robotics research laboratory 1 chapter 3 discrete system analysis - sampled data systems

Robotics Research Labo-ratory

1

Chapter 3

Discrete System Analysis- Sampled Data Systems


2

Linear Difference Equations

( )

0 0 1, , , , , ( )k k ku f e e u u

If f is linear and has a finite dimension, can be written as

1 1 2 2 0 1 1

1 0 1

( )

( ) ( 1) ( ) ( ) ( 1) ( )k k k n k n k k m k m

n m

u a u a u a u b e b e b e

u k a u k a u k n b e k b e k b e k m

is called a linear recurrence equation, or difference equation,

or ARMA (Auto Regressive Moving Average) model.

Remarks:i) Infinite-impulse response filter or recursive filter

ii) Finite-impulse response filter or non-recursive filter

iii) Difference equation1k k ku u u

General description of a discrete-time system is

0 1( ) ( ) ( 1) ( )mu k b e k b e k b e k m

( )


3

kkk

E z e k e z r z R

0 0( ) { ( )} , z

A sequence of infinite time sequence

Complex planezTime-domain Z-domain

(Frequency-domain)

If f is a linear time-invariant system, we have a discrete transform function.


4

2 2

2 2

)

( 2) 3 ( 1) 2 ( ) 0, (0) 0, (1) 1

( 2) ( ) (0) (1)

( 1) ( ) (0)

( ) ( )

( ) 3 ( ) 2 ( ) (0) (1) 3 (0) 0

ex

u k u k u k u u

u k z U z z u zu

u k zU z zu

u k U z

z U z zU z U z z u zu zu

U

zzz

2

1 1

( )1 23 2

1 1 ( 1) , ( 2)

1 1 2

( ) ( 1) ( 2) , 0,1,2,....

k k

k k

z z zz

z zz z

z z

u k k

- 1 - 1z z


5

Define the z-transform of the sequence {e(k)} as

( ) kkk

E z e z

Similarly,

( ) kkk

U z u z

can be written as

10 1

11

( )

( ) 1

mm

nn

b b z b zU z

E z a z a z

10 1

11

( )

( )

n n n mm

n nn

b z b z b zU z

E z z a z a

(backward)

(forward)

or

( )


6

Define1

0 1

11

( )

( )

( ) ( )( )

( ) ( )

( ) ( ) ( )

n n n mm

n nn

b z b z b z b z

a z z a z a

U z b zH z

E z a z

U z H z E z

H(z) is called a pulse(or discrete) transfer function or just a transfer function.


7

ke ku1z

1( ) ( )U z z E z

Derivation

1 ( 1)1 1( ) k k k

k k kk k kU z u z e z z e z

ex) unit delay

ex) unit discrete pulse (impulse)

1 , 0

0 , 0

( ) ( ) since ( ) 1

k

ke

k

U z H z E z


8

In general

( )k j k j j k jj ju e h h e discrete convolution sum

Take the z-transform

k k kk j k j j k jk k j j k

u z z e h e z h

( )( )

( ) ( ) ( ) ( ) ( )

Let

l j j lj l j lj l j l

k j l

U z e h z e z h z

U z E z H z H z E z


9

H(z)u(kT)e(kT)

E(z) U(z)

( )

( )

( )

kkk

kkk

kkk

U z u z

E z e z

H z h z

( ) ( )

( ) ( )

1 unit impluseE z

U z H z

For

If


10

Discrete Model of Sampled-data Systems

D/A output for unit pulse input ( Model of D/A converter)

G(s) A/Dy(t) y(kT)

D/Au(kT)

G(z)

2TT

1

0

{ ( ) ( )}

( )

1 1

1

Ts

Ts

u t u t T

es s

ezero order hold

s

L


11

Ts

Ts

G sY s e

s

G sG z y kT Y s

G

s

ss

e

G z z

1

1 1

1

( )( ) (1 )

(

( )( ) (1

)( )

) { }

{ ( )} { ( )} {(1 ) }z z z

z

Remarks:

i) The mathematical model of A/D converter is the same as a sampler (the train of unit impulse)

ii) The D/A converter is mathematically modeled as a zero-order hold

(1

) Tse

s


12

ex)

-1

1

(1

( 1)(

1 (1

( 1)(

1

( ) / ( )

( ) 1 1

( )

( ) 1( ) 1( )

)

)

)( )

)

( )

at

aT

aT

aT

aT

aT

aT

aT

z z

z z e

z e

z z e

z z e

z z z e

e

z e

G s a s a

G s a

s s s a s s a

G st e t

s

G z

G s

s

L

z


13

Block-Diagram Analysis of

Sampled-Data Systems

H(s) G(s)T( )R s *( )R s ( )E s *( )U s( )U s*( )E s

*

** *

* *

*

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

E s H s R s

E s H s R s

E s H s R s

E s H s R s

E z H z R z


14

-1 -1 *

*

0

00

00

0

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( )

t

t

k

t

k

k

L E s L H s R s

e t h t τ r τ dτ

h t τ r τ δ τ kT dτ

h t τ r τ δ τ kT dτ

h t kT r kT


15

0 0

( )

0 0

0 0

*

( ) ( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( )

(s) ( )

n

n k

k m T

m k

m k

m k

E z e t h nT kT r kT z

h mT r kT z

h mT z r kT z

H z R

s

z

R R

z

Remark:


16

* * * * * * *( ) ( ) ( ) ( ) ( ) ( ) ( )

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

( )( ) ( ) pulse transfer function

( )

U s G s H s R s H s G s R s

U z G z H z R z H z G z R z

U zG z H z

R z

From (1) and (2)

** *

* *

*

** *

* *

( ) ( ) ( )

( ) ( ) --- (1)

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ---(2)

E s H s R s

H s R s

U s G s E s

U s G s E s

G s E s


17

Remark:

G(s)( )x t

( )X s ( )Y s

( )y t

** * *

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

or

Y s G s X s

Y s G s X s G s

G s X s G z

X

X

s

zz


18

G(s)

H(s)

( )R s ( )C s*( )E s( )E s

*

* * * *

( ) ( ) ( ) ( )

= ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

E s R s H s C s

R s H s G s E s

E s R s GH s E s

*( )C s


19

**

*

* * *

* **

*

( )( )

1 ( )

( ) ( ) ( )

( ) ( )( )

1 ( )

( ) ( )( )

1 ( )

( ) ( )

( ) 1 ( )

R sE s

GH s

C s G s E s

G s R SC s

GH s

G z R zC z

GH z

C z G z

R z GH z


20

*( )D s 1 Tse

s

( )pG s( )Y s( )U s*( )M s*( )E s( )E s( )R s

( )H s

DigitalController D/A Plant

* *( ) ( ) ( ) ( )1 TS

p

eY s D s G s E s

s

( )H s

* * *

* * *

( ) ( ) ( )

( ) ( ) ( )

Y s H s E s

E s R s Y s

*( )Y s


21

* * * *

** *

*

*

* *

* *

( ) ( ) ( ) ( )

( )( ) ( )

( )

( ) ( ) ( )

( ) ( )

1

1where

Ts

p

Y s H s R s Y s

H sY s R s

H s

eH s D s G s

s

D s G s

( )( ) ( ) ( )

( ) ( )

1 1

1 1

Remark:

11 where

TspTs

p

Ts

G seG s e G s G s

s s

G s e G s


22

=

=

11 1

* * * *1 1

* *1

*

*

1

( ) ( )

( ) ( ) ( ) ( )

( ) 1 ( )

( )( ) 1

( )( ) ( )(1 )

Ts

Ts

Ts

pTs

p

e G s g t T

H s D s G s e G s

D s e G s

G sD s e

s

G sH z D z z

sz

L


23

, *

*

( ) ( ) 11

1( ) ( )

1

p

Ts

KG s D s

s

e KC s E s

s s

ex)

( )D s 1 Tse

s

( )pG s( )C s( )U s( )M s

( )E s

( )E s( )R s

( )H s


24

* * *

1

1

1

1

1

Since ( ) ( ) ( )

( ) ( ) 1( 1)

11

1

1

( ) ( )

( ) 1 ( )

1

1 ( 1)

Ts

T

T

T

T

C s H s E s

KH z H s e

s s

K Kz

s s

K e z

e z

C z H z

R z H z

K e z

K k e z

z

z z


25

Response between Samples :

i) Sub-multiple Sampling

ii) Modified z-transform

Sub-multiple Sampling

G(s)( )e t *( )e t

/

fictiti

ous sampler

T N

*( / )e t N

*( )NC t

( )C t

*( )C t

T

Td

/T N


26

t kTm

m

N

C t e mT g t mT

t kT N

C kT N e mT g kT N mT

t C kT N t kT N

0

0

*

( ) ( ) ( ) --- (1)

At any sampling instant / of the sampler

( / ) ( ) ( / ) --- (2)

C ( ) ( / ) ( / )

k

k NN N

k

k NN

k m

C z C t C kT N z

C z e mT g kT N mT z

0

* /

0

/

0 0

--- (3)

( ) ( ) ( / ) --- (4)

(2) (4)

( ) ( ) ( / )

z


27

/

0 0

/

0 0

( ) ( ) ( / )

Let / / where is an intege

(

r

( ) / ( )

) ( ) ( )

k NN

k m

l N mN

l

N N

m

C z e mT g kT N mT z

lT N kT N mT l

C z g lT N z e mT z

C z G z E z

ex)

( ) ( )

/

( ) ( ) ( )

( )

3 3

1

1We want to know the response at 3

Assume that (unit step) and 11

tG s g t es

t kT

C z G z E z

zE z T

z


28

13

13

3 , 3, 3

13

13

13

133

13

3

33 3

3 33 3

1 23 3

3 4 53 3 3

( ) ( )

= 0.717

10.717

Define z

( )0.717 1

+0.717 0.513

1

1.36 0.98 0.703

T Tz z TTz z T

zG z G z

z e

z

z

z zC z

zz

z

z zC z

z z

z z

z z z

6 7 83 3 3

93

1.504

1.55

1.08 0.773

z z z

z


29

Modified Z-transform Method– fictitious delay time

( )1 m T

G(s)( )E s

T*( )E s

( , )C z m

( )C s

*( )C s

( )1 m Tse

T

T( , )G z m

where 0 1m

Define

*1

ln

( ) ( , )

( , )

m

s zT

G s G z m

G s m

z


30

ln

ln

*

( )

( ( ) ) ( )

( ) ( )

( ) ( )

( , ) ( ) ( )

( )

( ) T s zT

mTp

Tk

T s zT

T

T

Ts

Tss p

g t m T tm

t t kT

g t mT t

G s m g t tmT

ee

T

G pj

e

e

G s

1

0

1

1

where

1

2 1

z L

L

L

( ) ( )Ts mT

c jw

c j

s

w

Ts

dp

zresidue of G s at pole of Ge e s

z e

Complex convolution theorem

( )( , ) ( )

T

mTs

s

G s zG z m residue of at pole of G

es

ez

z

1


31

Notes:

i) any specific point between two consecutive instants

ii)

1

( ) lim ( , )m

G z zG z m

ex)

=

=

1

1

-

1( )

1( , )

1lim ( )

mTsTs

mTsTss a

maT

aT

G ss a

zG z m z residue of e

s a z e

zz s a e

s a z e

e

z e


32

G(s)( )R s

( )E s

T

*( )E s( )C s

*( )C s

T

+ TN

H(s)

*( )NC s

TN

*( )NE s

1

N N

NN

C z G z E z

E z R z GH z E z

G zC z R z

GH z


33

G(s)( )R s ( )E s

T

*( )E s ( )C s

*( )C s

T

+ e-(1-m)T

T

H(s)

*( , )C s m

C z m G z m E z

E z R z GH z E z

G z mC z m R z

GH z

, ,

,,

1


34

State Space FormDifference equation

( )( 1) Φ ( ) Γ ( )

( ) ( ) ( )

x k x k u k

y k Hx k Ju k

Differential equation ( ) x t Fx t Gu t

y t Hx t Ju t

( ) ( ) ( )

( ) ( ) ( )

0

0

( ) ( )0( ) ( ) ( )

tF t t F t τ

tx t e x t e Gu τ dτ

0

( )

Let and

( ) ( ) ( )kT TFT F kT T τ

kT

t kT T t kT

x kT T e x kT e Gu τ dτ

( )

( ) (

( ) ( )

) ( )

0

Let for

1

when

TFT Fηx k e x k e dηG

u τ u kT kT τ kT T ZO

u k

η T τ

H

kT


35

Define0

Φ ,ΓTFT Fηe e dηG

Take the z-transform

1

1

[ Φ] ( ) Γ ( )

( ) ( ) ( )

( ) [ Φ] Γ ( )

( ) [ [ Φ] Γ ] ( )

zI X z U z

Y z HX z JU z

X z zI U z

Y z H zI J U z

Transfer Function H(z)


36

ex)

[ ]

[ ]

2 2

22 2 3

0 1 0

0 0 1

1 0

Φ2

1

0 1

2 Γ 2! 3!

1 0 1( ) ( ) 1 0

( ) 0 1 0 1

FT

x x u

y x

F Te I FT

T

TFT F TI T G

T

TY zH z z

U z

-

é ù éùê ú êú= +ê ú êúë û ëû

=

= = + + +

é ùê ú=ê úë û

é ùé ù ê úê ú= + + =ê úê ú ê úë û ë û

ì üé ù é ùï ïï ïê ú ê ú= = -í ýê ú ê úï ïë û ë ûï ïî þ

1 2 2

2

( 1)22 ( 1)

T T z

zT

é ù +ê ú=ê ú -ê úë û


37

P

P

G s H sI F

TzG s

x

Js

x

z

u

y x

12

2

0 1 0Remark:

0 0 1

1 ( ) ( )

Its di

1 0

( )

(wrong)( 1

)

et

scr

z

PG sz T z

z s zH z

2

2

e transfer function should be

( )1 ( 1) (correct)

2( 1) ( )

z


38

State Space Models for System with Input Delay

0

0

( ) ( )0

( )

0

0

( ) ( ) ( )

( ) ( ) ( )

( ) ( )( )

( ) ( )

tF t t F t τ

t

kT TFT F kT T τ

kT

FT Fη

T

TFT Fη

x t e x t e Gu τ λ dτ

x kT T e x kT e Gu τ λ dτ

e x kT e Gu kT T λ η dη

e x kT e Gu kT T λ η dη

λ lT mT

η kT T τ

where l is an integer , m is a positive number less than one.

, l m 0 0 1


39

( ) ( ) ( )TFT Fηx k e x k e Gu kT T lT mT η dη 0

1

mT TFT Fη Fη

mTx k e x k e Gdηu kT lT T e Gdηu kT lT 0( 1) ( ) ( ) ( )

Define

1

2 0

Φ

Γ

Γ

FT

T Fη

mT

mT Fη

e

e Gdη

e Gdη

kT-lT kT-lT+T kT-lT+2T

T

mT

η0


40

( ) ( ) ( )2Define a new state variable Γξ k x k u k

2

1

2 1

2 1

( 1) ( 1) Γ ( 1)

Φ ( ) Γ ( )

Φ ( ) Γ ( ) Γ ( )

Φ ( ) ΦΓ Γ ( )

Φ ( ) Γ ( )

ξ k x k u k

x k u k

ξ k u k u k

ξ k u k

ξ k u k

( ) ( ) ( ) ( )1 2

0 , 0 1 (no delay, prediction due to )

1 Φ Γ Γ 1

l m m

x k x k u k u k


41

1 2

1 1

1

( 1) ( )Φ Γ Γ( )

( 1) ( )0 0 1

( )( ) 0

( )

n n

n

x k x ku k

x k x k

x ky k H

x k

(n+1) dim

1 2

1

1 1 2

1

1 ( 1) Φ ( ) Γ ( 1) Γ ( )

Define ( ) ( 1)

( 1) Φ ( ) Γ ( ) Γ ( )

( 1) ( )

n

n

n

l x k x k u k u k

x k u k

x k x k x k u k

x k u k


42

n

n

n l

n

n

n l

l x k x k u k l u k l

x k u k l

x k u k l

x k u k

x k

x k

x k

x k

1 2

1

2

1 2

1

2

1 ( 1) Φ ( ) Γ ( ) Γ ( 1)

Define ( ) ( )

( ) ( 1)

( ) ( 1)

Φ Γ Γ 0

( 1)

( 1)

( 1)

( 1)

n

n

n l

n

n l

x k

u kx k

x k

x ky k

x k

x k

x k

H

1

2

1

00

0 0 1 0 00( )

0 0 0 1 0 0( )0( )

0 0 0 0 0 11( )

0 0 0 0 0 0

( )( ) 0 0

( )

( )

( )

robotics research laboratory 1 chapter 3 discrete system analysis - sampled data systems

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