Bili Near Transformation

7/28/2019 Bili Near Transformation

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Page 1

Bilinear Transformation

Control Engineering

by Dr. L. K. Wong


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Page 2

A Control System

Most plants are continuous-time systems

Power supply, power amplifier, motor Digital controllers are in discrete-time

Implemented by micro-controller

Controller PlantReference Output

+


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Page 3

Continuous-time Signals

f(t)

t


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Discrete-time Signals

f*(t)

tT2T

3T4T

=

= Otherwise0),(

)(*nTttf

tf


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Page 5

Transformation

Convert a continuous-time transfer functionto a discrete-time transfer function

H(s) H(z)


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Methods of Transformation

Backward difference Forward difference

Bilinear transformation z-transform


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Theoretical Background

as

b

sRsYsH

+

=

=)()()(

)()()( tbrtayty +=!

)()()(1 tbrtayty +=

dttytyty

t

t+=2

1 )()()( 112

LetTkt )1(1 =

kTt =2


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[ ])1()(2

)1()( 11 ++= kykyTkyky

y1(t)

tt1=(k1)T t2=kT


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[ ])1()(2

)1()2

1()()2

1( +=+ krkrbTkyaTkyaT

[ ]11 )()(2

)()2

1()()2

1( +=+ zzRzRbTzzYaTzYaT

[ ])1()(2

)1()( 11 ++= kykyT

kyky


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Page 10


azz

T

b

aT

z

z

bT

aT

z

zz

bT

z

zaT

bT

zzaTaT

zbT

zaTaT

zbT

zRzY

++

=

++

=

+++

=

++

=

+++

+=

++=

)1(12

2)1(

12

2)1(

212

)1(

2)

21(

2

2)2

1()2

1(

)1(2

)2

1()2

1(

)1(2

)()(

1

1

1

1

1

11

1

1

11

1

1

1


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Page 11


az

z

T

bzRzYzH

++

==

)1(

12)()()(

1

1

as

b

sR

sYsH

+==

)(

)()(

1

12

+

=z

z

Ts

Compare


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Example 1

Find a digital replacement of the followingcontinuous-time plant by bilineartransformation with sampling period of

T = 0.1s.

10010

1002.0

)( 2

2

1 ++++

= ssss

sH


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Answer

1

120

112

+

=

+=

z

z

zz

Ts

4286.08571.0

7086.08571.07200.0300600700

496600504)(

2

2

2

2

1

++

=+

+=

zz

zzzz

zzzH


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Frequency (rad/sec)

Phase(deg);Magn

itude(dB)

Bode Diagrams

-50

-40

-30

-20

-10

0

From: U(1)

10-1 100 101 102-100

-80

-60

-40

-20

0

To

:Y(1)

Frequency Warping10

5)(2 +=

ssH

9048.0

)1(0238.0)(2

+=

z

zzH


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Frequency Warping

Approximation has been taken place use a trapezoidal to approximate the area under

a curve

Frequency response ofH(s) deviates fromthat ofH(z)

Significant if it lies in critical frequency e.g. 3dB cut-off frequency


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Analytical Derivation

1

12

+

=

z

z

TsTj

A

Dezjs == andSubstitue

T

Tj

T

ee

ee

T

e

e

Tj

D

D

TjTj

TjTj

Tj

Tj

A

DD

DD

D

D

2

1cos

21sin

2

2

1

12

2

1

2

1

21

21

=

+

=

+

=

into


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Analytical Derivation

TT

DA

2

1tan

2=

DA

DD

D

TT

2

1

2

1tan

small,isIf


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Frequency Pre-warping

ModifyH(s) before applying transform Cancel out the warping effect exactly at a

frequency

Same frequency response ofH(s) andH(z)

at


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Step 1

Calculate the pre-warped frequency

TTP 2

1

tan

2

=


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Step 2

ReplacebyP

p

p

p

p

ss

ss

=

=


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Step 3

Applying bilinear transformation

112

+=

zz

TsP


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Example 2

Apply bilinear transformation withfrequency pre-warping at= 10 rad s1 tothe following continuous-time plant with

sampling period ofT = 0.1s. Calculate themagnitude and phase angle at for bothcontinuous-time and discrete-time plant.

10010

1002.0)(

2

2

3 ++++

=ss

sssH


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Answer

Step 1

Step 2

926.10

1.01021tan

1.02

=

=P

P

p ss

s 915.0926.10

10==

10015.984.0

100183.084.0)(

2

2

3 ++++

=PP

PPP

ss

sssH


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Answer

Step 3

Substitutes=j toH3(s) andz= ejT toH3(z),

Magnitude = 0.02

Phase = 0 rad

4077.07606.0

6979.07606.07098.0)(

2

2

3 ++

=zz

zzzH


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Question

Can we select two frequencies to pre-warp?


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w-transform

Transform az-plane transfer function into aso-calledw-plane transfer function

Inverse process of bilinear transformation

wwz += 11


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Design on thew-plane

Employs-plane design velocity error constant

gain and phase margin

No need to tackle the irrational functionz= ejT


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Further Modification

Large frequency distortion inw-transform Modify thew-transform as follows

2

1

21

wT

wT

z

+=


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Example 3

TransformH4(z) into a transfer functionH4(w) inw-plane by the given bilineartransformation. Sketch the bode plot of

H4(w).

)8187.0)(1(

9356.003746.0)(4

+=

zz

zzH

w

wz

1.01

1.01

+

=


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Answer

+

+

=

997.01

3001

1012

)(4ww

ww

wH


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Conclusion

Bilinear transformation Frequency pre-warping

w-transform


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Reference

M. Gopal, Digital Control Engineering.John Wiley & Sons.

I.J. Nagrath and M. Gopal, Control Systems

Engineering. 2nd edition. John Wiley &Sons.

Bili Near Transformation

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