298 SAMPLED-DATA CONTROL SYSTEMS 
Substituting the expression for u from (11.26), there results the so-called 
z form 
RPA Oza en 
~ 12° GQ =27) 
Higher orders of s~! may be similarly evaluated, leading to a table of z 
forms given in Table 11.1. It is seen that the expression for s~* will con- 
—2 
(11.33) 
TasBLeE 11.1* 



Ti+e2z 
svi — 
2m 
i; JP 1+ 1027! + 2-2 
12 (1 — 2712 
T3 gl + 22 
s—3 —— 
2) =e) 
Li T4271 + 42-2 + 273 T* 
i ae ele Shey 
6 (1 = 27)4 720 
ze T®z-) + llz-2 + 1lz-3 + 274 
8 een a OS ee 
24 (— 2) 
* Reproduced in part from R. Boxer and 8. Thaler, A Simplified Method of Solving 
Linear and Nonlinear Systems, Proc. IRE, vol. 44, no. 1, pp. 89-101, January, 1956. 
tain a pole of order k at zg = 1 in the z plane and at s = 0 in the s plane. 
Since the unit circle represents the other periodic strips of the s plane as 
well as the first strip, there are additional poles displaced by jkr/T as well 
on the corresponding map on the s plane. If the sampling period T is 
small enough, the additional poles on the s plane are sufficiently removed 
from the region of interest extending from —7z/T to 7/T to introduce 
negligible errors so long as 7/T is considerably greater than the highest 
frequency of interest in the system. It might appear that the inclusion 
of additional terms from (11.31) would improve the accuracy of the 
approximation. However, this would introduce additional poles in the z 
form, which would lead to increased rather than decreased errors. To 
make the approximation accurate, it is necessary to use a sampling fre- 
quency sufficiently higher than the maximum significant frequency passed 
by the system and then to employ the simpler forms. In the context of 
this discussion, this means that the frequency 7/7 must be high enough 
to place the spurious poles introduced by the periodicity of (11.25), when 
plotted in the s plane, far away from the region of interest. By doing so, 
the approximation to the exact Laplace transform is adequate. 
