150 SAMPLED-DATA CONTROL SYSTEMS 
would be desirable that the system error be continuously zero in the 
steady-state, although to assure this condition requires special treatment, 
as will be shown later. The discussions in this section will deal with the 
system error at sampling instants only. 
In specifying the requirements on K(z) to produce zero steady-state 
system error in response to a test function, a unity feedback system will be 
assumed. Modifications of the results obtained in this manner will be 
necessary if the feedback is not unity. From the system configuration 
given in Fig. 7.3, the system error sequence is equal to the control error 
sequence F(z), given by 
Ex,(z) = R(z) — Cz) (7.10) 
Since, by definition, K(z) is given by 
_ C&) 
then (7.1C) is expressed by 
Ex(z) = RL — K@)). (7.12) 
The second requirement on the form of K(z) is that the inverse of /,(z) 
have a final value of zero when R(z) is the z transform of the specified 
input function. Applying the final-value theorem, 
eto) = lim {(. — 2“) R@)L — KI (7.13) 
If the class of input test functions includes only steps, ramps, or constant- 
acceleration inputs, then A(z) takes the form 
Re) = (7.14) 
GS sze he 
where A(z) is a polynomial in z~! which does not contain factors of the 
form 1 — z-!. Itis readily seen from (7.13) that for inputs of this type, 
the steady-state error e;(~ ) will be zeroif 1 — K(z) satisfies the following 
relationship: 
Kee)" Gea Ee) (7.15) 
where F(z) is an unspecified ratio of polynomials in z—! and m is the order 
of the denominator of the input z transform R(z). 
The minimal prototype response function is defined such that F(z) in 
(7.15) is unity and the resultant order of K(z) in zis minimum. Thus, 
if the system is to follow a unit ramp input without steady-state error, 
K(z) for a minimal prototype response is given by 
1— K(z) = 0 — 2")? (7.16) 
Solving for K(z), 
K(z) = 221-2 (7.17) 
