1

數位控制（三）

2

z transform z transformation transforms linear difference e

quation into algebraic in s. Laplace transformation transforms linear time-i

nvariant differential equation into algebraic in z.

G

H

x y+

- GHG

sxsy

1)()(

3

G(s)

H(s)

x(s) y(s)+

- )()(1

)(

)(

)(

sHsG

sG

sx

sy

x(t) H(t) y(t)

0)()(y(t)

summationnsconvolutio

)()(y(t)

)()(y(t)

:integralnconvolutio

kkTthkTx

dhtxor

dthx

In time domain

In s domain

4

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

5

0

1

21

0

)()]([)(

transformLaplace

delay step timeone :

)()2()()0()(

)()]([)]([)(

transformz

dtetftfLsF

Z

zkTxzTxzTxxzX

zkTxkTxZtxZzX

st

k

k

k

6

Elementary Functions

Unit-step Unit-ramp Polynomial Exponential Sinusoidal Table of z transforms (Ogata p-29)

1)](1[)(

0

zz

ztZzXk

k

20 )1(

][)(

z

TzkTztZzX

k

k

)(][)(

0 azz

zaaZzXk

kkk

)(][)(

0aT

k

kakTat

ez

zzeeZzX

1cos2

sin)](

21

[][sin)(2

Tzz

Tzee

jZtZzX tjtj

7

Important properties Multiplication by a constant Linearity of z transform Multiplication by ak

Shifting theorem

Complex translation theorem Initial value theorem Final value theorem

)()]([)]([ zaXtxaZtaxZ

)()()]()([ zGzFkgkfZ

)()]([ 1zaXkxaZ k

)()]([ zXznTtxZ n

1

0])()([)]([

n

k

kn zkTxzXznTtxZ

)()]([ atat zeXtxeZ

)(lim)0( zXxz

)]()1[(lim)(lim 1

1zXzkx

zk

8

9

10

11

12

Poles and Zeros in the z plane

x(k). of sticscharacteri the

determine X(z) of zeros and poles theof locations The

)())(()())((

)(

)(

21

210

11

110

n

m

nnn

mmm

pzpzpzzzzzzzb

zX

or

azaz

bzbzbzX

13

Exercise 1 Ogata

B-2-1 B-2-2