THE Z-TRANSFORM ANALYSIS OF LINEAR SAMPLED-DATA SYSTEMS 63 
It is noted that the sequence terminates at N — 1 since the absolute 
time ¢ of truncation remains fixed at NT. Now, if a difference between 
F(z) and Fy(z) is formed and, further, if the value of z is set equal to 
unity, it is seen that the only term remaining will be f(NT). Thus, 
N N-1 
f(NT) = by f(nT)e" = 2) » f(nT)z | (4.30) 
n=0 n=0 
z= 
Now, as N is allowed to increase without limit and approaches infinity, 
it is seen that the two summations in (4.30) each converge to F(z) because, 
in the limit, N and N — 1 converge toward the same value. 
The final-value theorem can be stated as follows in consequence of the 
limit of (4.30) being as described. It is 
f(@) = lim (1 — 2-)F@) (4.31) 
where f() is the final value of the sample of the sequence f*(t) whose 
z transform is F(z). This theorem is of major value to the designer of 
sampled-data control systems since the specification of such systems 
generally contains a requirement of steady-state performance. 
EXAMPLE 
To illustrate the application of the final-value theorem, it will be 
applied to the example in Sec. 4.3. The z transform of the pulse 
sequence is 
1 
— 1.2271 + 0.22? 

Ca i 
It is noted that this transform can be factored into the following form: 


1 
KO) = en Ole) 
Applying (4.31), 
; 1 — 2} 
i(eo)) = ihren 
a CS) 0222 5)) 
Canceling the common factor in numerator and denominator, the limit 
is seen to be 
i(ea) = 125 
It is interesting to note that the inversion carried out in the example 
in Sec. 4.3 resulted in the value of the fourth sample at 1.248, as con- 
trasted to the final value of 1.25. The indication is that the system 
almost completely settles in four or five samples. 
