Dynamic Stochastic Estimation, Prediction and Smoothing … · 2019-09-25 · Dynamic Stochastic Estimation, Prediction and Smoothing DEEC/SSC, Spring 2003 PhD. Programme . ... Gram-Schmidt

Implementation Methods

Paulo Oliveira

Dynamic Stochastic Estimation, Prediction and Smoothing DEEC/SSC, Spring 2003

PhD. Programme

09/02/15 Paulo Oliveira 2

Implementation Methods Up to this point there has been presented the Kalman

filters as they are supposed to behave.

•  When implemented in computers, the observed mean-square estimation errors are (much) larger than the values predicted by the covariance matrix (EVEN with simulated data);

•  The variances of the filter estimation errors observed diverge from the theoretical values;

•  Solutions obtained for the Riccati equation have negative variance!

Facts from computer implementation examples:


Implementation Methods

A real problem: The Kalman filter is defined in terms of the real number

system, which has infinite precision... and then it is implemented on digital computers with finite

precision. Moreover, the REAL arithmetic of computers is not the

arithmetic of the real numbers. It is an arithmetic of floating-point number which are but a finite subset of the rational numbers.

Floats (4 bytes) can have a roundoff error in the order of 10-8


Error Analysis of Numerical Methods

For a procedure to solve a given problem, some features should be taken in consideration:

•  Numerical stability refers to robustness against roundoff

errors; •  Precision is also influenced by the procedural details of

the implementation method; •  Implementation methods cannot always be ordered

(results can depend on the problem at hand); •  Ill conditioned problems, or problems where the output of

a procedure (solution) is very sensitive to noise in the input data (problem).


Error Analysis of Numerical Methods

Example: The sensitivity of the solution to the linear problem

to uncertainties in A and b and roundoff errors is characterized by the condition number of A (for nonsingular A matrices)

It also equals the ratio of the largest and smallest singular value of A. Rule of thumb for the maximum relative error δ=||x-x||/||x||: where ε is the unit roundoff error and cA depends on the dimension of A. A problem is ill conditioned if cond(A)+1=cond(A) evaluates to true.

bAx =

xAxxAx

Acondx

x

/min/max

)( =

)(AcondcAεδ =


Ill conditioned Kalman Filtering problems

For Kalman filtering problems the solution of the associated Riccati equation should equal the covariance matrix of actual estimation uncertainty.

Factors that contribute to ill conditioning: •  large uncertainties in the values of A, Q, C and R. •  large ranges on the actual values of the systems matrices

due to poor choice of scaling or dimensional units; •  ill conditioning of the intermediate result R*=CΣCT+R; •  large matrix dimensions; •  poor machine precision.


Effects of Roundoff errors on Kalman Filtering

Floating-point Roundoff Errors b – base di – digit 0<di<b p – precision e – exponent

Intrinsic characteristics and errors associated to a representation: •  underflow limit bemin-1

•  overflow limit bemax •  rounding (to the nearest floating point number) •  chopping (towards zero)

ep bddd ±×± !21.

pb −= 1

21

ε

pb −= 1ε


Typical Floating-point Formats

Machine format ANSI/IEEE CRAY-1 IBM370

Single Double Single Double

Floating-

point

attributes

b p

emin

emax ε

2 24

-126

127 8106 −×≈

2 53

-1022

1023 1610−≈

2 48

-16384

8191 15103 −×≈

16 6

-64 63 7105 −×≈

16 14

-64 63 1610−≈


Bounds on Roundoff errors

Given a variable represented as a floating-point The magnitude of the roundof error will be bounded by The bound also holds for a m x n matrix A if the Frobenius

norm or the 2-norm are used.

;vv δ+

.vv εδ <

∑∑= =

=m

i

n

jijFAA

1 1

2

2

22 0

supxAx

xA

≠=


Roundoff Errors Propagation in Kalman Filters

Data flow:

(.)T

(.)T

(.)-1

z

C

A

GQGT

R

−− xCz ˆ )ˆ( −− xCzK

K

−x

−xC ˆ

TA

−P

+P

TAP+TAAP+

−CP

−KCP

TCP−

TCCP−

RCCP T +−

1)( −− + RCCP T

EST. LOOP

GAIN LOOP

+x



Data flow:

(.)T

(.)T

(.)-1

z

C

A

GQGT

R


K

−x

−xC ˆ

TA

−P

+P

TAP+TAAP+

−CP

−KCP

TCP−

TCCP−

RCCP T +−

1)( −− + RCCP T

EST. LOOP

GAIN LOOP

+x



Data flow:

(.)T

(.)T

(.)-1

z

C

A

GQGT

R


K

−x

−xC ˆ

TA

−P

+P

TAP+TAAP+

−CP

−KCP

TCP−

TCCP−

RCCP T +−

1)( −− + RCCP T

EST. LOOP

GAIN LOOP

+x


Numerical Analysis

The a priori value of the covariance P- is the one used in the Kalman gain computation. Its first order propagation error is considered next:

where Δ refers to the roundoff error added at each recursion

step. The norm of added roundoff errors is given by: Where ε1, ε2 are constant multiples of ε (the unit roundoff error).

( )[ ],

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

1

11

1

1TT

kkTT

kkkT

k

kkkT

kk

k

k

KCAPPAAPPAPKCAPKCAPKCAxCxzRCCPPxKCA

Px

−−−−+Δ+−−

−Δ+−++−

=

=−−−−

+−

−+

−−−−−

−+

−+

δδδδδ

δδδ

δ

δ

( ) ( )( ) ,

12

2

1

1

1−+

−

−−

−+

−+

+

+Δ+Δ++−

=

=Δ

kT

k

kk

k

k

PRCCPcond

zxCKzKzKxKCA

Px

ε

ε

δ


Example of filter divergence due to numerical errors

Given the scalar system where Q=0 and such that . P0 is so much greater than R that . •  Calculated variance equal to zero!

•  Actual variance

•  Theoretical variance in the exact case

,1

⎩⎨⎧

=

=+

k

k

k

k

xx

zx

RPo >>

0PR ε<

Exp Value

Exact Rounded

0

0)(

0

1)(

0

011

0

0111

0

00101

0

01001

000

000

000

RkPRPCPKPP

RkPPRCCPCPK

RPRPCPKPP

RPPRCCPCPK

PRPRCCPPPCCPPPCP

kkkk

Tk

Tkk

TT

T

T

T

+−=

++=

+−=

++=

++

−−

−−−

−

!!!

RRPRP

≈+0

0

kR

RkPRP

≈+0

0


An Overview of Factorization Tricks

The more numerically stable implementations of the Kalman filter use one or more of the following techniques to solve the associated Riccati equation:

1.  Factoring the covariance matrix of state estimation into Cholesky factors (triangular factors CCT=M);

2.  Modified Cholesky decomposition algorithms (M=DDUUT or M=LDLLT);

3.  Factoring the covariance matrices of the measurement and state noises, Q and R, respectively;

4.  Symmetric matrix square roots of element matrices (in the form );

5.  Triangularization (QR decomposition); 6.  Gram-Schmidt ortonormalization (ortho times triang.) ...

TvvI σ−


Cholesky Decomposition Methods and Applications

Objective: To find C such that where C is called Cholesky factor (not unique). If suitable constraints are used

a unique solution can be found. Decomposition methods Example: Solution:

MCCT =

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

+++

+=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

233

222

221322231213111

222

2212111

211

333231

2221

11

333231

2221

11

333231

322221

312111

000

000

ccccccccccccc

c

ccccc

c

ccccc

c

ppppppppp T

22122

1121

11

31

21

11

/cpcpp

ccc

−=

=

= ( )232

23133

22312132

33

32 /ccpcccp

cc

−−

−

=

=


Decorrelating the Components of Vector-Valued Measurements

Suppose that the observations are given by z=Hx+v

where E[vvT]=R is not a diagonal matrix, but can be factored as Where D is diagonal and U is an unit upper triangular matrix. Redefine Solve

,TUDUR =

.)()( 11

1

vxHvUxHU

zUz

+=

+=

=−−

−

[ ][ ][ ]

( )D

UUDUURUU

UvvEUUvvUEvvER

TT

T

TT

TT

T

=

=

=

=

=

=

−

−

−

−

−

−

−

−

1

1

1

1

1

1

1

1

HHUzzU

=

=


An Overview of Factorization Tricks

The more numerically stable implementations of the Kalman filter use one or more of the following techniques to solve the associated Riccati equation:

1.  Factoring the covariance matrix of state estimation into Cholesky factors (triangular factors CCT=M);

2.  Modified Cholesky decomposition algorithms (M=DDUUT or M=LDLLT);

3.  Factoring the covariance matrices of the measurement and state noises, Q and R, respectively;

4.  Symmetric matrix square roots of element matrices (in the form );

5.  Triangularization (QR decomposition); 6.  Gram-Schmidt ortonormalization (ortho times triang.) ...

TvvI σ−


Modified Cholesky (UD) Decomposition Algorithms

Objective: To find U (unit upper triangular matrix) and D (diagonal matrix) such that the mxm symmmetric matric M is factored as

With these constraints a unique solution can be found. Solution:

for j=m,...,1 for i=j,...1

σ=Mij

for k=j+1,...,m σ=σ-UikDkkUjk

if (i=j) Djj=σ Ujj=1 else Uij=s/Djj

TUDUM =

Computational complexity

flopsmmm )4)(1(61

+−

No square roots!


Symmetric matrix square roots of elementary matrices

The square of a symmetric elementary matrix verifies with The symmetric square root of a symmetric elementary matrix with

( ) ( )( )( )

T

T

TTT

svvIvvvIvvIvvIvvI

−

−−

−−

=

=

=−22

2

2 σσ

σσσ

.2 22 vs σσ −=

.112

2

v

vs−+=σ

( ) ( ),21

TT vvIsvvI σ−=−


Symmetric Matrix Square Factorization*

* Introduced by J. Potter in 1966.

Define the Cholesky factor of the covariance matrix P: so that the observation update Riccati equation can be factored as and define The previous expression can be written as For the case that the measurement is a scalar , Potter was able to write this expression as resulting that

a Cholesky factor !

( ) ( ) ,and TT PP +++−−− ΣΣ=ΣΣ=

( ) ( ) ( ) ( )( ) ( )TTTTTTT CRCCC −−−

−−−−−−++ ΣΣ+ΣΣΣΣ−ΣΣ=ΣΣ1

( ) .TT CV −Σ=

( ) ( )( )( ) .1 TTTn

T VRVVVI −−−++ Σ+−Σ=ΣΣ

( ) ,1 TTTn WWvRvvvI =+−

−

( ) ( ) .WorWW TTT −+−−++ Σ=ΣΣΣ=ΣΣ


Symmetric Matrix Square Factorization*

For the scalar case, an symmetric elementary matrix of the form where v is a column vector, is obtained. Using the results for the symmetric elementary matrices and The radicand verifies

,2vRvvIT

n+

−

,112

2

v

vs−+=σ .1

2vRs

+=

.011 22

22

≥+

=+

−=−vRR

vRv

vs


Bierman UD Factorization

Define the covariance matrix P using UD factors as: so that the observation update equation can be factored as and define as an n-vector, where n is the state

vector size. The previous expression can then be written as

( ) ( ) ,and TT UDUPUDUP ++++−−−− ==

( ) ( ) ( ) ( )( )( ) 1−−−−

−−−−−−−−−+++

+−=

RCUDCU

UDCUCUDUUDUDDUTT

TTTTT

( ) TT CUv −=

( )( )

( ) .1

T

T

TT U

RvDvDvvDDUDDU −

−−

−−−−+++

⎟⎟⎠

⎞⎜⎜⎝

⎛

+−=



The following unfactored expression is present: If it is possible to write it in the form: then D+ is the a posteriori D factor of P, because the resulting

equation can be solved for the a posteriori U factor as

.BUU −+ =

( ) ( )( )( ) ( )T

TT

BUDBUUBBDUUDU−+−

−+−+++

=

=

( ) −−−−− +− DvRvDvvDD TT 1

( ) BBDDvRvDvvDD TT +−−−−− =+−1



It suffices to find a numerically stable and efficient method for the UD factorization of a matrix of the form

Lemma – If

then for

,12

1

12

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

+

+=

∑∑

=

−

−

=

−−+

j

k kkk

j

k kkkjjjj

DvR

DvRDD

( ) .1 −−−−− +− DvRvDvvDD TT

( ) BBDDvRvDvvDD TT +−−−−− =+−1

,1forand1 njinj <≤≤≤≤

.1

12∑

−

=

−

−+

+−= j

k kkk

jijjij

DvR

vvDB


Numerical Analysis of Square Root Methods

The a priori value of the covariance P- is the one used in the Kalman gain computation. Its first order propagation error using a square root covariance factorization method is:



( )[ ],

)()()()()(

1

11

1

1

+−

−+

−−−−−

−+

−+

Δ+−−

−Δ+−++−

=

=

kT

k

kkkT

kk

k

k

PKCAPKCAPKCAxCxzRCCPPxKCA

Px

δ

δδδ

δ

δ

( )( ),

1 131

Cholesky

kT

kk

PRCCPcondP

Σ

++=

−+

−

−+

εδ

( ) ( )zxCKzKzKxKCAx kkk +Δ+Δ++−=Δ −−−+ 11 ε


Numerical Analysis

The a priori value of the covariance P- is the one used in the Kalman gain computation. Its first order propagation error is considered next:



( )[ ],

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

1

11

1

1TT

kkTT

kkkT

k

kkkT

kk

k

k

KCAPPAAPPAPKCAPKCAPKCAxCxzRCCPPxKCA

Px

−−−−+Δ+−−

−Δ+−++−

=

=−−−−

+−

−+

−−−−−

−+

−+

δδδδδ

δδδ

δ

δ

( ) ( )( ) ,

12

2

1

1

1−+

−

−−

−+

−+

+

+Δ+Δ++−

=

=Δ

kT

k

kk

k

k

PRCCPcondzxCKzKzKxKCA

Px

ε

ε

δ

TTkk

TTkk KCAPPAAPPA ))(()( −−−− −−−− δδδδ


•  Kalman Filtering: Theory and Practice - M. Grewal and A. Andrews, Prentice Hall, 1993;

•  Applied Optimal Estimation, A. Gelb, MIT Press, 1974;

•  Optimal Filtering, B. Anderson and J. Moore, Prentice Hall, 1979.

•  Factorization Methods for Discrete Sequential Estimation, G. J. Bierman, New York: Academic Press, 1977.

Bibliography

1!

H∞ Filtering and Smoothing"

PAULO OLIVEIRA!!

DEEC/IST and ISR!!

Last Revised: November 10, 2007!Ref. No. NLO#1!

2!

Theme"

•  KF only corresponds to the optimal filtering strategy under restrictive assumptions, and for some objectives (functionals)!

•  The requirement on the knowledge of the power spectral density of the disturbances is too restrictive. The WNG assumption too.!

•  Unknown multimodal and/or skewed pdfs are common!

However!

•  Optimality and stability still of great importance, in the presence of uncertainty (robustness)!

•  Other functionals / objectives can be used to formulate estimation problems. Minimization must be feasible!

3!

Norms of Signals"

L1 [0,T] norm !!!!L2 [0,T] norm !(energy) !!! ! ! ! ! ! ! ! ! !(1)!

!L∞ norm !(least upper bound)!

( ) ∞<= ∫T

dttuu0

1

( ) ( ) ∞<⎟⎟⎠

⎞⎜⎜⎝

⎛= ∫

21

02

TT dttutuu

( )( ) ∞<=∞

tuutsup

4!

Motivation for H∞ Filtering "

For finite energy signals in the input of system G, how much is the minimum energy on the output? !

!Possible interpretation as a Min-max Nash game in estimation:!!Maximum energy in the error is minimized.!! ! ! ! !!

For bounded systems, the H∞ norm is defined as!!! ! ! ! ! ! ! Denominated as the L2 !! ! ! ! ! ! ! induced norm.!

!For LTI systems corresponds to the peak in the Bode diagram.!

G u ∈ L2 y ∈ L2

2

2sup0, 22 uuLu

GuG

≠∈∞=

5!

Norms of Systems"

H2 norm!!!!!!!H ∞ norm!!

( )( )

( )( )

( )⎩⎨⎧ +

=

=Σ

tButCxtAx

tytx!

:

model time-Continuous LTI

G ( ) ( ) BAsICsG 1

functionTransfer −−=

( ) ( )( )

( )( )2/1

2

2/1

*2

21

21

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

∫∑

∫∞+

∞−

+∞

∞−

ωωσπ

ωωωπ

djG

djGjGtr

ii

G

( )[ ]ωσω

jGmaxsup=∞

G

6!

Input-output Relations*"

Stochastic!

Stochastic! ?!

?!

?!

( ) ( )ttu δ=

2u

2G

2y

∞y

2G

∞G

∞G

2G

*See [1] for details

∞u

1G∞"∞"

∞"

7!

Plant and Sensor Modeling "

( )( )

( ) ( )( ) ( )

( ) ( )( ) ( ) [ ]

( )( )( )

( ) ( ) ( ) ( )( ) ( ) ( ) estimated. be ointerest t ofquantity

functions. bounded continuous piecewise ,,,

,0:

txtLtztDtCtBtA

mRtw

pRty

nRtxTt

twtDtwtB

txtCtxtA

tytx

=

∈

∈

∈

∈+

+

=

=Σ

⎩⎨⎧ !

G

Assume that !!!

A, B( ) is stabilizable

A,C( ) is detectableD t( )

B t( )T

D t( )T

!

"

###

$

%

&&&= 0

I!

"#

$

%& ,

Independent plant/measurement noisesNormalized measurement noise

8!

H∞ Filtering"

( )( ) ( )

.0,

0'0

22

22ˆ

20sup22

2

22ˆ

20sup1

**

indeces case)(worst theminimizes that tz of tz estimatean obtain ,ty tsmeasuremen theusing and conditions initial

(unknown)known with systemFor *

>+

−

∈≠=

−

∈≠= R

Rxxw

zz

LwJor

w

zz

LwJ

G, :statement Problem

Important questions:!

•  Given γ>0, does there exist a filter with finite J1 (or J2)?

•  Under the assumptions , does it verifies J1<γ 2 (or J2<γ 2 )?!

•  How to find a realization for such filter?!* Considered 0 without loss of generality!** R-1 is a covariance matrix!

!

9!

H∞ FilteringFinite Horizon, Known Initial Conditions"

( ) [ ]

bygiven is for filter one exists,it if Moreover,)2

satisfies that 0for matrix symmetric a exists thereifonly and if such that filter aexist There)1

. andknown be conditions initial Let the

21

21

γ

γ

<

∈

<

∞<

J

,TttPJ

T :Theorem[2]

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

( ) ( ) ( ) ( ) ( ) ( ) ( ) . 00with 12 =++

−+=

PtBtBtPtLtLtP

tPtCtCtPtAtPtPtAtP

TT

TT

γ

!

x.

t( ) = A t( ) x t( ) + P t( )CT t( ) y t( ) − C t( ) x t( )"# $% with x 0( ) = 0.

(2) (3)

Null initial conditions considered without loss of generality.!

10!

Elements of Proof (I)"

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

( ) ( )

( ) ( )[ ] ( ) ( ) ( )( )∫ =

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎦

⎤⎢⎣

⎡⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

=−

−==

=−

T TTT

twtx

I

tLtLtwtx

,T][L

wtxtL

txtxtxtxtLtzΣ

w

zz

J

0

2

2

2

2

2

22

22

2

2

2

1

0~

0

01~

can write we(1) definition norm 0 theUsing

.0~1

ˆ~ and ˆˆ gintroducin From

.ˆ

as written becan functional theof valueThe

γ

γ

γ

,G

(4)

11!

Systems’ Theory Digression"

A system is dissipative if can not provide to the environment the !same energy that was suplied by the exterior – energy losses.!!Examples: electrical circuits, mechanical systems, thermodynamics...!

( )

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

system. theof ies trajectorallfor and 10 tallfor

such that function negative-non a exists thereif edissipativ

strictly is ratesupply with system The - [6]ity Dissipativ

101

0

,

t

RV: x

tsz: w

txVdttytustxVt

t

<

→

→

>+ ∫

G

( ) ( ) ( )

( ) ( )( ) ( )( ) ].,[any for ,0, From

.

101

0

holds e,iabldifferent is if Moreover,

tttdttxVtytus

tstVtVt

t∈<+−

<

∫ !

!

12!

Lyapunov Stability – Second Method"

•  Note that V(t)à ∞ as ||x||2 à ∞.!

•  Stability of dynamic systems can be studied, whitout solving the differential equations. Sufficient conditions.!

•  No systematic method to find a Lyapunov function exists.!

( )( )( )( ) S. of solutions all along ,0

0eshat verififunction t Lyapunov a exists thereif stable is 0

point mequilibriu The - method) (second theoremLyapunov

2

0

)))

≤

>

=

=

txVxxV

V

iiiiii

x

!α

( ) ( ) ( ) .tx0x:,0,

if stable is point mequilibriuAn -Stability Lyapunov

0

0εδε εδ <⇒<∃>∀ >

=

t

x

13!

Elements of Proof (II)"

( ) ( ) ( ) ( )

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

( )∫ =⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧+⎥⎦

⎤⎢⎣

⎡⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

−

= −

T TTT

T

dttVdtd

twtx

I

tLtLtwtx

txtPtxxV

0

2

1

0~

0

01~

used is ~~function candidate Lyapunov the conceptsity dissipativ toresorting and (4) nginterpreti-Re

γ

( ) ( )Therefore

~~is dynamicserror the(3), and of definition theFrom

wDPCBxCPCAx TT −+−=!G

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )txtPtxtxtPtPtPtxtxtPtx

txtPtxtxtPtxtxtPtxtVTTT

TTT

!!!!!!!

~~~~~~~~~~~~11111

111

−−−−−

−−−

+−

++

=

=

14!

Elements of Proof (III)"

Re-arranging the terms results

!x t( )Tw t( )

T!

"#

$

%&P

!x t( )w t( )

!

"

###

$

%

&&&

'

()

*)

+

,)

-)

dt0

T

∫ = 0,

where*

P =1γ 2

LT L + AT P−1 − 2CTC − P−1 "PP−1 + P−1A P−1B − CT D

BP − DTC − I

!

"

###

$

%

&&&.

Using (2) and cancelling terms

!x t( )Tw t( )

T!

"#

$

%& −CTC − PBBT P P−1B − CT D

BP − DTC − I

!

"##

$

%&&

!x t( )w t( )

!

"

###

$

%

&&&

(

)*

+*

,

-*

.*

dt0

T

∫ = 0

*Time-dependence omitted for simplicity.!

15!

Elements of Proof (IV)"

Using Schur complements

−CTC − PBBT P−1 + B − CT D( ) BP − DTC( ) = 0

−CTC − PBBT P−1 + PBBT P−1 − P−1BDTC − CT DBT P−1 + CTC = 0using the noises independence and normalization assumptions

0 = 0 . q.e.d.

. is matrix of

scomplementSchur the0 Z where,R Zand ,RW,RV ,R UmatricesGiven - sComplementSchur

1

mxmmxn

nxmnxn

WVZUZWVU −−⎥⎦

⎤⎢⎣

⎡

>∈∈

∈∈

Can be seen as a generalization to the matrix inversion lemma. !

16!

Visualization of the H ∞ Filter"

B(t) w(t) x(t) x(t) Δ y(t) C(t)

A(t)

r (t) x(t) x(t) z(t) C(t)

A(t)

y(t)

-

.

^ ^ ^ .

H ∞ gain

REAL

H ∞

Δ P(t)CT(t)

D(t)

17!

Discussion"•  Optimal structure obtained, similar to LTV Kalman filter!

•  Unbiased estimator obtained (otherwise J1 à ∞, J2 à ∞)!

•  Complete proof is out of scope, but can be obtained !i.  using systems’ theory [2, 5]; !ii.  using estimation tools in Krein spaces [3]; !!•  Stationary solutions can also be obtained (finite or infinite

horizon cases)!

•  Modified Riccati equation that!For γ ◊ ∞ degenerates on the Riccati equation in KF Provides more robust solutions, for smaller γ Unfeasible for γ< γ min !!!!

18!

H∞ SmoothingFinite Horizon, Known Initial Conditions"

( ) [ ]

is verifiesand minimizeshat smoother t One)2

satisfies that 0for matrix symmetric a exists thereifonly and if such that smoother aexist There)1

. andknown be conditions initial Let the

211

21

γ

γ

<

∈

<

∞<

JJ

,TttXJ

T :Theorem[2]

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

( ) ( ) ( ) ( )

( ) .0with

12

=

+−

−+=−

TX

tCtCtLtL

tXtBtBtXtAtXtXtAtX

TT

TT

γ

!

( )( )

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

( )( ) ( )

( )

( ) ( ) .0 ,00ˆwith

0ˆˆ

==

⎥⎦

⎤⎢⎣

⎡+⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡

−=⎥

⎦

⎤⎢⎣

⎡

tλx

tytCt

txtAtCtCtBtBtA

ttx

TT

T

λλ !

!

!

!

19!

Remarks"

•  Proof is omitted, see [2] for details.!

•  The H ∞ smoother structure is equal to the H2!!

•  Smoothers for all 4 cases are well known.!

•  Much more recent results than the H2 solutions!

•  Other functionals have already been solved, e.g. mixed H2/H ∞ Also, solutions for nonlinear cases available!

•  Now a couples of examples from [5] are included to document some of the results outlined!

20!

Examples, from [5]"

21!

Examples, from [5]"

22!

References"

[1] J. Doyle, B. Francis, and A. Tannenbaum, Feedback Control Theory, Macmillan Press, 1990 !

[2] K. Nagpal and P. Khargonekar, “Filtering and Smoothing in the H∞ Setting,” IEEE TAC, vol 36, n2, Feb. 1991!

[3] B. Hassibi, A. Sayed, and T, Kailath, “Linear Estimation in Krein Stpaces – Part I Theory,” IEEE TAC, vol 41, n1, Jan. 1996!

[4] C. Scherer, P. Gahinet, and M. Chilali, “Multiobjective Output-feedback Control Via LMI Optimization,” IEEE TAC, vol 42, n7, 1997!

[5] U. Shaked and Y. Theodor, “H∞-Optimal Estimation: A Tutorial,” Proc. CDC, pp. 2278-2286Arizona, USA,1992. !

[6] J. Willems, “Dissipative dynamic systems,” Arch. Rational Mech. Anal, Vol. 45, pp. 321-393, 1971.!

1!

New Methods for State Estimation!

PAULO OLIVEIRA!!

DEEC/IST and ISR!!

Last Revised: December 14, 2007!Ref. No. NLO#2!

2!

Key Challenges in Estimation!

Characteristics of the envisioned Estimator!!•  Reduced computational requirements!•  Causal (to be used during the mission)!•  Possible to be refined in post-processing!!In the linear case, all relevant features !are obtained together: exponential stability, !optimal performance and robustness !(gain and phase margins).!!!

Stability

Performance

Robustness

In the nonlinear case no optimal common!solution is available.!

!(e.g. EKF is the performance tentative solution).!

3!

Theme!

•  Stochastic H2 filtering, prediction, and smoothing problems are only optimal for linear time-varying systems under Gaussian disturbance assumptions with known power spectral densities!

•  H∞ allows to lift the noise assumptions for LTV systems!

•  Real world systems are nonlinear!!!•  In general, EKF does not guarantee stability, performance, nor

robustness!

•  Nonlinear observers can outperform linear or linearized versions of observers (EKF / SOF), both for structured and unstructured disturbances [1, 2]!

4!

Exponential Observers for Linear Systems!

Consider the linear system !!!!!!The Luenberger observer, in a deterministic setup, is given by !!!Exponential stability can be proven resorting to the Lyapunov equation!!!That is, for a positive definite matrix Q there exists a unique positive

definite P, such that the above equation is verified.!

( )( )

( )( )

( ) ( )( )( )

( ) .observable is pair thewhere

:

A,C

mRtu

pRty

nRtxtBu

tCxtAx

tytx

∈

∈

∈+

=

=Σ

⎩⎨⎧ !

L

( ) ( ) ( ) ( ) ( )( )txCtyKtButxAtx ˆˆˆ −++=!

( ) ( ) QKCAPPKCA T 2−=−+−

5!

Observers for Nonlinear Systems!

Consider the class of affine nonlinear systems!!!!!!!!The suggested Luenberger-like nonlinear observer would be!! ( ) ( )( ) ( )( ) ( ) ( ) ( )( )( )txhtytutxgtxftx ˆˆˆˆ −++= K!

( )( )

( )( )

( ) ( ) ( )( )( )

( ) ( ) ( ) are known.where

mRtu

pRty

nRtxtuxg

xhxf

tytx

. h, .,g.f and

:∈

∈

∈+

=

=Σ

⎩⎨⎧ !

N

What fails in the stability proof for this nonlinear observer?

6!

Lipschitz Nonlinear Observers!

PAULO OLIVEIRA!!

DEEC/IST and ISR!!

Last Revised: December 14, 2007!Ref. No. NLO#2!

7!

Thau’s Observers @ 1973!

Consider the class of autonomous nonlinear systems !!!!!!!!!!!Proposed observer (motivated by Luenberger’s and Kalman’s work)!!

ΣG :!x t( )y t( )

==

Ax t( )Cx t( )

+ f x t( )( )"#$

%$

x t( )∈Rn

y t( )∈R p

A,C, and f .( ) are known; the pair A,C( ) is observable, u is a

deterministic input, and f .( ) is a Lipschitz time-invariant

function, i.e.

( ) ( ) ( )( ) ( ) ( )( )txCtyKttxAtx xf ˆˆˆ ˆ −++=!

(1) (2)

( )( ) ( )( ) ( ) ( )txtxLtxftxf ˆˆ −≤−

8!

( )

stable.cally assymptoti is (2)observer Thau then the allfor

satisfy oequation t Lyapunov theofsolution themakecan it such that selected is gain theif (1), systems of class For the

min

x

LPQ

K

>λ

:Theorem[2]

. where verified,is,2

equation Lyapunov thesuch that ,matrix definite positive unique a exists there,matrix definite positiveany for (1) class

of systemsfor condition ity observabil Given the

0

00

KCAAQPAPA

PQ

T

−=

−=+

:nPropositio

Main result:!

(3) (4)


9!

Elements of Proof (I)!

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

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

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

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

andtxftxfxAtxftxfxKCAtx

txPtxtxPtxtxPtxdtdtV

PtxPtx

txtxtx

TTT

TTT

T

ˆ~2~~

,ˆ~ˆ~~by given is dynamicserror The

~~~~~~

method second sLyapunov'apply lets matrix,constant symmetric a is 0 where,~~tVfunction

candidate Lyapunov for the and,ˆ~ Defining

00

0

−++=

−+=−+−=

+==

>=

−=

!

!

!!!

10!

Elements of Proof (II)!

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

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )[ ] ( )( ) ( ) verified.be (4)

t enough tha isit i.e. ,0 then if

~2

~~2~2

~~2~~2

condition Lipschitz theFromˆ~2~~2

can write one (3)equation Lyapunov theFrom

min

2min

2min

<>

−−≤

+−≤

+−≤

−+−=

tVPLQ

txPLQ

txPtxLtxQ

txPtxLtxQtxtV

txftxfPtxtxQtxtV

T

TT

!

!

!

λ

λ

λ

results. veconservati very toleadcan it Moreover, process.error and triala sconstitute and on y indirectel impacts of choice The PK

11!

Visualization of the Filter!

x(t) x(t)

Δ y(t) C(t)

A(t)

r (t) x(t) x(t) z(t)

C(t)

A(t)

y(t)

-

.

^ ^

^ .

Gain

REAL

Δ L

f(x(t))

f(x(t)) ^ NLO

12!

Example with difficulties [4]!

13!

The minimum occurs when Q=I.!

Example with difficulties [4]!

The method is not effective, however given L and P, is of some use to proove stability of the observer.

14!

Consider the class of nonlinear systems !!!!!!The proposed observer has the structure!!

( )( )

( )( )( )( )

( )( )

( ) ( ) able.differentily continuous are and where

:

.h.f

pRty

nRtxtxhtxf

tytx

∈

∈=

=Σ

⎩⎨⎧ !

K

( ) ( )( ) ( ) ( )( )( )( ) able.differentily continuous also is . where

ˆ,ˆ ˆg

gxf txhtyttx +=!

(5) (6) (7)

Exponential Observers @ 1975!

( ) ( )( )( ) ( )( ) ( )( )( ) ( ).00ˆ that guarantee toimpossible isit however

ˆ if 0ˆ, thatgeneralin verify shouldobserver An

xxtxhtxhtxhtyg=

==

15!


( )( ) ( )( ) ( )

( ) ( ) tketxtx

RxxVxVb

VRxxcxVxxxxVg(.)

n

n

λ

λλ

ρ

−≤−

>∈−≤

=∈≥

>−=

ˆ

and stablelly exponentia is (6)observer then the.0 somefor and ~ allfor ,~~)

00,~ allfor ,~~ a)such that 1 and ˆ~ where,~

functionscalar a exists thereif (7), satisfying function some and(6) and (5) systems autonomousnonlinear Given the

!

:Theorem[3]

The generic proof for autonomous systems is an immediate consequence of the Lyapunov second method.!

16!


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

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

( ) ( ) ( ) ( ) ( )

ly.respective , of seigenvaluelargest andsmallest theare and where, allfor

ˆˆ

verifyingobtained is (6)observer lexponentiaan ,ˆany From2

such that 0 exits there0given that,,matrix gain x an exists thereif

,ˆˆ with (6)observer proposed theand (5) system autonomousnonlinear Given the

210

000

2/1

1

2

0

Qqqtt

txtxtteqqtxtx

txQxgKxfPPxgKxf

PQKmn

txhtyKtx,htyg

T

>

−−−

⎟⎟⎠

⎞⎜⎜⎝

⎛≤−

−=∇−∇+∇−∇

>

>

−=

λ

:Theorem[3]Main result:!

(8)

17!


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

( ) ( ) ( )

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

( )( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )xhxhKxfxfPx

xPxhxhKxftxf


txPtx

txhtyKtxftxftx

txtxtx

T

T

TTT

T

ˆ(ˆ~~ˆ(ˆ

~~~~~~

method) (2matrix constant symmetric a is 0P where ,~~tVfunction candidate Lyapunov For the

ˆ(ˆ~by given is dynamicserror the,ˆ~ Defining

nd

−−−+

−−−=

+==

>

=

−−−=

−=

!!!

!

* Explicit dependence on t will be omitted. !

18!


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

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

( ) ( ) .0for ,2~~2

results ~~~ fact that theand (8) Using

~~

Therefore .ˆ1for

~ˆˆ

calculus integral of theoremlfundamenta the toResorting

1

2

2

1

0

1

0

>−<−<

>

∇−∇+∇−∇=

−+=

∇−∇=−−−

∫

∫

εεε tVqqxQxtV

xεxPx

xdswhKwfPPwhKwfxtV

xs)(sxw

dsxwhKwfxhxhKxfxf

T

T

ssT

ssT

s

ss

!

!

19!

Example, from [3]!

20!


Consider the class of autonomous nonlinear systems !

( ) ( )( )

( )

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

( ) .0.,..

,,

2

22

131

321

such that and , where

verifiesandarguments itsin Lipschitz is andknown are and where

:

:,,:

=∇

∈∈

+∇+=

⎪⎩

⎪⎨⎧

∈

∈+

=

=Σ

yyCC

xyyyyyx

.

C

CA

pRtyy

nRtxxyyxCxAx

yx

!

!!

!!

φ

φφφ

φφφφ

φ

φG (9)

(10)

21!


( ) ( )

( )( )

(9). systemfor observer lexponentiaan exist then there

and

such that r gain vecto a and ,00exist there)

observable is pair thea)if (10) verifying

(9), systems autonomousnonlinear For the

3max

min

2

∞∇>

>>

−=−+−

φλλ

PQ

K, QPb(A,C)

QKCAPPKCA T

:Theorem[3]

(11)

22!


( )( )

( ) ( ) ( ) ( )

( ) ( ) ( )( ) observer the, that ideringCons

tosimplified becan That

verifiessignal new thisof time, torelative ,derivative The

thatnote and ,y variablenew theDefine

21

2321

2321

22

2

yCyCwywyAyAww

yyxyyyAxw

yy

xw

φ

φφφφ

φφφφ

φφ

φ

−==

++++=

∇−+∇++=

∇=

−=

!

!!!

!!

( ) ( ) ( ) ( )wCyKwyAywAw ˆˆˆˆ 12321 −−++++= φφφφ!

proposed. is

23!


( ) ( ) ( )( )

( )

( ) ( )( ) ( ) ( )( )23232323

nd

2323

ˆˆ

2

method) (2matrix constant symmetric a is 0P where ,tVfunction candidate Lyapunov For the

ˆby given is dynamicserror the,ˆ Defining

φφφφφφφφ

φφφφ

+−+++−+

+−=+=

>

=

+−++−=

−=

wwPePeww

QeeePePeetV

PeewweKCAe

wwe

TT

T

TT

T

!!!

!

24!

Elements of Proof (III)!

( ) ( )( )

( )( )

( )[ ] 23min

1

023

1

023min

22

2

ePQ

dseywPe

PedseywQV

sT

T

s

∞∇+−≤

⎟⎟⎠

⎞⎜⎜⎝

⎛+∇+

⎟⎟⎠

⎞⎜⎜⎝

⎛+∇+−≤

∫

∫

φλ

φφ

φφλ!

From this relation (11) is immediate.!

25!

Example, from [3]!

26!

Lipschitz Observers @ 1998!

Consider the class of non-autonomous nonlinear systems !!!!!!!!!!!Proposed observer (motivated by Luenberger’s and Kalman’s work)!!

( )( )

( )( )

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

( ) ( )( )

i.e. function,invariant - timeLipschitz a is and input, ticdeterminis

a isu ,observable is CA, known; are and ,,

,:

..fCA

mRtu

pRty

nRtxtutx

tCxtAx

tytx

φ

φ

∈

∈

∈+

=

=Σ

⎩⎨⎧ !

G

( ) ( ) ( ) ( )( ) ( ) ( )( )txCtyKtttxAtx ux ˆ,ˆˆ ˆ −++= φ!

(12) (13) (14)

( ) ( )( ) ( ) ( )( ) ( ) ( )txtxtutxtutx ˆ,ˆ, −≤− γφφ

27!

( )

stable.ally asymptotic isit then

minensure toassuch that chosen becan gain the)

observable is pair thea)satisfies (14)by given observer theif (13), g verifyin(12),systems autonomous-nonnonlinear For the

min γIjωKCAσKb(A,C)

Rω>−−

+∈

:[5] Theorem

(15)

Lipschitz Observers @ 1998!

( ) ( ) ( ) ( ) ( )( ) ( ) ( )( )[ ]tutxtutxtxKCAtx ,ˆ,~~ φφ −+−=!

The estimation error dynamics is!!!!and the following major result holds:!!

28!


The proof is done in three parts (see [5] for details):!i)!!!!!!!!ii) !

( )

( )( )

s.eigenvalueimaginary no has

matrix thesuch that 0

exits e then ther,min

2

min If

⎥⎦

⎤⎢⎣

⎡

−−−=

>

>−−+∈

T

R

A-KCεIIIγA-KC

H

IjKCA

ε

γωσω

( )

( ) ( ) 0

equation theosolution t a exits theresuch that 0P a exists e then therstable is

2

If

=+++−+−

>−

εIIPPγKCAPPKCA

KCA

T

29!


cont...!iii)!!!!!!!!!!Using the properties introduced before, results:!!!!!

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

( ) ( ) ( )

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

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

tutxtutxtxKCAtx

PxKCAPxxPKCA


txPtx

txtxtx

T

TT

TTT

T

,ˆ,

,ˆ,~~

~2

~~

~~~~~~

method) (2matrix constant symmetric a is 0P where ,~~tVfunction candidate Lyapunov For the

by given is dynamicserror the,ˆ~ Defining

nd

φφ

φφ

−

−+−=

+−+−=

+==

>

=

−=

!!!

!

!V t( ) ≤ "xT A− KC( )T P + P A− KC( ) + γ 2 PP + I$

%&'()"x

30!

References!

[1] P. Kokotovic , The Joy of Feedback: Nonlinear and Adaptive, Macmillan Press, 1991 Bode Prize Lecture, 1992.!

![2] F. Thau, ”Observing the state of nonlinear dynamical systems,”

International Journal of Control, pp. 471-479, 1973. !![3] S. Kou, D. Elliot, and T. Tarn, ”Exponential observers for

nonlinear dynamic systems,” Journal of Information and Control pp. 204-216, 1975. !

![4] Wen Yu, ”Nonlinear Observer,” class notes, CINVESTAV-IPN.!![5] R. Rajamani, ”Observers for Lipschitz Nonlinear Systems,”

Transactions on Automatic Control, pp. 397-400, 1998.!!

1!

Nonlinear Observers with Linearizable Error Dynamics!

PAULO OLIVEIRA!!

DEEC/IST and ISR!!

Last Revised: January 8, 2007!Ref. No. NLO#3!

2!

Theme!

The search for a “very special” property...

Given a nonlinear system,with nonlinear measurements of the state available, find a coordinate transformation that renders the dynamics and the output linear on the new coordinates!!!

(except for a nonlinear output injection term)

( )( )

( ) ( )( )

. and ,, as abb. andtor column vec a as expressed outputs system theare and inputs, theare

,states system of vector theis where

,:

yux

pRty

mRtunRtx

xhyuxfx

∈

∈∈

=

=Σ

⎩⎨⎧ !

G

Consider the class of non-autonomous nonlinear systems !

3!

Theme!

•  Challenge for the control problem set at IFAC 1978 (Helsinki) by Roger Brockett to Arthur Krener [1]!

•  Control problem well understood (during the 80s), see [1, 2] for a survey on the new techniques: feedback linearization, input-output linearization, backstepping , zero dynamics, …!

•  Harder to be solved for nonlinear observers!

Relevant questions:!•  Conditions for the existence of such transformation!

•  Synthesis methods (complexity)!

•  Robustness relative to unmodelled dynamics...!!!!!

4!

•  First systematic approach [3] that resorts to a nonlinear state transformation to linearize the original system up to an additional output injection term!

Krener and Isidori @ 1983!

( )( )

( )

( ) ( )

( )zKCAz

BuzAz

CzuyBuAz

wz

xhuxf

yx

(A,C).zCwKuy

~~

ˆˆ

form theofobserver an Design 2)

!,,

Find )1

) ofity observabil (assume z-zz~ dynamicsError )3ˆ,

−=

++=

++

=

=

=

=

=

−+

⎩⎨⎧

⎩⎨⎧

!

!

!!

α

α

( )( )ywxz

γ

ϕ

=

=

5!

Krener and Isidori @ 1983!

The proposed solution proposed is composed of three steps (see [1, 3] for details):!

!

1)  A set of partial differential equations (PDE) must be solved to find γ(y)!

2)  The integrability of conditions for this PDE involve the vanishing of a pseudo-curvature!

3)  A coordinate transformation z=φ(x) can be obtained after a set of PDEs is solved, resorting to conditions on the Lie derivatives of the outputs!

“The process is more complicated then feedback linearization and even less likely to be successful...” in [1]!

6!

Kazantzis and Kravaris @ 1997!

Slitghly different objective:

Given a nonlinear system, with nonlinear measurements of its state available, find a nonlinear state transformation that renders the observer error dynamics linear!!!

(except for a nonlinear output injection term)

( )( )

( ) ( ) .00 :,:

and ,00 point, mequilibriuan is 0origin Thefields. vector analytic are where

systemsnonlinear autonomous of class heConsider t

=

→→

⎩⎨⎧

==

=

=

hRRhRRf

fx

xhyxfx

mnnn

!(2)

7!


Motivated by Luenberger's original ideas on the linear observer

design problem, the proposed approach will try to reconstruct a

nonlinear invertible function z = θ x( ).

( )

verifiedbemust PDE following thedynamics,intended for the and (2) system theof definition theUsing

fies that veriderivative with time

yAzxxdt

dxx

z βθθ

−=∂

∂=

∂

∂= !!

( ) ( ) ( ) ( )( ) ( ),yAzxhxAxfxx

ββθθ

−=−=∂

∂ (3)

8!


}k{kConvexHull

,..., n, ikxf(x)F

n,...,

i

10

with 1 seigenvalue has 0at evaluated field vector theof Jacobian The

∉

==

:A1 Assumption

( ) ( )[ ]

.rank has

1

matrix x

matrix x theDenoting

that theassumed isit

001

n

n-HF

HFH

O

nm

nmT

xmh...

T

xh

H

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

∂

∂

∂

∂=

!

:A2 Assumption

stable.locally is (3) that states essecially 2A

9!


( ) ( )

( ) ( ) ( )

.0 of odneighborho in the solution, analytic unique a admits PDEs of system above the

then , of ones the torelative of seigenvalue

theof ceindependen and , sassumption Under unknown.

theis wherewith

equations aldifferentilinear -quasi of

systemorder -first heConsider t :

(0,0) (0,0)

,00,0 and 0,0,0 0,0,0

,,

=

∂

∂

∂

∂

=∂

∂==

=∂

∂

x

w

wx

w

wxwxxw

ψϕ

ϕψϕ

ψϕ

A2,A1

Theorem AuxiliarysLyapunov'

The novelty in [4] was the use of this result app to (3) to guarantee the existence and uniqueness of solutions.!

(4)

10!


( ) ( ) ( ) ( )yxAxfxx

βθθ

−=∂

∂ PDEs of system thefor Solution

( ) ( )

( )( )

s.eigenvaluecommon havenot do and hen solution w Unique.

ofsolution theis where, is (3) ofsolution

unique Then the matrices.constant and )0( with

caselinear heconsider t :MethodLinear

,,

,,,

AFBHATTF

TTxw

H,x

BA,F,

For

BHxAwwxFxwx

wxwxxw

=+

=∂

∂=

−=

=

=∂

∂

β

ψ

ϕ

ψϕ

(5)

11!


( )

observer.order full a above system dynamic themakesthat mapnonlinear invertiblelocally a exists Then there

e.controlabl is and ),0( Hurwitz, is where

form theof system dynamicorder th -n theand hold and (2) system dynamic for thehat Consider t :

(x)z

(A,B)x

BA

yAzz

θ

β

β

=∂

∂=

−=!

A2A1Theorem

Why is this method or structure acttractive?...!

12!


(3)

( )

( ) ( ) ( ) ( )( )

( ) ( ) ( )( ) ( ) ( )( ) ( ).ˆˆˆˆ

such that (2)for observer order -full asymptotican is

ˆˆ

ˆˆ

system The (3). ofsolution invertiblean be Let :

ˆ1

zzAxxAxxdtdzz

dtd

xx

xfx

xθz

xhy

−=−=−=−

∂

∂−=

=

⎟⎠⎞⎜

⎝⎛

−

−⎥⎦⎤

⎢⎣⎡

θθθθ

θββ!

Theorem

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

( ) ( )( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )( )xxAyxAxhyxhxA

xfxxhyxx

xfxxxxxxxdtd

θθβθβββθ

θββ

θθθθθθ

−=+−−−−

=∂∂

−−−

∂∂

−∂∂

=∂∂

−∂∂

=−

ˆˆˆˆ

ˆ1

ˆˆ

ˆˆˆˆˆ

:(brief) Proof

!!

13!


( )( )⎩

⎨⎧

=

=

xhyxfx!

( ).xz θ=

( )yAzz β−=!

( ) ( ) ( ) ( )( )⎟⎠⎞⎜⎝⎛

−

−⎥⎦⎤

⎢⎣⎡∂

∂−= xhyx

xxfx ˆ

1

ˆˆ

ˆˆ ββθ!

( ).ˆˆ xz θ=

( )yzAz β−= ˆ!

zAz ~~ =!

Observer design!In original coordinates!

Observer in new coordinates!

Linear error dynamics!

14!

Visualization of the Filter!

x(t) x(t)

Δ y(t)

r (t) x(t) x(t) y(t)

-

.

^ ^ .

Gain

REAL

Δ

f(.)

NLO

f(.)

h(.)

h(.) ( )1

ˆˆ

−

⎥⎦⎤

⎢⎣⎡∂

∂− x

xθ

β(.)

β(.)

15!

Krener and Xiao @ 2002 [5]!

( )( )

( )

( ) ( )( )( ) ( )yzAzg

zhAzzg

nRpβ: R

zAzzz-z

yzg

.)h()g(hgzhy

β

β

−=

−=

→

==

=

==

=

=

ˆˆˆ

such that field vector continuous

a exists re then the,~~ i.e. linear, is ˆ~ dynamics

error thesuch that ,ˆˆzobserver nonlinear a exists thereIf

000 and fields vector continuous are and where

zgz systemsnonlinear of class heConsider t :

!

!

!Theorem Converse

16!

Example I from [5]!

17!

Example I from [5]!

18!

Example I from [5]!

Run demo!

19!

Example II from [5]!

20!


21!


22!


( )( )

( ) ( ) .00 :,:

and ,00 point, mequilibriuan is 0origin Thefields. vector analytic are where

,,

systemsnonlinear autonomous-non of class heConsider t

=

→→

⎩⎨⎧

==

=

=

hRRhRRf

fx

uxhyuxfx

mnnn

!

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

( ) ( ) ( ) ( )

( ) ( ) ( ).0 and 00Let

.000 and 00,,00,

00

0011

10

10

where

,,,,

verifiedare relations following that theAssume

xB,

xh,H

xfF

hfxhxfuxhxhuxhuxfxfuxf

∂

∂=

∂

∂=

∂

∂=

====⎩⎨⎧

+=

+=

β

23!


( )

( ) ( ) ( ) ( )( )

( ) ( ) ( )( )⎟⎠⎞⎜⎝⎛

−

−⎥⎦⎤

⎢⎣⎡∂

∂−=

−=

−=∂

∂

=

uxhyx

uxfx

BHTTATF

xhxAxfxx

eixz

,hf

,ˆ1

00

00

ˆ,ˆˆ

obtained isobserver nonlinear following The.

equation theofsolution therequires

. ,tion transforma

coodinatenonlinear for the and ,pair the toconditions technicalsame under the results, previous theApplying

ββϕ

βϕϕ

ϕ

!

24!


( ) ( )

( ) ( )

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

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )( )xhxAxfxx

xhxAxfxx

ex-φxφe

,hf

uxfxfxφuxhyuxfxf

xφ

uxfxφuxhyuxf

xφe

ˆˆˆˆ,

equation aldifferenti the verifiesThen .ˆLet

0000

00

1010

i.e form, PDEin pairs for the verifiedrelations Given the

,,ˆ,ˆˆˆ

,,ˆ,ˆˆ

βφ

βφ

φφ

ββ

ββ

−=∂

∂−=

∂

∂

=

+∂

∂−−−+

∂

∂=

∂

∂−−−

∂

∂=

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ ⎟

⎠⎞⎜

⎝⎛⎟

⎠⎞⎜

⎝⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛ ⎟

⎠⎞⎜

⎝⎛⎟

⎠⎞⎜

⎝⎛!

( ) ( )uxNuxNAee ,,ˆ −+=!

( ) ( ) ( ) ( )( ) ( )( ).,,, where 01 iiiii xhuxhuxfxxφuxN ββ −+∂

∂=

25!


( ) ( ) ( )

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

( )( ) ( )

stable.ally asymptoticlocally is 0then max

min

maxmin

2121

1

if Hence

22

thathave wefunction candidate Lyapunov for the And

2

such that0 aexist e then ther0any for then Hurwitz, for i.e. possible,

islesson previous in the introduced ones thesimilar todesign A

,,

then Lipschitzlocally is f that assumefurther weIf

=

>

+−≤

=

=+

≥≥

−≤−

e

uLPλQλ

ePλuLQλeV

PeeeV

Q-PAPA

PQA

xxuLuxNuxN

T

T

!

26!

•  Other methods to solve the PDE could be used [5]!

•  Design method easier to be accomplished than [3]!

•  The authors of [4] claim “to be able to do so for all linearly observable, real analytic systems whose spectrum of the linear part lies wholly in the right half complex plane”.!

•  Krener and Xiao extended the method to arbitrary specta [5] (the Siegel domain) and showed that the sufficient conditions were also necessary.!

•  Discrete time [6] and state and disturbance estimation design [7] versions became available!


27!

References!

[1] !Mitter S., Baillieul J., and Willems J., “Mathematical Control Theory,” 1998.!!

[2] ! Kokotovic P. and Arcak M., “Constructive Nonlinear Control: A Historical Perspective,” Elsevier, 2000.!

!

[3] Krener A. and Isidori A. , “Linearization by Output Injection and Nonlinear Observers”, Systems and Control Letters, nº3, pp. 47-52, 1983.!

!

[4] Kazantzis N. and Kravaris C., “Nonlinear Observer Design using Lyapunov’s Auxiliary Theorem,” CDC, San Diego,1997. !

!

[5] ! Krener A. and Xiao M., “Nonlinear Observer Design in the Siegel Domain,” SIAM Journal of Control and Optimization, v. 41, nº 3, pp. 932-953, 2002. !

!

[6] Xiao M., Kazantzis N., Kravaris C., and Krener A., “Nonlinear Discrete-time Observer Design with Linearizable Error Dynamics,” IEEE Trans. Automatic Control, v.48, nº4, pp. 622-626, 2003. !

!

[7] !Kravaris C., Sotiropoulos V., Georgiou C., Kazantzis N., Xiao M., and Krener A., “Nonlinear Observer Design for State and Disturbance Estimation,” Systems and Control Letters, v. 57, pp. 730-735, 2007.!

!

Documents

Dynamic Stochastic Estimation, Prediction and Smoothing … · 2019-09-25 · Dynamic Stochastic Estimation, Prediction and Smoothing DEEC/SSC, Spring 2003 PhD. Programme . ... Gram-Schmidt