BEHAVIOR OF SYSTEMS BETWEEN SAMPLING INSTANTS 209 
where the order of the polynomial depends on the highest order of pole at 
the origin contained in G,(s)/s. The evaluation of the various constants, 
L, M, N, etc., in (8.24) can be handled by considering simultaneously all 
the channels in Fig. 8.7 containing terms of the form 1/s%. Their various 
contributions are contained in (8.24). 
Assuming for the sake of illustration that the highest-order term in 
1/s of G,(s)/s is the third, then (8.24) will contain terms of the second 
order in time. The constant L is the initial value of the polynomial 
at time mT and is the sum of the various terms of c,(¢) at that instant. 
Thus, if ci(¢), co(t), and c3(¢) are the continuous outputs from the ele- 
mentary channels whose transfer functions are 1/s, 1/s?, and 1/s*, respec- 
tively, L is given by 
L = ex(mT) + co(mT) + c3(mT) (8.25) 
The second term of (8.24) has the constant M, which is the slope of the 
polynomial at the time m7’. Since the slope of the term from 1/s is zero, 
only the contributions of the other two channels are considered. Thus, 
if the z transform corresponding to sC;(s) is formed for these two chan- 
nels and is inverted at the time m7’, the value of the initial derivative is 
obtained. Thus, 
M = c(mT) + ci(mT) (8.26) 
Finally, the third term in (8.24) has the constant N, which is one-half the 
second derivative of c,(t) at the beginning of the interval at time mT. 
Since the first and second channels containing 1/s and 1/s? have zero 
initial second derivatives, the only contribution is given by the channel 
containing 1/s*. Thus, the z transform corresponding to s?C;,(s) is 
formed for the third channel and is inverted at the time m7 and the value 
of the initial second derivative obtained. Thus, 
N = 4e'!(mT) (8.27) 
In the manner outlined previously, it is seen possible to obtain the time 
function which describes the ripple in any chosen sampling interval. 
This is very valuable since ripple may be undesirable only at given inter- 
vals and trivial at others. For instance, when a step input is applied to 
a sampled system, it is only the first overshoot which may be excessive, 
and the procedure outlined above permits its calculation without the 
necessity of calculating all other intervals. Thus, by adding the con- 
tributions of all channels, an exact determination of the ripple is possible. 
In addition, by having divided the contributions to the ripple in terms of 
each of the poles of the plant transfer function, it is possible to localize the 
more significant effects, should that be necessary. 
