Page 806 - The Mechatronics Handbook

0066_Frame_C25 Page 5 Wednesday, January 9, 2002 7:05 PM

The series (25.10) converges if it is assumed that there exist values r l and r u with r l < |z| < r u as bounds
on the magnitude of the complex variable z.
The z-transform has the same role in discrete time systems that the Laplace transform has in continuous
time systems. In case of sampling, the complex variable z of the z-transform is related to the complex
variables s in the Laplace transform via

z = e s∆T (25.11)

where ∆Τ is the sampling time used for sampling. Both the Laplace and z-transform are linear operators
and satisfy

{
{
L aut() + byt()} = aLu t()} + b yt()} (25.12)
{
Using the definition in (25.8) and the linearity property in (25.12), the transform of most commonly
used functions has been precalculated and tabulated.
Of particular interest for the analysis of linear differential equations such as (25.1) and (25.2) is the
Laplace transform of a derivative:

  ∞
d
d
L -----ut()  = ∫ -----ut()e – st t d

 dt  t=0 dt
= ut()e – st ∞ 0 ∫ ∞ u t()e – st t d
+
s
t=0
= su s() u 0()
–
With u(0) = 0 it can be seen that the Laplace transform of the derivative of u(t) is simply s times the
Laplace transform of u(s). This result can be extended to higher order derivatives and the result for the
nth derivative is given by

 d n  n n−j d j−1
L -------ut()  = s us() – ∑ s -----------ut()
n

 dt n  j=1 dt j−1 t=0
In case the signal u(t) satisfies the initial zero conditions

d j−1
-----------u(t) = 0 for j = 1,…,n
j−1
dt t=0
the formula reduces to

 n 
d
L -------ut()  = s us()
n

 dt n 
n
and the Laplace transform of an nth order derivative is simply s times the transform u(s).
For discrete time systems the interest lies in the z-transform of a time-shifted signal. Similar to the
Laplace transform, the z-transform of an n time-shifted signal can be computed and is given by

n−1
Lq uk(){ n } = z uz() – ∑ z n−j uj()
n
j=0
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