294 SAMPLED-DATA CONTROL SYSTEMS 
approximation depends on the contributions of the sidebands being small, 
a condition which will be true if the sampling frequency is chosen high 
enough. Thus, 
uence 7 C(s) (11.11) 
Substituting the central term only in (11.9), the Laplace transform of the 
system error becomes 
_ G(s)R(s) G(s) R(s) 
(9) =TFE@ ~ TFId/MBWOCO oa 
Simplifying this expression and making use of the fact that G(s) and 
B(s)G(s)/T are approximately equal, the approximate expression for e(s) 
becomes 
_ [C1/T)B(s) — 11G(s)?R(s) 
e(s) = [i + Gs) 2 (11.13) 
It is noted that if B(s)/T were exactly equal to unity, a situation which 
would obtain if the data reconstruction were perfect, there would be no 
computational error. In order to make (11.13) more tractable, B(s), 
Pe €q(2) 
Pp Seo) 
(s) 7 C's) C),(s) 2.2 
=; z we Kis) : 
ie (t) c,(t) Eq(s) 
Fic. 11.10. Reduced sampled model for determination of computational error. 
which is given in (11.5) in terms of exponential operators, can be expanded 
into a power series in s._ If this is done by expanding each exponential 
into its power series and if the terms of the series so obtained are substi- 
tuted in (11.13), the sees transform of the error becomes 
eo)— TS K(s)C(s) Poe lrs 360 s” K(s)C(s) +: (11.14) 
where K(s) is defined by 
by niGAs) 
EAS) 7 T4466) 
For the likely case that the sampling interval is small, and consistent 
with the approximation made in (11.11), only the first term of (11.14) 
need be considered to obtain an estimate of error. Even with these 
simplifications, the inversion of the first term of (11.14) involves the 
same difficulties as the main problem itself. Applying the same tech- 
niques, the sampled error may be obtained from the model shown in Fig. 
11.10. Cy(s) is the Laplace transform of the reconstructed output func- 
tion, using the polygonal approximation. For purposes of error compu- 
