linear fractional transformation · 2010 spring me854 - ggz linear fractional transformation page...

2010 Spring ME854 - GGZ Linear Fractional TransformationLinear Fractional Transformation

Linear Fractional TransformationLinear Fractional Transformation

Linear Fractional Transformation Linear Fractional Transformation (LFT)(LFT)

.,, somefor

)1()(

zero,not is If (LFT).ation transformfractionallinear a called , and ,,,with

)(

form theof : mappingcomplex variableoneConsider

1

C

sssF

cCdcba

dsc

bsasF

CCF

∈

−+=

∈

+

+=

−

γβα

γβα

֏

Definition 9.1

exist. )( provided

)(),( with :),(

LFTupper an define ,exist )( provided

)(),( with :),(

LFTlower a Define matrices.complex other twobe and let and

as dpartitionematrix complex a be Let

1

11

12

1

112122

1

22

21

1

221211

)()(

2221

1211

2211

1122

2211

2121

−

−××

−

−××

××

+×+

∆−

∆−∆+=∆⋅

∆−

∆−∆+=∆⋅

∈∆∈∆

∈

=

u

uuuu

qpqp

u

l

llll

qpqp

l

qp

u

qp

l

qqpp

MI

MMIMMMFCCMF

MI

MMIMMMFCCMF

CC

CMM

MMM

M

֏

֏




121

1

2212111121111

121

1

2211221211

11

22

2

2

2221

1211

2

2

11

1

1

2221

1211

1

1

])([

)(

, ,

),( and ),( ngrepresenti diagram following

thefromclear be should upper LFT andlower of ologiesfor termin motivation The

122

wMMIMMuMwMz

wMMIuuMwMy

yu

yuw

u

MM

MM

z

yyu

u

w

MM

MM

y

z

MFMF

ll

ll

yM

l

ul

uull

l

−

−

∆

∆−∆+=+=

∆−∆=

+=

∆=

∆=

=

∆=

=

∆∆

��

M

∆l

1z1w

1u1y

M

∆u

2z 2w

2u2y

),(1 llzw MFT ∆= ),(2 uuzw MFT ∆=




.invertible is )( if posed)-(or well defined- wellbe tosaid is , ),( LFT,An1

22

−∆−∆ MIMFl

Definition 9.2

Lemma 9.1

, ,

with

),()()(

),())((

Then .invertible is Suppose

11

11

11

11

1

1

−−=

−

−=

=++

=++

−−

−−

−−

−−

−

−

DCADCB

CACN

DCC

DACBACM

QNFBQADQC

QMFDQCBQA

C

l

l

1

111

)(

1111

))((

1

21

1

221211

))((

))((

)()(

)()(

:Note

111

−

−−−

+

−−−−

++

−

−

++=

+−++=

+−+=

−+=−−−

DQCBQA

DQCDQACBQDQACA

CDQCIQDACBAC

MQMIQMMM,QF

DQCDQCDQCAC

l

��




Lemma 9.2

.matrix r nonsingulaany for

,00

0

,

0

00

0

isThat . , , ,with

))((),(

then,invertible is If )

.matrix r nonsingulaany for

,0

00

,

0

0

00

isThat . , , ,with

)()(),(

then,invertible is If )

. with LFTgiven a be ),(Let

1

2122

1112

1

21

22

1112

22

1

21

1

2122

1

211112

1

2111

1

21

1

1211

2221

1

12

11

2221

1

1222

1

1211

1

12222111

1

12

1

12

2221

1211

E

E

EMMI

I

MM

FM

MI

I

MM

FDC

BA

MMDMCMMMMBMMA

DQCBQA QMF

Mb

E

E

EMIM

MM

I

FM

IM

MM

I

FDC

BA

MMDMCMMMMBMMA

BQADQC QMF

Ma

MM

MMMQMF

ll

l

ll

l

l

+

=

−

=

−==−==

++=

+

−

=

−

−

=

−==−==

++=

=

−−

−−−−

−

−−

−−−−

−




Note:

[ ]

[ ]IMEEMIEM

I

ME

EMMI

I

MM

F

MMMMMM

MMMIMMIM

M

I

M

M

IM

MM

I

FDC

BA

MMD

MMMMB

MC

MMA

DCM

ADCBM

CM

ACM

M

l

M

l

11

11

12

1

2221

1

2122

1112

1

122211

1

122221

1

1211

1

12

11

11

12

1

12

2221

1

12

11

2221

1

1222

11

1

122221

1

12

11

1

12

1

22

1

21

1

12

1

11

112

112

))((0

00,00

0

)0(0

00

,

0

0

00

have we9.1, Lemma Using

��

��

−

−

−

−−−−

−−

−−−−−

−

−

−

−

−

−

−

−

−

+−

−+

=

+

=

−−=⋅+

−−

=

−

−

=

−=

−=

=

=

−=

−=

=

=




Note:

( )

( ) [ ] [ ]

11

)(

1

)(

1

1

1

)(0

)_(

1

0

1

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

)()(

)(

)_()

0

0

0

0(

0

0,

Note

0

0 ,

0

0

where

))(( ,

11

1

1

−−

+

−

+

−

−

−

+

−

+

−

−

++=+−−++=

+−+−+=

+

+−−=

−

−

−

−−

−

−+=∆

−

−=∆

−−=

++=∆

−−

−

−

QDIQBAQDIAQDQBQDQDIQBA

DDQIAQDDQIQBQQBA

DDQI

DDQIBQBAQQA

D

B

Q

Q

D

BI

Q

QAIANF

Q

Q

DD

BB

AIA

N

QDIQBANF

QDIDQDID

DQI

DQIBQI

DQI

BQI

l

l

��

��

��




Lemma 9.3

.

by given is

where),,()],([ Then r.nonsingula is and Let

1

2221

1

22

1

221221

1

221211

1

22

2221

1211

−−=

∆=∆

=

−−

−−

−

MMM

MMMMMMN

N

NFMFMMM

MMM uu

( )

( )

( )∆=

−∆−−∆+=

∆+∆−∆−=∆

−+=−

∆−∆+=∆

−−−−−

−−−−−−

−−−−−−−

−

−

,

)())((

)(],[

have we

))(())((

identity theUsing

)( , Note

12112122

21221211

1

2212

1

21

1

22121121

1

22

1

22

1

2212

1

21

1

22121121

1

22

1

22

1

1

1121

1

12

1

11212212

1

11

1

11

1

21

1

221211

12

1

112122

NF

MMMMMMIMMM

MMMMMMIMMMMF

AAAAAAAAAAAAA

MMIMMMF

u

NNNN

u

AAAA

u

��

��




Simple Block Diagram and LFT

( ) [ ]

+−+−

+−+−=

−−+

+

=

−−−

=

=

=

−−

−−

−

FFPKIKWFPFPKIKW

FFPKIPKWFPFPKIKIPW

FFPFPKIKW

PWPWKGF

FPFFP

W

PWPW

Gu

vz

n

dw

l

f

1

1

1

1

1

2

1

2

1

1

22

1

22

)()(

)())((

)(00

0,

LFTan as writenbe can system loop closed thewhere

00

0

, ,

with

LFTan as rearranged be can diagram

blockright the withsystemfeedback Afu

− y

1W

2WPK

F

u

d

n

v

G

K

z w

uy




Parametric Uncertainty

[ ].1.01

where

),()1.01(1.01

)1.01(

1.01.01

m

1

. in LFTanby drepresenteeasily be can case, thisin that Note

)3.01( ),2.01( ),1.01(

s variationfollowing thehave parameters that Suppose

.

by described be can motion system theof equation dynamic

The side.right theon systemdamper -spring-massa Consider

1.01

1

1

1

m1

ij

mm

mlmm

m

mm

m

kcm

mM

MFmmm

kkccmm

m

Fx

m

kx

m

cx

=

−

−=

=+−=+

−+=

+=+=+=

=++

− δδδδ

δδ

δ

δδδ

ɺɺɺ

m

F

ck




Parametric Uncertainty (cont’d-1)

have we

equations, above into of elements Substitute

and

)(

2.0

3.0

)(

:equations dynamic following list the Now

1

221221

2

1

1212112

21

M

y

y

y

y

y

y

u

u

u

umxkuxcuFmy

xcy

xky

umxkuxcuFmx

xx

m

c

k

m

c

k

m

c

k

m

c

k

mkcm

c

k

mkc

∆=

=

+−−−−=

=

=

+−−−−=

=

δ

δ

δ

ɺ

ɺ

m

F

ck




Parametric Uncertainty (cont’d-1)

−−−−−

=

∆=

−−−−−

=

−

−=

−−−−−

−−−−−

1.0111

00002.00

000003.0

000010

where,),(

Therefore

1.0111

00002.00

000003.0

000010

havecan we,1.01

that Recall

1.0111

2

1

2

1

2

1

1.01112

1

1.01

1

ck

c

kM

F

x

x

MFx

x

u

u

u

F

x

x

ck

c

k

y

y

y

x

x

M

mmmmmc

mk

l

m

c

k

mmmmmc

mk

m

c

k

mm

ɺ

ɺ

ɺ

ɺ

M

∆

=

2

1

x

xx

ɺ

ɺɺ

=

F

x

x

w 2

1

u

y



Basic PrincipleBasic Principle

matrix. transfer ingcorrespond the

gcalculatinby obtained becan matrix

theand s,' out the pulling :principles basic The

LFT plant andy uncertaintPlant b)

LFT plant and Controller a)

:examples previous Two

M

GK

∆

⇒

⇒

out. taken s' with s' and of in terms and Writec)

s' and s'

as s' theof outputs and inputs Mark the b)

seperated each with

relationut input/outpfor diagramblock a Draw a)

:process general a is following The

δ

δ

δ

uwyz

uy




.

w

u

u

u

u

M

w

u

u

u

u

aadae

bcbdbe

de

z

y

y

y

y

=

−−

+−−

−−

=

4

3

2

1

4

3

2

1

4

3

2

1

10

00

00001

00001

100

Then

one.right see drawn, be can diagramblock a Then,

)1(

)(

Let

:1

:relationut input/outp anConsider

1

2

1121112

1211

1

2

212

2

121

2

212

yeydwywedy

ycbaz

Gwwed

cbaz

δδδδδδ

δδδ

δδδ

δδδ

−−=

++=

++=

=++

++=

−

.I

IwMFz u

=∆∆=

22

21 ,),(

where

δ

δ




−−−

−−

−−−−

=

=== +

+

+

+

+

+

003.57000

00003.570

000100

4.228.77063.27.110123.

0414.0983.9.10

001.329.186.360226.

, ,

where toolbox,Control Robust in

aircraft) emaneuverbl(highly HIMATheConsider t

0

320

)28.1(2

203.0

)3(5.0

210000

)100(50

2

P

IWIWIWs

s

ns

s

ps

s

del

6

y2

5

y1

4

e2

3

e1

2

z2

1

z1

0.5s+1.5

s+0.03

Wp22

0.5s+1.5

s+0.03

Wp11

2s+2.56

s+320

Wn22

2s+2.56

s+320

Wn11

50s+5000

s+10000

Wdel22

50s+5000

s+10000

Wdel11

x' = Ax+Bu

y = Cx+Du

State-Space

8

u2

7

u1

6

n2

5

n1

4

d2

3

d1

2

p2

1

p1

Example 9.1

=

2

1

2

1

2

1

2

1

2

1

2

1

2

1

)(ˆ ,

u

u

n

n

d

d

p

p

sG

y

y

e

e

z

z

where

)aircraft'linmod('D]C,B,[A,

by calculated be can )(ˆ and

=>>

sG

Ku

)(ˆ sG

∆z p

=

2

1

2

1

n

n

d

d

w

e

y



Redheffer Star ProductRedheffer Star Product

−

−=∗

−

=

=

−

−

),()(

)(),(

as defined is and ofproduct star theThen .invertible is

)(at further th assume square, and defined wellis product matrix that theSuch

,

matrces patitioned compatibly are and that Suppose

2221

1

112221

12

1

22111211

11221122

2221

1211

2221

1211

PKFPKPIK

KPKIPKPFKP

KP

KPIKP

KK

KKK

PP

PPP

KP

u

l

P

z w

wz K

P�K

z w

wz

−+−

−−+=

+−=

++=

++=

+−=

+=

+=

+=

+=

−

−

−+=

−−

−

−+=

−

−

−

w

w

KPKPIKKPKPIK

KPKIPPKPKIPP

z

z

wKwPKPKIz

wKzPKwPKz

wKPzKPwPz

wKPwPKPIz

wKzKz

wKzKz

zPwPz

zPwPz

KPKIPKKPKF

PKPIKPPKPF

u

l

ˆ)()(

)()(

ˆ

Therefore

)ˆ()(ˆ

ˆˆˆ

ˆ

)ˆ()(

ˆˆ

ˆˆ

ˆ

ˆ

Note

121

221122212222

211

112211121111

)(),(

1222

1

1122212221

1

112221

12

1

221112

)(),(

2111

1

22111211

122111

1

22111

121221121111

122221122212

122221

1

11222

22221

122111

122212

11211

��

��



Redheffer Star ProductRedheffer Star Product

where,tion representa a has ˆˆ

:matrix transfer Then the

,

nsrealizatio space state with matrices transfer are and that Suppose

22212

12111

21

22212

12111

21

=∗

∗

=

=

DC

BAKP

z

z

w

wKP

DDC

DDC

BBA

K

DDC

DDC

BBA

P

KP

KKK

KKK

KKK

֏

22111122

2221

1211

2221

1211

1222

1

212221

1

21

12

1

1221

1

111211

221

111

222

121

122

1

2122

1

21

1

1

122

1

11121

21

1211

2221

21

1222

1

1221

1

1

12

1

22111

1

21

1

111

222

2

122

1

12

1

1

1

1

2211

1

2

~ , where

~

~

~~

~~

DDIRDDIR

DD

DD

DD

DD

DDRDDDRD

DRDDRDDDD

CD

CD

DC

DC

CDRDCCRD

CRDCRDDCC

BB

DD

DD

BB

DDRBBDRB

DRBDDRBBB

AB

CD

DC

BA

CDRBACRB

CRBCDRBAA

KK

KK

KK

KKKK

KK

KK

KK

KKKK

KK

KK

KK

KKKK

KK

KK

KK

KKKK

KK

−=−=

∗

=

+

+=

∗

=

+

+=

∗

=

+

+=

∗

=

+

+=

−−

−−

−−

−−

−−

−−

−−

−−

linear fractional transformation · 2010 spring me854 - ggz linear fractional transformation page...

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