reduced order observers

Reduced Order Observers for Linear Systems

SOLO HERMELIN

Updated: 08.12.08

Table of Content

SOLO Reduced Order Observers for Linear Systems


∈∈∈+=∈∈∈∈+=

pxmpxnpx

nxmnxnmxnx

RDRCRyuDxCy

RBRARuRxuBxAx1

11Plant:

We want to construct a Observer such that it’s output will asymptotically convergeto .x

x̂


∈∈∈+=∈∈∈∈+=

pxmpxnpx

nxmnxnmxnx

RDRCRyuDxCy

RBRARuRxuBxAx1

11Plant:

Assume: npCrank pxn ≤=

Find:( ) xnpnRC −

⊥ ∈ such that:

nxnC

C

⊥is nonsingular.

Solution: Find the Singular Value Decomposition (SVD) of C

( )[ ] HCpnpxCCpxn nxnpxppxpVUC −Σ= 0

where H means Transpose of a matrix and complex conjugate of it’s elements, and:

nCH

CH

CCpCH

CH

CC IVVVVIUUUU ==== ;( )

( ) ( ) nxnnpxpp

ppC

diagIdiagI

diagpxp

1,,1,1,1,,1,1

0,,,, 2121

==

>≥≥≥=Σ σσσσσσ

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

CCxppnCxnpn nxnpnxpnpnxpnVUC

−−⊥−−⊥Σ= −−⊥ 0Then:

UC is any orthogonal matrix and ΣC is any non-zero diagonal matrix.


Define:

We have : xC

C

p

uDy

=

−

⊥

( ) 1: xpnRpxCp −⊥ ∈=

( )

−=

−

= ⊥

−

⊥ p

uDyCC

p

uDy

C

Cx ††

1

or : ( ) pCuDyCx ††⊥+−=

where: ( ) nxpTT RCCCC ∈= −1† is the Right Pseudo-Inverse of C or pICC =†

( ) ( )pnnxTT RCCCC −−⊥⊥⊥⊥ ∈= 1† is the Right Pseudo-Inverse of C or pnICC −⊥⊥ =†

Then:

( ) ( )

( ) ( )

=

=

−−

−

⊥⊥⊥

⊥⊥

⊥ pnxppn

pnpxp

I

I

CCCC

CCCCCC

C

C

0

0††

††††

( ) nICCCCC

CCC =+=

⊥⊥

⊥⊥

††††


We have:

+=+=

uDxCy

uBxAx

( ) pCuDyCx ††⊥+−=and:xCp ⊥=

( ) ( )[ ]{ }uBpCuDyCACuBxACxCp ++−=+== ⊥⊥⊥⊥††

or:

( ) uBCuDyCACpCACp ⊥⊥⊥⊥ +−+= ††

We want to obtain an estimation of . If we add we can see that:pp̂ ( )uBxCyL −−

( )[ ] ( )

( )

0ˆ

ˆˆ

0

††

††

=−−−−=

−+−−=−−

−

⊥

⊥

uDpCCuDyCCy

uDpCuDyCCyuDxCy

pnpxpI

Apparently does not contain any information on , but let compute .py y

( ) ( )[ ]{ } uDuBpCuDyCACuDuBxACuDxCy +++−=++=+= ⊥ˆ††


We have:

Therefore contains the information on .py

( ) ( )[ ]{ } uDuBpCuDyCACuDuBxACuDxCy +++−=++=+= ⊥ˆ††

( ) uDuBCpCACuDyCACy +++−= ⊥ˆ††

Let estimate by using:p

( )( )[ ]uDuBCpCACuDyCACyL

uBCuDyCACpCACp

−−−−−+

+−+=

⊥

⊥⊥⊥⊥

ˆ

ˆˆ

††

††

or:

( )[ ] ( ) ( )[ ]uBuDyCApCACLCuDyLptd

d +−+−=−− ⊥⊥†† ˆˆ

( ) pCuDyCx ††⊥+−=


We have:

( )[ ] ( ) ( )[ ]uBuDyCApCACLCuDyLptd

d +−+−=−− ⊥⊥†† ˆˆ

( ) pCuDyCx ˆˆ ††⊥+−=


We also have:

( )[ ] ( ) [ ]uBxACLCuDyLptd

d +−=−− ⊥ˆˆ



One otherform: ( )[ ] ( )

( ) ( ) ( ) uBCLCuDyCACLC

pCACLCuDyLptd

d

−+−−+

−=−−

⊥⊥

⊥⊥

†

† ˆˆ



Andanotherform:

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

uDyLpCACLCuDyLptd

d

−+−+−+

−−−=−−

⊥⊥⊥

⊥⊥

††

† ˆˆ

( )[ ] ( ) ( )uDyLCCuDyLpCx −++−−= ⊥⊥††† ˆˆ


We have:


uBCuDyCACpCACp

−−−−−+

+−+=

⊥

⊥⊥⊥⊥

ˆ

ˆˆ

††

††

Subtract those equations:

Define the estimation error:


uBCuDyCACpCACp

−−−−−+

+−+=

⊥

⊥⊥⊥⊥††

††

( ) ( )ppCACLppCACpp ˆˆˆ †† −−−=− ⊥⊥⊥

ppp ˆ:~ −=

( ) pCACLCp ~~ †⊥⊥ −=

p~We can see that ( the estimation error) is uncontrollable and is stable iff.

( )[ ] iCACLCi ∀<− ⊥⊥ 0Real †λ ppp →→ ˆ&0~


Note:

Define:

( )

( )

( ) ( ) ( )

( )

( ) ( ) ( )

HC

pnxpnCxppn

pnpxC

pnxpnCxppn

pnpxC

xnpn

pxn

nxn

pxppxp

VU

U

C

C

Σ

Σ

=

−−−

−

−−−

−

−⊥ ⊥⊥

0

0

0

0

( )

( )

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

−

=

−=

−−⊥−⊥−−−

−

− pxnxppnxnpn

pxn

xnpn

pxn

pnxpnxppn

pnpx

xnpn

pxn

CLC

C

C

C

IL

I

T

Cpxp

0:

Since:

( )[ ]

( ) ( )

=

− −−

⊥⊥− pn

p

pn

p

pn

p

I

I

IL

ICC

C

C

IL

I

0

000††

Define:[ ] [ ]

( )[ ]†††††

0: ⊥⊥

−⊥ +=

= CLCC

IL

ICCMH

pn

p


Note (continue – 1):

Define: CLCT −= ⊥:

Then:

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

uDyLpCACLCuDyLptd

d

−+−+−+

−−−=−−

⊥⊥⊥

⊥⊥

††

† ˆˆ

( )[ ] ( ) ( )uDyLCCuDyLpCx −++−−= ⊥⊥††† ˆˆ

[ ] [ ]†††: ⊥⊥+= CLCCMH

( )uDyLpz −−= ˆ:

( )

( )

−+=

+−+=

uDyHzMx

uBTuDyHATzMATtd

zd

KF

ˆ

( )( )

( )

( ) ATC

THMAT

C

THATMAT

IT

CMH

I

IMH

T

C

KF

npn

p

=

=

=

=

−

0

0

Those are the well knownReduced Order Observer

Equations


Note (continue – 2):

Then:

( )

( )

−+=

+−+=

uDyHzMx

uBTuDyHATzMATtd

zd

GF

ˆ

( )( )

( )

( ) ATC

THMAT

C

THATMAT

IT

CMH

I

IMH

T

C

GF

npn

p

=

=

=

=

−

0

0

( ) CGCHATTMIAT

TMATATTFAT

DHSuSyHzMx

DGBTJuJyGzFtd

zd

n ==−=−=−

−=++=

−=++=

:ˆ

:

=+=+

−==−

−

0DHS

ITMCH

DGBTJ

CGTFAT

valueseigenstablehasF

n

nxpnxmnxq

qxpqxmqxq

nq

HSM

GJF

xz

xxyHuSzMx

yGuJzFz

RRR

RRR

RR

∈∈∈∈∈∈

∈∈

→

++=

++=

,,

,,

,ˆ,

ˆˆ

SOLO ObserversGeneric Observer for a Linear Time Invariant (LTI) System

pxmpxnnxmnxn

pmn

DCBA

yux

uDxCy

uBxAx

RRRR

RRR

∈∈∈∈∈∈∈

+=+=

,,,

,,

Observer

nxpnxmnxq

qxpqxmqxq

nq

RSM

GJF

xz

xxyRuSzMx

yGuJzFz

RRR

RRR

RR

∈∈∈∈∈∈

∈∈

→

++=

++=

,,

,,

,ˆ,

ˆˆ

A Necessary Condition for obtaining an Observer is that (A,C) is Observable.

The Observer will achieve if and only if:

xx →ˆ

=+=+

−==−

−

0DRS

ITGCR

DGBTJ

CGTFAT

valueseigenstablehasF

n

L.T.I. System

[ ] [ ]†††: ⊥⊥+= CLCCMH CLCT −= ⊥:

HATGMATF == :&:


Let use a constant feedback from theReduced Order Observer to control the plant:

xK ˆ

The control is

xKrpCKxKru ˆ~† −=+−= ⊥

( )[ ]( ) ( )[ ]ppCpCuDyCKr

pCuDyCKrxKru

ˆ

ˆˆ

†††

††

−−+−−=

+−−=−=

⊥⊥

⊥

The augmented system is

( )

[ ]

[ ]

+

−=+=

−+=+−=

+

−

=

⊥

⊥⊥

⊥⊥

rDp

xCKKDCuDxCy

p

xCKKrpCKxKru

uB

p

x

CACLC

A

p

x

†

††

†

~

00

0~

SOLO Reduced Order Observers for Linear SystemsThe augmented system is

( )[ ]

[ ]

+

−=

+

−

+

−=

⊥

⊥⊥⊥

rDp

xCKKDCy

rB

p

xCKK

B

CACLC

A

p

x

†

†

† 000

0~

or

The poles of the closed loop system are given by:

( )

[ ]

+

−=

+

−

−=

⊥

⊥⊥

⊥

rDp

xCKKDCy

rB

p

x

CACLC

CKBKBA

p

x

†

†

†

00~

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

ObservertheofPoles

pn

ControllertheofPoles

n

pn

nCACLCIsKBAIs

CACLCIs

CKBKBAIs †

†

†

detdet0

det ⊥⊥−⊥⊥−

⊥ −−⋅+−=

−−

−+−

Hence the Reduced Order Controller has the “Separation Property” of the Controller andObserver.

SOLO Reduced Order Observers for Linear SystemsCompensator Transfer Function

By tacking the Laplace Transform of the compensator dynamics we obtain:

( )[ ] ( ) [ ]uBxACLCuDyLptd

d +−=−− ⊥ˆˆ


xKu ˆ−=( ) ppCCuDyCCxC

I

ˆˆˆ †

0

† =+−= ⊥⊥⊥⊥

( )[ ] ( ) ( ) xKBACLCuDyLxCs ˆˆ −−=−− ⊥⊥

( ) ( ) ( )[ ] ( ) xKBACLCKDLCsyLs xnpnˆ

−⊥⊥ −−−−=

( ) ( ) ( )[ ] ( ) ( ) 1†ˆ

mxxmpnpnnx yLsKBACLCKDLCsx −−⊥⊥ −−−−=where

Therefore

( ) ( ) ( )[ ] ( ) ( ) ( ) ( )[ ] ( ) ( )pnpnnxxnpn IKBACLCKDLCsKBACLCKDLCs −−⊥⊥−⊥⊥ =−−−−−−−− †

( ) ( ) ( )[ ] ( ) ( ) 1†

mxxmpnpnnx yLsKBACLCKDLCsKu −−⊥⊥ −−−−−=

SOLO Reduced Order Observers for Linear SystemsHow to Find K & L For a Stable Closed Loop

This can be done by solving the followingAsymptotic Minimum Variance Control Problem:

xKu ˆ−=

( ) ( ){ } ( ) ( ) ( ){ }

+=−===++=

uDxCy

tWwtwEtwExwuBxAx T τδτ 0,0,00

( ) ( ) ( ) ( ){ } )(0lim definitepositiveRtuRtutxQtxEJ TT

t>+=

∞→

System with no output noise to allow us to use a Reduced Order Observer.

The solution to this problem is:

where: PBRK T1−=and P is the solution of the Algebraic Riccati Equation:

01 =−++ − PBRBPQAPPA TT or:HT

T

ARicAQ

BRBARicP =

−−−

=−1

Minimize:

A stabilizing solution (and unique) exists iff:

1 (A,B) is stabilizable

2 AH has no jω axis eigenvalues

If Q ≥ 0 then P ≥ 0

SOLO Reduced Order Observers for Linear SystemsHow to Find K & L For a Stable Closed Loop (continue – 1)

( ) wCuBCuDyCACpCACpinputknown

⊥⊥⊥⊥⊥ ++−+=

1

††

( ) pCuDyCx ††⊥+−=and:xCp ⊥=

and

( )wuBxACxCp ++== ⊥⊥

( ) ( )[ ]{ }( ) wCuDuBCuDyCACpCAC

uDwuBpCuDyCACuDwuBxACuDxCy

inputknown

+++−+=

++++−=+++=+=

⊥

⊥

2

††

†† ˆThe measurements are given by (instead of )y y

Let define:

†*†*

****

:,:

:,:,:,:

⊥⊥⊥

⊥

==

====

CACCCACA

wCvwCwyypx


( ) wCuBCuDyCACpCACpinputknown

⊥⊥⊥⊥⊥ ++−+=

1

††

( ) wCuDuBCuDyCACpCACyinputknown

+++−+= ⊥

2

††

Define:

†*†*

****

:,:

:,:,:,:

⊥⊥⊥

⊥

==

====

CACCCACA

wCvwCwyypx

The Estimation Problem becomes:

( ) ∗∗∗∗ ++= winputknownxAx 1

( ) ∗∗∗∗ ++= vinputknownxCy 2

[ ]

=

=

⊥

⊥⊥⊥

**

**

00

00**

*

*

:RS

SP

CWCCWC

CWCCWCvw

v

wE

TTT

TTTT


The Estimation Problem:

( ) ∗∗∗∗ ++= winputknownxAx 1

( ) ∗∗∗∗ ++= vinputknownxCy 2

[ ]

=

=

⊥

⊥⊥⊥

**

**

00

00**

*

*

:RS

SP

CWCCWC

CWCCWCvw

v

wE

TTT

TTTT

The Solution to the Estimation Problem is:

( ) ( ) ( )[ ] ( ) 10†

0

1 −⊥⊥

−∗∗∗ +=+= TTTTCWCCACYCWCRCYSL

or

( )[ ] ( ) 10†

0

−⊥⊥ += TTT

CWCCCAYWCL

where

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

−−−−

−−=∗−∗∗∗∗−∗∗∗

∗−∗∗∗−∗∗∗

CRSASRSP

CRCCRSARicY

T

TT

11

11


In explicit form (the Algebraic Riccati Equation) is:

But

( )[ ] ( )[ ] ( ) ( )( ) 01111 =−+−−+− ∗−∗∗∗∗−∗∗∗−∗∗∗∗−∗∗∗ TTT

SRSPYCRCYCRSAYYCRSA

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

TTnxnpn

TT

CACCWCCWIC

CACCWCCWCCACCRSA

−⊥−

−⊥

⊥−

⊥⊥⊥∗−∗∗∗

−=

−=−†1

00

†1

00†1

( ) ( ) ( )( ) ( ) †1

0†

†1

0†1

⊥−

⊥

⊥−

⊥∗−∗∗

=

=

CACCWCCAC

CACCWCCACCRC

TTTT

TT

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

n

TTTTT

CWCCWCCWIC

CWCCWCCWCCWCSRSP

⊥−

⊥

−⊥⊥⊥

∗−∗∗∗

−=

−=−

0

1

00

0

1

000

1


Therefore Y(n-p)x(n-p) is given by the following (n-p) Algebraic Riccati Equation:

( )[ ] ( )[ ]{ }TTTn

TTn CACCWCCWICYYCACCWCCWIC †1

00†1

00 ⊥−

⊥⊥−

⊥ −+−

( ) ( ) YCACCWCCACY TTTT †1

0†

⊥−

⊥− ( )[ ] 00

1

00 =−+ ⊥−

⊥TTT

n CWCCWCCWIC

Note:

1 ( )[ ] ( ) ( ) ( )

( ) PCCWCCWI

CCWCCCWCWCCWCCWCCWICCWCCWIP

TTn

T

I

TTTTTn

TTn

=−=

+−=−=

−

−−−−

1

00

1

00

1

00

1

00

21

002 2:

This is a Projection, since P2 = P, but oblique because P isnot symmetrical.

2 For W0 = In we get: ( ) ( ) ⊥⊥−−

=−=−=− CCCCICCCCICCWCCWI nTT

nTT

n††11

00

( ) ( ) ( ) ††††1

0†

⊥⊥⊥−

⊥ = CACCACCACCWCCAC TTTTTT

( )[ ] ††††1

00 ⊥⊥⊥⊥⊥⊥⊥−

⊥ ==−−

CACCACCCCACCWCCWIC

pnI

TTn

( )[ ] TT

I

TTTn CCCCCCCWCCWCCWIC ⊥⊥⊥⊥⊥⊥⊥

−⊥ ==−

†

0

1

00


Hence, for W0=In, Y(n-p)x(n-p) is given by the following (n-p) Algebraic Riccati Equation:

Notice (continue – 1):

3 With this L , A*- L C* will have stable eigenvalues, but

2

( ) ( ) ( ) 0†††††

†

=+−+ ⊥⊥⊥−

⊥⊥⊥⊥⊥

⊥⊥

T

CCI

TTTCCYCACCACYCACYYCAC

n

( )[ ] ( ) ( ) ††0

1††

†

CACYCCCCAYCL TTCC

C

TTT

⊥

=−⊥⊥

⊥⊥

=+=

( ) †††** ⊥⊥⊥⊥⊥ −=−=− CACLCCACLCACCLA

Therefore has stable eigenvalues, and the Reduced Order Estimator is stable

( ) †⊥⊥ − CACLC


Notice (continue – 2):

4 Following P.J. Blanvillain and T.L. Johnson(IEEE Tr. AC., Vol. AC-23, No.1, June 1978) this Problem is equivalent to the following

( ) ( )

=+=xCy

WNxuBxAx 0,0~0Given

Find the Dynamic Compensator Parameters (F, G, H, M)

+=+=

yMzHu

yGzFzCompensator

Which minimizes the Quadratic Performance Index:

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

+= ∫∞

0

,,, dttuRtutxQtxEMHGFJ TT


Let append to the Reduced Order Observer the Stable Transfer Matrix

( ) ( )

=+−=

−

DC

BADBAIsCsQ

ˆˆ

ˆˆ:ˆˆˆˆ 1

=−−− uBCpCACyCACy ˆ††

The input to the Stable Transfer Function will be the same as for the Reduced Order Observer.

References

SOLO

Kwakernaak, H., Sivan, R., “Linear Optimal Control Systems”, Wiley Inter-science, 1972, pg.335

Reduced Order Observers for Linear Systems

Gelb A. Ed, “Applied Optimal Estimation”, The Analytic Science Corporation, 1974, pg.320

August 13, 2015 30

SOLO

TechnionIsraeli Institute of Technology

1964 – 1968 BSc EE1968 – 1971 MSc EE

Israeli Air Force1970 – 1974

RAFAELIsraeli Armament Development Authority

1974 –2013

Stanford University1983 – 1986 PhD AA

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