46 SAMPLED-DATA CONTROL SYSTEMS 
where the various b’s are constants which differ from the various a’s 
of (3.24). In the feedback implementation indicated by Fig. 3.17, a 
correction or difference polynomial 6@(¢) must be generated and added to 
Q(t) to obtain P(t). This difference polynomial 6(¢) may be written as 
follows: 
O(t) = Ado + Aailt — (n + 1)T] + Aa,[t — (n + 1)T]? 
+ Aa,[t —(n+1)T]? +--+ (8.26) 
The various Aa’s are coefficients which are so chosen that when Q(t) 
and 6(¢) are added they result in P(é). It is recalled that the requirement 
on P(t) is that it pass through the latest number of data samples equal 
to the order of P(t). This requirement is satisfied if the added poly- 
nomial @(t) is such that at a time [t — (n + 1)7] equal to zero, 6(¢) has a 
value equal to e(n + 1)T and has zero value for all previous instants 
since Q(t) already passes through these points. Summarizing, the 
requirements on 6(t) are that 
at t—(n+1)T=0 6(t) = e(n+1)T 
t—(n+1)T = —-T Ay = 0 
t—(n+1)T = —2T 8) —) 
etc. (3.27) 
It is evident by inspection that a polynomial having zeros at —7T, —2T, 
ete., will satisfy the zero conditions of (3.27) and that the form of this 
polynomial is 
A(t) =e(n + 1)T iE a0 Scileg 1 a oe 
E ate gt ae | (3.28) 
It is seen that when [t — (n + 1)T] is zero, the value of @(¢) is e(n + 1)T. 
To illustrate the evaluation of the various coefficients of (8.26), a 
third-order polynomial will be considered. In this case, P(t), Q(¢), and 
6(¢) are third-order polynomials, and (3.28) will contain all three of the 
factors explicitly stated. Including these factors and multiplying the 
polynomial out, there results the following: 
Wi = @-+- 1)7] 
67 
[se UMP 
[2 
[Gus Uae 
ot ete 
Now, if 6(t), as given by (3.29), is added to the polynomial Q(¢) at the 
sampling instant (n + 1)T, the new polynomial so generated, P(t), 
will pass through four sample points, ranging from the most recent at 
a(t) =e(n+1)T {1+ 
(3.29) 
