Page 471 - Mechanical Engineers' Handbook (Volume 2)

462 Closed-Loop Control System Analysis
u (k) 1 k 0

s
0 k 0
From Eq. (30)
Z{u (k)} 1 z k z k 1 z 1 z 2

s
k 0 k 0
1 z
(34)
1 z 1 z 1
Note that the series converges for z 1. In finding the z-transform, the variable z acts as
a dummy operator. It is not necessary to specify the region of z over which Z{u (k)} is
s
convergent. The z-transform of a function obtained in this manner is valid throughout the z-
plane except at the poles of the transformed function. The u (k) is usually referred to as the
s
unit step sequence.
Exponential Function
Let
ƒ(t) e at t 0

0 t 0
The sampled form of the function with sampling period T is
ƒ(kT) e akT k 0, 1, 2,...
By definition
F(z) Z{ƒ(k)}
ƒ(k)z k

k 0
e akt k

z
k 0
z
1 e at 1 e 2at 2
z
1

z
1 e at 1
z

z e at

4.4 Some Important Properties and Theorems of the z-Transform
In this section some useful properties and theorems are stated without proof. It is assumed
that the time function ƒ(t)is z-transformable and that ƒ(t) is zero for t 0.

P1. If
F(z) Z{ƒ(k)}
then