MISCELLANEOUS APPLICATIONS OF SAMPLED-DATA THEORY 295 
tation, c,(t) differs only slightly from the actual output c(t). Thus, it 
may be stated that 
CAS) C26) BG GG) (11.15) 
From which it follows that the Laplace transform of the error function 
€,(s) may be represented with reasonable accuracy by 
a{@) = TS Bis) K(s)C*(8) (11.16) 
Taking the z transform of both sides of (11.16), 
al) = 75 @-2- = )K@C®) (11.17) 
where B(s) is taken as the transfer function of the triangle hold. 
If the original solution being computed was the response of the system 
to a unit step function, K(z) can be approximated by obtaining the first 
back difference of the approximate output already computed. This 
introduces a delay of one-half a sampling interval, but this is readily 
eliminated by shifting forward the first difference by one-half a sampling 
interval. Thus, 
iO) = ae F Gee cu) | (11.18) 
where the term z’”” is used loosely to indicate the advance of the resultant 
sample points by one-half a sampling interval. 
If this technique is applied to the illustrative example worked out pre- 
viously, the resultant error samples result in the points shown in Fig. 
11.86. The dashed line is an exact computation of the error. It is seen 
that a remarkable accuracy in error computation exists in this case. The 
procedure used to evaluate the computational error was related closely to 
the physical properties of the system. It would have been much more 
difficult for the engineer to arrive at these results had only pips 
mathematical procedures been employed. 
11.3 Approximation of the Inverse Laplace Transform 
In the previous sections, the numerical process was correlated with a 
sampled physical system which simulates the mathematical operation. 
It has been shown?!” that numerical computation methods can also be 
related to the approximate evaluation of the inverse Laplace transform 
without using a physical model. The inverse Laplace transform is given 
by 
c+jw 
f® = =i ‘ F(s)e* ds (11.19) 
