robotics research laboratory 1 chapter 4 discrete equivalents

Robotics Research Labo-ratory

1

Chapter 4

Discrete Equivalents


2

Goal: to obtain a discrete-time controller ( filter, equalizer, compensator ) which provides transient and frequency re-sponse characteristics as close as possible to those of the original continuous-time controller

analog controller

digital controller D/AA/D

( )x t

( )x t * ( )x t * ( )m t ( )y t

( )y t

Three approaches:

i) numerical integration

ii) pole and zero mapping

iii) hold equivalence


3

Method 1 : Numerical Integration

( )x t ( )y tC

R

0 0 0

( ) 1 ( )

( ) 1

t t t

Y s a

X s RCs s a

y ay ax

dydt a ydt a xdt

dt


4

0 0 0

0 0

( 1) ( 1)

0 0

( 1) ( 1)

At

( ) (0) (1)

( 1)

(( 1) ) (0) (2)

From (1) and (2)

( ) (( 1) )

kT kT kT

kT kT

k T k T

kT

k T k

t kT

dydt a ydt a xdt

dt

y kT y a ydt a xdt

At t k T

y k T y a ydt a xdt

y kT y k T a ydt a xdtkT

T


5

time

( )y t

0 1k 1kk

forward

backward

trapezoid

( 1) ( 1)

( ) (( 1) )kT kT

k T k Ty kT y k T a ydt a xdt


6

i) Backward Difference Method

( Backward Rectangular Rule )

1

( ) (( 1) ) ( ) ( )

( ) ( ) ( ) ( )

y kT y k T aT y kT x kT

Y z z Y z aT Y z X z

1

1 1

( ) ( )

( ) 1 1

or1

1

z

Y z aT a

X z z aT

z

z

s sT

aT

zT


7

1

2 22 22

2 2

Since Re( ) 0 for the stability

1 1Re Re 0

Since > 0 and

1Re 0

1 1 < 0 ,

2 2

s

z z

T Tz

T z σ jω

σ jω

σ jω

σ σ ωσ ω

σ ω


8

s plane

j

0 1 Re z

Im z z plane1

1z

sT

stable but considerable distortion


9

ii) Forward Difference Method

( Forward Rectangular Rule / Euler Method )

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

( ) ( ) ( ) ( )

1 1 1

1 1 1y kT y k T aT y k T x k T

Y z z Y z aT z Y z z X z

( )

( )

1

1

1

1

1 1

1

1 1

or 1 1

z z

s sTT

Y z aTz a

X z z aTz za

Tz

z


10

For Re( ) 0

-1+j Re 0 , <1

s

σ ωσ

T

may be unstable, cannot be used in practice

0 1

s plane z plane

1z sT


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iii) Trapezoid Integration Method– Tustin Transform Method– Bilinear Transform Method

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

( ) (( ) )

( ) ( ) ( ) ( ) ( ) ( )1 1 1

11

12

2

y kT y k TaTy kT y k T

x kT x k T

aTY z z Y z Y z z Y z X z z X z

1

1 1

1

1

1( ) 2( ) 1 1

2

2 1

1

aTzY z

aTX z z z

a

za

T z


12

1

1 or

2 1

2 1

11

z zs s

T T zz

2 2

2 2

2 2 2 2

2 1Re 0

1

1 1 2Re Re 0

1 ( 1)

1 0 1

z

T z

σ jω σ ω j ω

σ jω σ ω

σ ω σ ω

1

z plane

1 21 2

sTz

sT

s plane

stable but still noticeable frequency distortion


13

1ln perfect

1( 1) forward

1 1 backward

2 1 trapezoid

1

s zT

s zT

zs

T zz

sT z

Remark:


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( ) forward ( )( 1) /

( ) backward ( )( 1) /

( ) trapezoid ( )(2 / ) ( 1) /( 1)

F

B

T

a aH s H z

s a z T a

a aH s H z

s a z Tz a

a aH s H z

s a T z z a


15

iv) Bilinear Transformation Method

with Frequency Pre-warping

1

1

1

22

2 2 2

2

1

( ) ( )2 1

1

1( ) = ( )

2 111

( )2

tan2

T

jω TT jω T

jω T

T

a aH s H z

zs a aT z

a aH jω H e

ω a ω eaa T e

aH z

ω Tj a

T


16

2

21 1

11

1

1At ,

22 1

At tan , . . tan , 2 2 2 2

2 tan

2

If 1 ( 1 or ) , 2

T

ss

ω a H

ω T ω T aTa i e H

T

aTω

T

aT πaa ω ω a

ω


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Procedure:

1

1 1

1

Let be the frequency of interest

1. Write the desired filter characteristic with transform variable

and critical frequency in the form ( / )

22. Replace by such that tan

ω

s

ω H s ω

ω a aT

1

22 1

3. Substitute 1

ω T

zs

T z

1 1

1 11tan( / 2) 1

( ) ( )ps z

ω ω T z

sH z H

ω


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1

1

1 1

1 1

2tan

2. ., ( )2

tan2

2tan tan

2 2( ) 2 1 2 1

tan tan1 2 1 2

p

ω Ta Ti e H s

ω Ts a sT

ω T ω T

TH zω T ω Tz z

T z T z


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1

10) ( )

10

10 / sec 0.2 sec ( 10 / sec)

tan1 ( 1) tan1 ( )

1 1 ( 1) tan1tan11

s

p

ex H ss

ω rad T ω π rad

zH z

z z zz

Remarks:

1. Approximation will be correct if

2. However, we must have if a stable filter

is to remain stable after warping

1 ( 2 / ) or s sω π T a ω ω

1/ π T ω


20

Method 2 : Pole and Zero Mapping

Rule 1. A pole or zero at is mapped to

Rule 2. An infinite pole or zero is mapped to 1

2

aT

sT s

s a z e

z

ωz e ω

2

2 2 1sω π

j T j TTz e e

0 1

Rule 3. Select the gain of ( ) at a critical point

. . low-pass filter DC gain

( ) ( )

zp

s zp z

H z

i e

H s H z


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0 1

0 1

) ( )

1 ( )

2 ( ) 1 ( )

1

( ) ( )

1

2

1 1 ( )

2

zp aT

s zp z aT

s zp z

aT

aT

zp aT

aex H s

s az

H z Kz e

KH s H z

e

H s H z

eK

e zH z

z e


22

0 1

0 1

Other rule ( Rule 2*)

1 ( )

( ) 1, ( )1

( ) ( )

1

1 ( )

zp aT

s zp z aT

s zp z

aT

aT

zp aT

H z Kz e

KH s H z

eH s H z

K e

eH z

z e


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Method 3 : Hold Equivalent

H(s)( )e t ( )u t

( )( )ho

H sH z z

s

z11

sampler samplerH(s)hold( )e t

( )hoH z

( )u k( )e k


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1

1

1 1

1

11

1 1

) ( )

( ) 1( )

1

aT

aT

ho

aT

aT

e zz

z e z

aex H s

s a

aH z z

s s a

e

z e

z


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Frequency (rad/sec)

Pha

se (

deg)

; M

agni

tude

(dB

)

Bode Diagrams

-50

-40

-30

-20

-10

0From: U(1)

100 101 102-100

-80

-60

-40

-20

0

To: Y

(1)

1

) , 0.011

ex G s Ts

G s

backwardG z

G s

forwardG z

Frequency (rad/sec)P

hase

(de

g);

Mag

nitu

de (

dB)

Bode Diagrams

-50

-40

-30

-20

-10

0From: U(1)

100 101 102-200

-150

-100

-50

0

To: Y

(1)


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G s

trapezodalG z prewarpingG z

Frequency (rad/sec)

Pha

se (

deg)

; M

agni

tude

(dB

)

Bode Diagrams

-60

-40

-20

0From: U(1)

100 101 102-100

-80

-60

-40

-20

0

To: Y

(1)

Frequency (rad/sec)

Pha

se (

deg)

; M

agni

tude

(dB

)

Bode Diagrams

-60

-40

-20

0From: U(1)

100 101 102-100

-80

-60

-40

-20

0

To: Y

(1)

G s


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) , .1

0 0110

ex G s Ts

Frequency (rad/sec)

Pha

se (

deg)

; M

agni

tude

(dB

)

Bode Diagrams

-50

-40

-30

-20

-10

0From: U(1)

101 102-100

-80

-60

-40

-20

0

To: Y

(1)

Frequency (rad/sec)P

hase

(de

g);

Mag

nitu

de (

dB)

Bode Diagrams

-50

-40

-30

-20

-10

0From: U(1)

101 102-150

-100

-50

0

To: Y

(1)

G s

backwardG z

G s

forwardG z


28

Frequency (rad/sec)

Pha

se (

deg)

; M

agni

tude

(dB

)

Bode Diagrams

-60

-50

-40

-30

-20From: U(1)

101 102-100

-80

-60

-40

-20

0

To: Y

(1)

Frequency (rad/sec)

Pha

se (

deg)

; M

agni

tude

(dB

)

Bode Diagrams

-60

-50

-40

-30

-20From: U(1)

101 102-100

-80

-60

-40

-20

0

To: Y

(1)

G s

trapezodalG z prewarpingG z

G s


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ex) The third order low-pass Butterworth filter designed to have unity pass bandwidth ( = 1 ), Use sampling periods

(T = 0.1, T = 1, and T = 2 ). Example in page 195

3 2

1

2 2 1H s

s s s

0 0.5 1 1.5 2 2.50

0.5

1

1.5magnitude and phase of discrete equivalents

0 0.5 1 1.5 2 2.5-250

-200

-150

-100

-50

0bilinr = o,warp = +, backwd = *, forwd = x

normalized frequency w/wp

i) T = 0.1

bilinear = o

warped = +

backward = *

forward =

1ω


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0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

magnitude and phase of discrete equivalents

0 0.5 1 1.5 2 2.5-250

-200

-150

-100

-50



ii) T = 1

bilinear = o

warped = +

backward = *

forward =


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0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1

magnitude and phase of discrete equivalents

0 0.5 1 1.5 2 2.5-250

-200

-150

-100

-50



iii) T = 2

bilinear = o

warped = +

backward = *

forward =


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Forward Rule

1

x Ax Be

u Cx De

sX AX BE

U CX DE

zX AX BE

TU CX DE

( ) ( ) ( ) ( ) 1x k x k TAx k TBe k


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( ) ( ) ( ) ( )

( ) ( ) ( )

1x k I TA x k TBe k

u k Cx k De k

Backward Rule

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )

zX AX BE

Tzx k x k TAx k TBe k

x k TAx k TBe k x k

w k

1

1 1 1

1 1 1

1


34

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

1

1

1

1 1

1

Solving for in terms of a

1

nd

w k I AT w k I AT TBe k

u k C I

x w e

I AT x k w k T

AT w k D C

Be k

x k I AT w k I AT TBe k

I AT BT e k

Simlilarly, we can have an equivalent discrete-time state equation

using a bilinear rule (Refer . 198-200 in Franklin's)pp

robotics research laboratory 1 chapter 4 discrete equivalents

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