32 SAMPLED-DATA CONTROL SYSTEMS 
is a polynomial in time whose order is equal to or lower than the assumed 
order, the reconstruction will be perfect. On the other hand, if the actual 
function is some other form, the polynomial extrapolation will be an 
approximation whose accuracy must be reviewed from the standpoint of 
engineering considerations. 
A theoretical basis for the polynomial extrapolation based only on 
samples taken at equally spaced prior sampling instants is the Gregory- 
Newton extrapolation formula®4 used in the numerical integration of 
differential equations. In view of the practical importance of the 
Gregory-Newton formula in the design of sampled-data systems, its 
derivation will be given. 
A convenient technique for the derivation of the formula is to employ 
the expression for prediction in the frequency domain. The transfer 
function of a system whose output is the value of the input 7 sec in the 
future is 
Fis) =e" (3.4) 
This expression may be rearranged in the following manner: 
Puls) (l=) = e-yhee (3.5) 
Expanding the expression enclosed in the brackets by means of the 
binomial expansion, there results 
fly 
ze mre (T + 
di 2 


Pe te a Us 0 es 
As will be shown later, the number of terms which are used depends on 
the order of polynomial which is to be extrapolated. 
The transfer function /’;,(s) expresses the transfer function of a device 
whose output is the future value of the input, provided that an infinite 
number of terms are taken. The Laplace transform of this output C(s) 
is given by 
C(s) = F,(s) R(s) (3.7) 
where R(s) is the Laplace transform of the input. Substituting for F;,(s) 
the series form of (3.6), C(s) becomes 
— tS — e-Ts)\2 (T 
C@) = 2) + RQ = 1 + kOe 
Inverting (8.8), the output of the extrapolator becomes 
r(é + 7) — r(t) plate, 
ns r(t) — 2r(t — T) + r(t — 27) (T + 
or) eee 
— tar + (3.9) 

