case, 
2 an 

R(s) = = (9.52) 
Rie ee ee 9.53 
(Zn) a n We =e pee ( . ) 
From (9.53) and (9.54) it is obvious that the numerators of R(z,) and 
R(én”) are 
IAG p= a 
TEC Sa yas (9.55) 
Applying the first of the steady-state constraints, given by (9.47), and 
the physical-realizability constraint one can write 
1 
K (én) = eo (fren 1 + foen) (9.56) 
Only two terms are included in the arbitrary F(z,) in (9.56) since the 
design is to be a minimum settling time, and a look ahead shows that 
the constraint given by (9.51) requires only two free constants. From 
(9.55) the function Q(2n) is formed as 
Q(2n) an ae a D2, aahiene =F feen) (9.57) 
The constraint equations obtained from the application of (9.51) to 
the function given in (9.57) are 
fh 
as T (fi + fe) = 0 
F_ Tin + Uf — Tat fr = 0 (9.58) 
