22 SAMPLED-DATA CONTROL SYSTEMS 
By expanding the exponential e-7* into a power series in s, P,»(s) becomes 
Ps) = f@l) vex" (1 — a a5 os SS? ‘ (2.19) 
This result shows that the response of a linear system to a finite-width 
pulse is the sum of the impulsive response of the system and its various 
derivatives. Thus, if the pulse is applied to a relaxed system whose 
transfer function is G(s) and whose impulsive response is g(¢), the output 
of this system, c(t), to a pulse given by (2.17) becomes 
c(t) = fn Tv | 9 -~t7O@+4e'@--- | (2.20) 
It is important to note that if the various derivatives of the impulsive 
response are small, as is usually the case in low-pass systems, all but the 
first term of (2.20) may be negligible. In applying the impulse approxi- 
mation, this assumption is made. An estimate of the error so introduced 
may be obtained by evaluating some of the terms in (2.20) which are 
neglected. 
EXAMPLE 
The system transfer function G(s) is 1/(s + a), and the impulsive 
response is e~“. The various terms in (2.20) are 
git) =e 
OND) a 
g(t) = ave 
etc. 
From (2.20), the output of the system in response to a finite-width 
pulse is 
—at 272 p—at 
c(t) = f(nT)y (« ao ate er +7 = aia ) 
The summation is factored and cara to the following: 


c(t) = yf(nT)e +s soy aa aaa ep .) 
The first term of the series is the ae response of the system to an 
impulse whose area is equal to that of the actual pulse. The remain- 
ing terms are corrections to this first estimate. For instance, it is 
readily seen that if ay is 0.1, that is, if the duration of the pulse is 
10 per cent of the time constant of the system, 1/a, the response differs 
from the impulsive response by about 5 per cent. 
The approach used in the preceding discussion and example assume 
