296 SAMPLED-DATA CONTROL SYSTEMS 
For practical stable systems, the poles of F(s) are contained within the 
left half plane so that the path of integration can be the imaginary axis, 
which means that c is equal to zero. The integral (11.19) can be sepa- 
rated into two component integrals 
il +74/T 1 je 
a st wee, st ek, —st 
f(t) Daj ee F(s)e* ds + oe; i F(s)e* + F(—s)e* ds (11.20) 
The path of integration for the first integral of (11.20) is shown in Fig. 
11.11a, where the heavy line on the jw axis represents this portion of the 
Im 


Im 
ju /T 
s-plane 
z-plane 

Poles of 
pe al 
—ju/T 
(a) (6) 
Fig. 11.11. (a) Truncated integration path for inversion of Laplace transform. (6) 
Map of truncated integration path on z plane. 
path. If 1/7 is chosen sufficiently large so that the limit of integration 
jr/T is well-removed from the location of the poles of F(s), the contribu- 
tion of the second integral of (11.20) may be discarded. 
Assuming that the error produced by the omission of the second integral 
of (11.20) is acceptable, then f.(£), an approximation of f(¢), is given by 
1 Hi +in/T 
a(t) = == F(s)e* ds 11.21 
fall) = 55 J, FC) (11.21) 
If the output at sampling instants only is desired, then ¢ = nT in (11.21) 
and 
fl +ix/T 
—— nTs 
fa(nT) Oa / Eee F(s)e"T* ds (11.22) 
A change of variable from s to z is made, where z is defined by the usual 
expression, 2 = e’7. Replacing s by 1/T Inz in (11.22), 
fa(nT) = 5 | F G In :) ent(T) nz d(1/T In z) (11.23) 
F 
which simplifies to 
= ee i es n—1 
fa(nT) = [ 7 fi G In 2) 2"! dz (11.24) | 
