MISCELLANEOUS APPLICATIONS OF SAMPLED-DATA THEORY 299 
A property of the z forms which are listed in Table 11.1 is that the 
expressions obtained by replacing z by e’? have the same initial terms 
when expanded into a power series in s about zero. For instance, con- 
sidering the z form for the second integral given by (11.33), an error 
transform can be formed 
—s3T —2sT 
e(s) = ue Ural Men inate ents il (11.34) 
If e(s) is differentiated twice, it will be found that the first two derivatives 
are zero for s=0. Higher-order z forms have this property also, 
except that more derivatives are 
involved. The significance of this R 
result is that, in the frequency do- 
main, there will be a close coincidence 
in frequency response between the 
p/P RONAN and exact forms in the Fig. 11.12. Feedback system used in 
region of zerofrequency. Asthefre- example. 
quencies under consideration become 
higher, there will be more and more deviation between the frequency- 
response characteristics of the exact and approximate forms. This is 
another manifestation of the fact that sampled models are most accurate 
at frequencies which are well below the sampling frequency. 
Application of the z forms listed in Table 11.1 is fairly simple. The 
Laplace transform of the desired variable, including initial conditions, is 
obtained and is expressed in terms of powers of s~! as follows: 

CT ae Ce SIN ae eae 0 
BOYS Tes tt eo SEG 

(11.35) 
The significance of (11.35) in the time domain is that an equality is estab- 
lished between the weighted sum of the time integrals of the input and 
output variables. The approach is to approximate each of these integrals 
by azform from Table 11.1 and to invert the resultant z transform. This 
is done directly by substituting for each of the terms in (11.35) the 
appropriate z form obtained from Table 11.1, as illustrated by means of an 
example. 
EXAMPLE 
A simple type I servomechanism is represented by the block diagram 
shown in Fig. 11.12. The input to the system r(¢) is a unit step func- 
tion. The over-all transfer function K(s) is given by 
Com 1 
Eg) = R(s) y Cea eag 
and R(s) = 1/s 
