數位控制（三）
1
z transform


z transformation transforms linear difference
equation into algebraic in s.
Laplace transformation transforms linear timeinvariant differential equation into algebraic in
z.
x +
-
G
y
y( s)
G

x ( s ) 1  GH
H
2
convolution integral :
In time domain
x(t)

y(t)   x ( )h (t   )d
y(t)
H(t)

or y(t)   x (t   )h ( )d
convolutions summation

y(t)   x ( kT )h (t  kT )
k 0
In s domain
x(s) +
-
G(s)
y(s)
y( s)
G( s)

x( s ) 1  G ( s ) H ( s )
H(s)
3
The z transform method allows

Conventional analysis and design techniques



Root-locus
Frequency response analysis (convert z to w)
Z transformed characteristic equation allows
Simple stability test
4
z transform

X ( z )  Z [ x (t )]  Z [ x ( kT )]   x ( kT ) z
X ( z )  x (0)  x (T ) z
1
k 0
2
 x ( 2T ) z
k
   x ( kT ) z
k

Z 1 : one time step delay
Laplace transform
F ( s )  L[ f (t )] 

0
f (t )e  st dt
5
Elementary Functions

z
z 1
Unit-step
Tz
X ( z )  Z [t ]   kTz 
 Unit-ramp
( z  1)
z
X ( z )  Z [a ]   a z 
 Polynomial
( z  a)
z
 Exponential X ( z )  Z [e ]   e z 
(z  e )
1
z sin T
X
(
z
)

Z
[sin

t
]

Z
[
(
e

e
)]

 Sinusoidal
2j
z  2 z cosT  1
 Table of z transforms (Ogata p-29)

X ( z )  Z [1(t )]   z k 
k 0

k
k 0

at
k
2
k k
k 0

akT k
aT
k 0
jt
 jt
2
6
Important properties




Multiplication by a constant Z[ax(t )]  aZ[ x(t )]  aX ( z)
Linearity of z transform Z[f (k )  g(k )]  F ( z)  G( z)
k
1
k
Z
[
a
x
(
k
)]

X
(
a
z)
Multiplication by a
Shifting theorem Z [ x(t  nT )]  z n X ( z)
n 1
Z [ x (t  nT )]  z [ X ( z )   x (kT ) z k ]
n



k 0
at
Complex translation theorem Z [e x(t )]  X ( zeat )
X ( z)
Initial value theorem x(0)  zlim

lim x(k )  lim[(1  z 1 ) X ( z )]
Final value theorem k 
z 1
7
8
9
10
11
Poles and Zeros in the z plane
X ( z) 
b0 z  b1 z
m
z  a1 z
n
m 1
n 1
   bm
   an
or
b0 ( z  z1 )( z  z2 )( z  zm )
X ( z) 
( z  p1 )( z  p2 )( z  pn )
The locations of the poles and zeros of X(z) determine
the characteristics of x(k).
12
Exercise 1

Ogata


B-2-1
B-2-2
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