Well-Posedness of Feedback Loop: Consider t he … of Feedback Loop: Consider t he following...

2010 Spring ME854 - GGZ Internal Stability

Internal StabilityInternal StabilityWell-Posedness of Feedback Loop:

systemfeedback following heConsider t

++ K P +

+

r -

u

di

up

dy

n

( )

.realizable physicallynot is systemfeedback theHence, proper.not are )( to)(

and )()()( signals external thefromfunction transfer that theNote

)(3

1)()()(

3

2)(

However, functions.nsfer proper traboth be

1)( ,2

1)(

Let

susd

sdsnsr

sds

sdsnsrs

su

sKs

ssP

i

i

−−

−+−−

+=

=+

−−=



proper. and defined- wellare

matrices transfer loop-closed all if posed- wellbe tosaid is systemfeedback A

Definition 5.1

+

+K̂

P

2e 2w

1e1w

.

and as

signalsinput regroup and and

as loopfeedback theinto signalinput

external theregroup and :ˆLet

2 1

21

−=

ee

ww

KK

Lemma 5.1

.invertible is

)()(ˆ

ifonly and if posed- wellis above systemfeedback The

∞∞− PKI

Figure 5.2



+=

+=

122

211ˆ

by drepresente be can system ThePewe

eKwe

Proof:

Note:invertible

)(

)(ˆ invertible )()(ˆ

∞−

∞−⇔∞∞−

IP

KIPKI

.invertible is ˆ function transfer theof ermconstant t that toequivalent is this

Note proper. is and exists )ˆ( that toequivalent is posedness- wellThus,

ˆ)ˆ(

as rewriten be can equation Then

1

.211

PKI

PKI

wKwePKI

−

−

+=−

−

invertibleˆ

)(

)(ˆ invertible ˆ )()(ˆ

ˆ)(ˆ and )( ,ˆˆ

ˆˆˆ and nrealizatio space stateFor

−

−=

∞−

∞−⇔−=∞∞−

=∞=∞

=

=

ID

DI

IP

KIDDIPKI

DKDPDC

BAK

DC

BAP


Internal StabilityInternal StabilityInternal Stability:

∞

−−

−−

−−

−−−

−−

−−+=

−−

−−=

−

−

RHeeww

KPIPKPI

KPIKPKPIKI

KPIPKIP

KPIKPKI

IP

KI

tobelongs ),( to),( from

)ˆ()ˆ(

)ˆ(ˆ)ˆ(ˆ

)ˆ()ˆ(

)ˆ(ˆ)ˆ(ˆ

matrix transfer theif stable internally be tosaid is 5.2 Figurein defined system The

2121

11

11

11

111

Definition 5.2

=

+

+=−

−−=

+

−=

++

+−

+−

+−

++

−

2

1

21

21

)2)(1(

)1(

21

2

1

1

Then,

2

1)ˆ( ,

1

1ˆ ,1

1

:exampleFor stability. internal

for matricesfer four trans ofeach test tosufficient andnecessary isit that Note

w

w

e

e

s

sKPI

sK

s

sP

ss

ss

ss

s

ss



bounded. are

locations)any (at signals injected that theprovided bounded system ain signals allthat

guaranteesIt system.feedback practical afor t requiremen basic a isstability Internal

Remark 5.1

.)ˆ( and posed- wellisit if

only and if stable internally is 5.2 Figurein system Then the .ˆ Suppose

1

∞

−

∞

∈−

∈

RHPKIP

RHK

Corollary 5.2

Proof:

.)ˆ( and posed- well ˆ

1

1

∞−

∞

−

∈−∈

−

−⇒ RHPKIPRH

IP

KI֏

∞−−

∞−−

−−

−−−

∈−=−∈−+=−

−−

−−=

−

−⇒⇐

RHKPKIPPKPIRHKPKPIIPKI

KPIPKIP

KPIKPKI

IP

KI

ˆ,)ˆ()ˆ( since ˆ)ˆ()ˆ(

invertible )ˆ()ˆ(

)ˆ(ˆ)ˆ(

ˆ posed- well

1111

11

111



plane. halfright

closed thein zeros no has ))(ˆ)(det( ly,equivalent or, ,)ˆ( if

only and if stable internally is 5.2 Figurein system theThen .ˆ, Suppose

1sKsPIRHKPI

RHKP

−∈−

∈

∞−

∞

Corollary 5.4

stable. is ))(ˆ)(( ii)

ly);respective ,)(ˆ and )( of poles

rhpopen are and ( )(ˆ)( of poles rhpopen ofnumber thei)

and posed- wellisit ifonly and if stable internally is system The

1−−

+=

sKsPI

sKsP

nnnnsKsP kpkp

Corollary 5.3

.)ˆ(ˆ and posed- wellisit if

only and if stable internally is 5.2 Figurein system Then the . Suppose

1

∞−

∞

∈−

∈

RHKPIK

RHP

Theorem 5.5:



[ ]

[ ]

5.2) Problem (See stable is if stable internally is system theHence,

)ˆ(

ˆ)ˆ(

)ˆ(ˆ

ˆ

ˆ)ˆ(ˆ

ˆ

ˆ0

ˆ

where, ))(ˆ)((

1

1

1

1

1

A

DDID

CDCDDIC

DDIB

DBB

CDCDDIB

DB

A

CBAA

DC

BAsKsPI

−=

−=

−

=

−

+

=

=−

−

−

−

−

−

Proof:

−−−

=−

=

=

=⇐

DDICDC

BA

DBCBA

sKsPI

DDCDC

BA

DBCBA

sKsP

DC

BAsK

DC

BAsP

ˆˆ

ˆˆ0

ˆˆ

))(ˆ)(( and ,

ˆˆ

ˆˆ0

ˆˆ

)(ˆ)(

then,ˆˆ

ˆˆ)(ˆ and )( that Assume



[ ]

).(ˆ)( ofon cancellati pole/zero unstable

no is theres,other wordin or mode, lecontrollab and leunobservab unstable no is Or there

are. ˆ0

ˆ,ˆ and

ˆ

ˆ,

ˆ0

ˆ

iff detectable is ),( and lestabilizab is ),( since that note and

;))(ˆ)(( ii) have then westable, internal is system that theAssume 1

sKsP

A

CBACDC

B

DB

A

CBA

ACBA

RHsKsPI

∈−⇒ ∞−



2

2

2

2

1

)1(

)2()ˆdet(

where

2

10

)1()2(

)1(

2

1

)ˆdet(

1

20

1

1

1

2

)ˆ( ,

1

20

1

1

1

2

)ˆ(

and

1

10

1

1

1

1

)(ˆ)( then ,

10

11

1

)(ˆ ,

1

10

01

1

)(

:matrices transfer 22 be )(ˆ and )(Let

+

+=−

+

+−+

+

+

+

=−

+

+−

−

+

+

=−

+

+−+

+

=−

+

−−

−

+

−

=

−

+

−=

+

−=

×

−

s

sKPI

s

s

ss

s

s

s

KPI

s

sss

s

KPI

s

sss

s

KPI

s

sssKsPs

s

sK

s

ssP

sKsPAn example:

)(ˆ)(for on cancellati pole/zero no is therecase, In this

stable internal RHP closedat zeros havenot does )ˆdet( :Note

sKsP

KPI ≠−


Internal StabilityInternal StabilityCoprime Factorization over RH∞:

identity).(Bezout 1)()()()(such that )( and (s)

spolynomial exists thereifonly and if coprime are )( and )( spolynomial Two

=+ smsysnsxsyx

smsn

zeros).common no

is therely,equivalent(or 1 isdivisor common greatest their if coprime be tosaid

are ts,coefficien real example,for with,),( and )( spolynomial woConsider t snsm

1)()()()(

such that )( and (s) exists thereif

over coprime be tosaid are in )( and )(function transfer two,Similarily

=+

∈ ∞

∞∞

smsysnsx

RHsyx

RHRHsnsm


Internal StabilityInternal Stability

Definition 5.3

[ ]

[ ]

[ ]

ly.respective ,invertibleright and

left are ~~

and matricessay that toequivalent are sdefinition e that thesNote

.~~

~~

such that in and matrices exists thereif and rows ofnumber

same thehave they if over coprimeleft are ~

,~

matrices two,Similarily

.

such that in and matrices exists thereif and columns

ofnumber same thehave they if over coprimeright are , matrices Two

NMN

M

IYNXMY

XNM

RHYX

RHRHNM

INYMXN

MYX

RHYX

RHRHNM

ll

l

l

ll

rrrr

rr

=+=

∈

=+=

∈

∞

∞∞

∞

∞∞

Coprime Factorization over RH∞:



Right and left coprime factorizations

. ~~

such that and

,, and ,~~

ion factorizat coprimeleft a and ion factorizat coprime

right a exists thereifion factorizat coprime double have tosaid is )()(matrix A

.over coprimeleft are ~

and ~

where,~~

ion factorizat a is of (lcf)ion factorizat coprimeleft A ,Similarily

.over coprimeright are and where,

ionfactorizat a is of (rcf)ion factorizat coprimeright A matrix. realproper a be )(Let

11

1

1

IXN

YM

MN

YX

RHYX

YXNMPNMP

sRsP

RHMN

NMPP

RHMNNMP

PsP

l

lrr

ll

rr

P

=

−

−

∈

==

∈

=

=

∞

−−

∞

−

∞

−




Theorem 5.6

satisfied. is

(5.7) equation re,fiurthermo and, ly,respective lcf, and rcf are ~~

Then

0

)(

~~

and

0

define and stable, both are

and that such be and Let n.realizatio detectable and lestabilizaba is

)(

andmatrix rational realproper a is )( Suppose

11 NMNMP

IDC

IF

LLDBLCA

MN

YX

IDDFC

IF

LBBFA

XN

YM

LCA

BFALF

DC

BAsP

sP

rr

l

l

−− ==

−

+−+

=

−

+

−+

=

−

+

+

=




Proof:

[ ] [ ]LBI

IAsIII

C

FBAsIC

BFAsI

BFAsILCBFBFAsILCAsIAsI

ICC

IFF

LBBFA

LBLCBFLCA

DDCDC

BA

DBCBA

GGXN

YM

MN

YX

IDDFC

IF

LBBFA

XN

YMsG

IDC

IF

LLDBLCA

MN

YXsG

sGsGsG

l

lrr

l

lrr

−

−−

=−

−−

−−−−−−−−=−

−+

−+−+

=

==

−

−

+

−+

=

−=

−

+−+

=

−=

=

−−

−

−−−−

11

1

111

1

21211

22

21211

21

21

21

)()(

thatNote Also

)(0

))(()()()(

thatNote

0

0

00~~

Then,

0 )(,0

)(

~~)(

let ),()()( connection system serial Recall




Proof:

[ ] [ ]

{ }{ }

1 1 1

1

1

1 1 1

1 1

Since

( ) ( ) ( )( )( )

( )

( ) ( )( ) ( )

( ) ( ) ( ) ( ) ( )

0

Therefore,

I sI A LC sI A LC BF LC sI A BFI I sI A I I

I sI A BF

sI A LC I BF LC sI A BF sI A BF

sI A LC sI A BF BF LC sI A LC sI A BF

− − −−

−

− − −

− −

− − − − − − − − − − − =

− −

= − − − − − − − + − −

= − − − − − − − + − − − −

=

[ ] [ ]1

0

( )lr r

l

M YX Y F II I sI A B L I I

N XN M C I

−− −

= − − + = − ɶ ɶ

��


Remark 5.2

.~

and ,~

,0 ,

takecan we thenstable, is if that Note

IMMPNNYYIXX

P

lrlr ========



Remark 5.3

feedback. statea by variablecontrol changing fromnaturally

out comes ionfactorizat coprimeright instance, For tion.interpreta control

feedback a given be can function a transfer of ionfactorizat coprime The


).()()( is,that );()()()( that such

)( and ,)()(

Then,

)(

)(

Then stable. is that such

:feedback state with:

plant a for equation space-state heConsider t

11sMsNsPsusMsNsy

DDFC

BBFAsN

v(s)

y(s)

IF

BBFAsM

sv

u(s)

DvxDFCy

vFxu

BvxBFAx

BFA

vFxuDuCxy

BuAxxP

P

−− ==

+

+==

+==

++=

+=

++=

+

+=

+=

+=

ɺ

ɺ



Lemma 5.7

)(~

)(~

)()()(

)(~

)(~

)()()(

)( and )( of slcf' and srcf' theRecall

11

11

sUsVsVsUsK

sNsMsMsNsP

sKsP

−−

−−

==

==


. in invertible is ~~

)

in invertible is ~~

)

in invertible is ~~

~~

)

in invertible is ii)

stable. internally is systemfeedback The i)

:equivalent are conditions following The 5.2. Figin described ystem heConsider t

∞

∞

∞

∞

−

−

−

−

RHNUMVv

RHUNVMiv

RHMN

UViii

RHVN

UM

s



Proof:


since coprime are

,0

0

matrices that theNote

0

0

ˆ

thatso

0

0

ˆ

Now ,ˆ

lyequivalentor ˆ

if stable internally is system that theNote

11

1

1

1

1

1

1

=

=

=

∈

∈

−

−

−−

−

−

−

−

∞

−

∞

−

VN

UM

V

M

VN

UM

V

M

IP

KI

V

M

VN

UM

INM

UVI

IP

KI

RHIP

KI

RHIP

KI



1 1

and are coprime, there exist matrices , , , s.t.

and

and

0 0

0 0 0

0

P P K K

P P K K

P P P

K K K

NM UV X Y X Y RH

X M Y N I X V Y U I

M U

Y X Y N V I

X X Y M

V

− −

∞∈

+ = + =

− = −

0and , . This proves the equivalence of 1 and 2. Note

0

that 1 and 3 can be proved in a similar way.

P P P

K K K

I

Y X YRH

X X Y∞

− ∈ −



Conditions 4 and 5 are implied by conditions 2 and 3 by

0

0

Since left-hand side is invertible in , so in right-hand side. Therefor

M UV U VM UN

N VN M MV NU

RH∞

− − =

− −

ɶ ɶ ɶ ɶ

ɶ ɶ ɶ ɶ

1 11

1

1

e,

conditions 4 and 5 are satisfied. We need to show either condition 4 or 5

implies condition 1. Consider

ˆ

0 0

0 0

I V UI K

NM IP I

M VM U V

I N I I

− −−

−

−

=

=

ɶ ɶ

ɶ ɶ ɶ

1

if (i.e., ) or condition 5 is satisfied.

RH

VM URH VM UN RH

N I

∞

−

∞ ∞

∈

∈ − ∈

ɶ ɶɶ ɶ



Corollary 5.8


−

+−+

=

−

−

+

−+

=

++

=

−

−

∈

==

==

∞

−−

∞

−−

IDC

IF

LLCBLCA

MN

UV

IDFC

IF

LBBFA

VN

UM

LCABFALF

I

I

VN

UM

MN

UV

RHVUVU

UVVUK

RH

NMNMPP

0

)(

~~

~~

and

0

0

by given be can matrices for these nrealizatio space state ofset particular

a Then stable. are and that such be and let e,Furthermor

0

0 ~~

~~

thatsuch ~

,~

,, with

~~ˆ

controllera exists thereThen .over lcf and rcf ingcorrespond

thebe ~~

andmatrix rational realproper a be Let

00

0

0

0

000

0000

0

1

0

1

000

11



Process of finding a coprime factorization for a scalar transfer function


( )i) suppose ( )

( )

ii) let (s) be any stable polynomial with the same order as (s)

( ) ( )iii) Then (s) ( ) / ( ), where (s) and (s) is a

(s) (s)

coprime of

num sP s

den s

den

num s den sP n s m s n m

α

α α

=

= = =

factorization ( ).P s



Scalar Example:


2Let ( ) and select (s) ( 1)( 3). Then

( 3)

( ) 2 ( ) where ( ) and ( )

( ) ( 1)( 3) ( 1)

is a coprime factorization of ( ). To find ( ) and ( ) such that

sP s s s

s s

n s s sP s n s m s

m s s s s

P s x s y s

α−

= = + ++

−= = =

+ + +

( ) ( ) ( ) ( ) 1.

1ˆ ˆConsider a stabilization controller for : , Then ( ) / ( ), where10

ˆ( ) ( ) and ( ) 1 is a coprime factorization and

( 11.71)( ( ) ( ) ( ) ( )

x s n s y s m s

sP K K u s v s

s

u s K s v s

s sm s v s n s u s

+ =

−= =

+

= =

+ +− =

2.21)( .077): ( )

( 1)( 3)( 10)

Then,

( 1)( 1)( 3) ( ) ( ) / ( )

( 11.71)( 2.21)( .077)

( 1)( 3)( 10) ( ) ( ) / ( )

( 11.71)( 2.21)( .077)

ss

s s s

s s sx s u s s

s s s

s s sy s v s s

s s s

β

β

β

+=

+ + +

− + += − =

+ + +

+ + += =

+ + +


Internal StabilityInternal StabilityCoprime Factorization over RH∞:Systematic process of finding a coprime factorization

i) Let ( ) be a stabilizable and detectable realization of ( )

ii) Find a stabilization state feedback gain ( stable) using either pole

assignment or LQR design

iii) Find a stable o

A,B,C,D P s

F A BF+

1

bserver gain ( stable) using either pole assignment

or Kalman filter design

iv) Let

( )

0 & 0

0

where

L A LC

A BF B L A LC B LC LM U V U

F I F IN V N M

C DF I C D I

P NM−

+

+ − + − + − = = − + −

=

ɶ ɶ

ɶ ɶ

1 and

and

M N

VM UN I MV NU I

−=

− = − =

ɶ ɶ

ɶ ɶ ɶ ɶ


Internal StabilityInternal StabilityCoprime Factorization over RH∞:MIMO example:

4 3 1 2 0 2 0 0Let , , , be a stabilizable and

0 2 0 3 1 4 0 1

detectable realization.

The stabilization gain can be found by

2.0 .333 F -place(A,B,[-2 -3]),

0 1.67

and sta

A B C D

F

F

− = = = =

>> = = −

ble estimation gain can be found by

.0.1714 0.6411L -lqr(A',C',eye(2),eye(2)),

0.4974 1.3679

(eig(A LC) { 4.5538 0.3454 })

L

j

− − >> = = − −

+ = − ±



−

+−+

=

−

−

+

−+

=

IDC

IF

LLCBLCA

MN

UV

IDFC

IF

LBBFA

VN

UM0

)(

~~

~~

&

0

0

Then

2

22 2

2

2 22

2 2

2 2

6 127 12 5 14

s 5 6s 5 6 s 5 6

3 9 204

s 5 6 s 5 6s 5 6

1.07 3.07 2.47 6.96

s 5 6 s 5 6

.707 1.41 2.18 4.37

s 5 6 s 5

0where , ,

0

ss s s

ss s

s ss

s ss

s s

s s

s s

s

M N

U

++ + +

+ ++ + + +

+ +−

+ + + ++ +

+ +

+ + + +

− − − −

+ + +

= =

=

2

2 2

2

2 2

5.85 7.70 2.62 5.24

s 5 6 s 5 6

1.45 3.37 9.07 15.2

6 s 5 6 s 5 6

,

s s s

s s

s s s

s s s

V

+ + +

+ + + +

+ + ++ + + + +

=

2

2 22 2

2

2 2

2.74 13.68 6.93 29.78.85 20.18 2.97 13.7

s 9.11 20.86 s 9.11 20.86s 9.11 20.86 s 9.11 20.86

5.11 25.82.28 11.4 3.33 3.89

s 9.11 20.86 s 9.11 20.86

and ,

s ss s s

s ss s

ss s s

s s

V N

− − − −+ + +

+ + + ++ + + +

− −− − + −

+ + + +

= =

ɶ ɶ2

2 2

2

2 2

2 2

8 6.67 42.9

s 9.11 20.86 s 9.11 20.86

0.509 2.31 1.738 7.805 8.113 1

s 9.11 20.86 s 9.11 20.86

0.829 3.46 2.28 9.12

s 9.11 20.86 s 9.11 20.86

,

,

s s

s s

s s s s

s s

s s

s s

U M

− −

+ + + +

+ + + ++ + + +

− − − −

+ + + +

= =

ɶ ɶ 2 2

2

2 2

6.7 2.74 10.94

s 9.11 20.86 s 9.11 20.86

2.16 9.11 2.995 4.02

s 9.11 20.86 s 9.11 20.86

s

s s

s s s

s s

− −

+ + + +

− − + −

+ + + +

Well-Posedness of Feedback Loop: Consider t he … of Feedback Loop: Consider t he following...

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