Continuous Time Adaptive Filtering

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NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

CONTINUOUS TIMI3 AD- F'ILWZRING

k Yashin

December 1984 WP-84-105

Working h p e r s a r e interim reports on work of t h e international lnstitute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily r e p r e s e n t those of t h e Institute or of i t s National Member Organizations.

INTERNATIONAL

INSTITUTE

FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

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This paper deals with t h e problem of adaptive estimation in t h e continuous time Kalrnan filtration scheme. The necessary and sufficient conditions of t h e convergence of t h e p a r a m e t e r estimators are discussed. For systems which a r e characterized by constant but unknown parameters. t h e conditions of convergence can be checked before t h e observation start. The method of proof is based on the relations between singularity property of some probability measures a n d convergence of t h e Bayesian estimation algorithm.

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CONTINUOUS ADAPTIVE

FIL7TRlNG

k Yashin

1. INTRODUCTION

In m a n y cases, t h e coordinates of partially observed plants subjected t o random noise c a n now be successfully e s t i m a t e d (filtered) [1,2,3,4.5], by using good a @ o r i d a t a on t h e s t r u c t u r e a n d numerical values of t h e p a r a m e t e r s .

In r e a l life, however, some c h a r a c t e r i s t i c s of t h e plants m a y prove impossible t o determine before t h e experiment or observations begin. This leads t o c e r t a i n difficul- ties i n solving I!le filtering problem. U n c e r t a i n t y in t h e coefficients of t h e equation t h a t describe t h e plant subjected t o r a n d o m disturbances, m a y lead t o substantial performance dekerioration of coordinate estimating algorithms adjusted t o c e r t a i n fixed values of t h e p a r a m e t e r s [I].

Fortunately, t h e values of unknown p a r a m e t e r s c a n be updated reasonably often with t h e arrival of observed data. However, simultaneous estimation of both t h e p a r a m e t e r s and plant coordinates may prove t o be t h e nonlinear problem. Lack of

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appropriate computer methods for t h e solution of nonlinear problems brings to life a host of heuristic estimation algorithms, which work in many actual situations but need more thorough analytical investigation [2.3.4].

The temptation to apply t h e available coordinate estimation algorithms (which are only effective if t h e parameter values are known) t o t h e solution of these nonlinear problems leads t o adaptive filtering, whereby t h e filter equations use c u r r e n t parameter estimates which a r e obtained from processing of observations. The equations or algorithms which lead t o parameter estimates a r e referred t o as adaptation, or adjustment, algorithms [5,6,7,8].

While in t h e case of available parameters the coordinate estimation algorithm is a kind of Kalman filter, with unknown parameters this a r r a n g e m e n t is referred to as adaptive Kalman filtering [9, lo].

The choice of t h e algorithm for t h e parameter estimation is somewhat arbitrary

[I].],

but one common property is frequently very important in all of these algorithms:

t h e r e s u l t a n t p a r a m e t e r estimates should, in some sense, t e n d t o t h e i r t r u e values a s the number of observations grows. Such estimates a r e referred t o a s consistent. The plant with unknown p a r a m e t e r s is identifiable if t h e parameters' estimates are consistent.

It would be natural t o investigate t h e property of parameter estimate consistency through studying only t h e properties and characteristics of t h e initial plant dynamics equations, t h e properties of noises, and t h e specifics of t h e filtering algorithms and adaptation procedure. Such an a t t e m p t for systems described by discrete time equations has been reported in [12.13,14].

The conditions sufficient for consistency of parameters' estimates, t h a t take on values from a c e r t a i n finite set, follow, in those papers, from t h e singularity of proba-

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bilistic measures associated with various values of t h e parameter.

This paper provides t h e necessary and sufficient conditions for consistency of these parameter estimates in adaptive Kalman filtering for t h e case of t h e denumerable s e t of the parameter values.

This proof relies on a close link between consistency a n d absolute continuity and singularity of a certain family of probabilistic measures. In t h e cases t o be discussed, singularity entails consistency of estimates; t h e conditions for consistency may be t h e conditions for t h e family of measures t o be singular. Recent research concerning absolute continuity a n d singularity of probabilistic measures associated with random processes [15] has made formulating t h e necessary and sufficient conditions for these properties possible. Representing some processes as solutions of stochastic dif- fererential equations permits formulating t h e singularity conditions in t e r m s of t h e characteristics of these equations. In adaptive Kalman filtering, t h e characteristics of t h e initial equations c a n be expressed as filter parameters. In this way, t h e condition for consistency of estimates can be tested in each specific case before t h e observations are made. A study of consistency has been performed for Bayesian estimates in discrete time adaptive Kalman filtering in [16, 171.

2. ETATEXJZNT

OF

THE PROBLEM

The problem of p a r a m e t e r estimation in continuous time adaptive Kalman filtering can be investigated in t h e framework of the following formal description.

On probabilistic space

(R,H,P)

a random variable ~ ( w ) is specified which takes on values from a certain denumerable s e t

jg,j,

ⁱE N with apriori probabilities

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Let

fl

be a-algebra in Q t h a t is g e n e r a t e d by t h e values of t h e p a r a m e t e r @

.

On t h e s a m e space t h e random process (6,c)

=

61,c1, t

>

⁰is specified. Denote as

H:.(

a n d

HIC

the a-algebras

t h a t a r e g e n e r a t e d by values of t h e processes (zP,[) a n d

c

u p t o time t , t r 0 , respec- tively. Let

~ t e ~ v ( =

H P V H:-(,

t

²0 a r e u-algebras g e n e r a t e d by t h e union of a-algebras

H P a n d H ~ Cwhile

H P ~ =

V H),

t r

0 a r e u-algebras generated by t h e union of a - algebras

H@

and HI(

.

^t²⁰

^.

^Let

^{Xdn( =}

^V_t

^H:-(.

^HPnd.(

⁼

^V^HPQ+(

⁼

^{H .} ^HPB(

⁼

^vtHP-(

^.

1

Denote by I # , I@6.( a n d HP*( t h e respective nondecreasing right-continuous families of u-algebras.

Assume t h a t on probability s p a c e ( Q , H , P ) t h e process (19.6) c a n be r e p r e s e n t e d a s a s y s t e m of stochastic differential equations:

where ( 1 9 ~ ) ~ ~ is a sequence of k-dimensional vectors; ( [ t ) t , o is a s e q u e n c e of 1- dimensional vectors and W l a t , W Z a t a r e independent k

,

a n d k2-dimensional Wiener processes. respectively, independent of t h e initial values of

zPo,t0

a n d t h e random value of

8.

The m a t r i x a ( @ , t ) is (k k ) , A ( B , t ) is (1 k ) and t h e m a t r i c e s b , ( ~ , t ) , b z ( @ , t ) . and B ( t ) a r e ( k k , ) , ( k k 2 ) , a n d ( 1 k z ) , respectively, a n d a r e t h e bounded functions of t i m e for a n y value of @

.

The process 19 describes t h e t i m e variation of t h e unobservable coordinate of a c e r t a i n dynamic plant, while t h e process

6

models t h e m e a s u r e m e n t of t h e coordinate 19 with random noise.

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Introduce on ( ~ , H P . ~ ' C ) a denumerable family of probabilistic measures

P, icN:

Assume t h a t matrix B ( t

**) ~ * ( t )**

is nonsingular for any t ¹0. Let

Pt . ^P

^and

^pt

^denote

t h e restrictions of t h e m e a s u r e P to the u-algebras f f ! e d . C , f f C a n d

f f f ,

respectively, while

P;,

and

R,

restriction of the measures

Pi

to

ffB.'.C,

^{f f C} ^and

^{f f f .}

Denote by

z.j

the Radon-Nicodim derivative of t h e measure with respect t o t h e measure when i t exists.

Definition. HC-adapted p r o c e s s

B t , t

¹0 is c o n s i d e r e d t o b e a s t r o n g l y c o n s i s t e n t e s t i m a t e of t h e p a r a m e t e r

B

i f

In this paper we will study t h e consistency conditions of t h e estimates

Bt

^,^t

^r

^{0 where}

B t

=

E(B

I

f f f ) .

3.

RESULT

Assume t h a t the initial conditions go and

to

a r e such t h a t

where

E,

denotes mathematical expectation with respect t o t h e measure

P.

From the fact t h a t the coefficients of t h e equations (1) are bounded a n d from the condition (2) i t follows t h a t with any

t <

= and i EN

Assume also t h a t t h e joint distribution of t h e random variables do and

to

is Gaussian with respect t o each measure

P',

EN. Hereafter b i t ) b 2 ( i , t ) . b , ( i . t ), b 2 ( i , t ) , a ( i , t ) and A ( i . t ) will replace b , ( B i , t ) , b z ( P i , t ) , a ( B i . t ) , A ( B , , ~ ) , respec-

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tively, for convenience of t h e notation.

The main r e s u l t of this paper is t h e following t h e o r e m .

Theorem 1 . The strong consistency property o f the estimate

pl,

^t²⁰ ^takes

place if and onlyfor any i . j E N

where

and

einj

( t ) m e the solutions o f the following linear differential equatkns:

and initial conditions

where

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K ( t ) =

( b 2 ( i , t ) B e ( t )

+

y,(t

)A(i.t))(B(t )Be(t))-I*

a n d yi

( t

) s a t i s f y

the

f o l l o w i n g e q u a t i o n s

Corollary. A s s u m e t h a t t h e c o e f f i c i e n t s of

the

s t o c h a s t i c e q u a t i o n s ( 1 ) do n o t d e p e n d o n t i m e t , a n d a s t a t i o n a r y s o l u t i o n of e q u a t i o n (7) e z i s t s . T h e n f o r t h e e s t i - m a t e

En

t o b e c o n s i s t e n t

it

^iss u f f i c i e n t t h a t f o r a n y i , j E

N ,

i

= j

w h e r e

ai,,

^isa s t a t i o n a r y s o l u t i o n of e q u a t i o n ( 1 0).

The Corollary follows from equalities (4) and (5), and from t h e stationarity assumption.

Remark. h t h e c a s e of a f i n i t e n u m b e r o f possible v a l u e s of the p a r a m e t e r

B

a n d c o n s t a n t c o e f f i c i e n t s in e q u a t i o n

( I ) ,

c o n d i t i o n (4) c a n be v e r i f i e d b e f o r e t h e o b s e r u a -

tiom

a r e m a d e .

4. PROOF

OF

THEOREX 1

The proof will be preceded by several auxiliary lemmas.

kmma 1. I h e p r o c e s s ( 1 9 . f ) o n probabzlistic s p a c e s

( o M , H ~ . ~ , P )

c a n be r e p r e s e n t e d in t h e f o r n :

dzPt = a ( i , t ) d t d f

+

b l ( i , t ) d W l , t + b z ( i , t ) d W ~ . ~ , $0 ( 1 3)

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where W , , , ^,W z , , are independent Wiener processes independent of d o , # o .

Proof. E r s t note t h a t for any i ^E

N

measures

P'

a r e absolutely continuous with respect t o m e a s u r e P a n d consequenty P,'

<<

Pi for any t ²0. Denote

&'

.

= - .

Let u s

d Pt

consider t h e t r a n s f o r m a t i o n of t h e local c h a r a c t e r i s t i c s of t h e process $,[ with absolutely continuous transformation of t h e probabilistic m e a s u r e P i n t o t h e m e a s u r e

P,

i E

N

on (Q,HB~*-F). From t h e definition of the m e a s u r e s P:, i E

N,

t r 0 it follows t h a t for a n y t r 0

&* =

l ( 8

=

^{P i )} Pi

a n d consequently,

121, iEN,

t r 0 a r e independent on time. In compliance with t h e analog of t h e Girsanov t h e o r e m on transformation of t h e local c h a r a c t e r i s t i c s of t h e processes with absolutely c o n t i n u o u s transformation of t h e probabilistic m e a s u r e s , t h e local c h a r a c t e r i s t i c s of t h e process d , [ , in t h e case of a t i m e independent Radon- Nicodim derivative, r e m a i n u n c h a n g e d a n d consequently t h e process $,[ m a y be r e p r e s e n t e d by equation (1). Note now t h a t following t h e definitions of t h e m e a s u r e s

PL,

i E

N,

t h e n e x t equalities hold

P

-a.s.

while t h e processes W l , , , W Z . , r e t a i n t h e i r properties with r e s p e c t t o t h e m e a s u r e s

P,

i E

N.

Thus on ( Q , H B . ~ . ~ ) t h e process ( 3 , ~ ) c a n be r e p r e s e n t e d by stochastic differential equation (13).

Consider now t h e transformation of t h e local c h a r a c t e r i s t i c s of t h e process d , [ with t h e restriction of o-algebra H B . ~ . ~ t o

H d . 6 .

Since t h e coefficients a ( i . t ) d t ,

A ( i , t ) d t , a r e Hb.4-adapted t h e innovation process ( d , ( ) which r e s u l t s from restricting

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t h e o-algebras

H $ . ~ . ~ ~

t o

H P . ~

^,^t²⁰can be represented on

( R , P . C , P ) ,

ⁱ^EN in form ( 1 3 )

.

I t is a well known f a c t t h a t with t h e above assumptions t h e problem of estimation of the coordinates of t h e process 19 from observations of t h e p a t h s of t h e process

#

on each space

( R , P J , P )

c a n be solved in conditional Gaussian t e r m s . Denoting by

t h e m e a n s q u a r e optimal e s t i m a t e of filtering for t h e process 29 and by 7 ,

( t

)

= m;

((*,

-

^{m m ;}

^-

^mf

¹ l

^{H f )}

t h e conditional variance of t h e e s t i m a t e i E

N,

t r 0 we have, for t h e s e variables, t h e well known equations [ l a ] :

Denote

mg= E(d;

I

Hg.C), y f ( t )

=

E((dt

-

ma)(*;

-

m!)*

I HBJ)

The following assertion is t r u e .

]Lemma 2. m e process

#

c a n be r e p r e s e n t e d o n ( R , H ~ ~ . ( , P ) in the f o r m

d t ,

=

A ( @ , t ) m $ d t

+

B ( t ) d W t ,

to.

and W; is I?*€-adapted Wiener process,

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The proof of this lemma can be done using the same arguments as in [ l a ] .

Lemma

3. On probabilisistic spaces

( R , H ~ . [ , P ) ,

i

E N

the p r o c e s s [ c a n be r e p r e s e n t e d as

Proof. Using the absolute continuity of the measure P:, i

E N ,

t

r

0 in relation to d P f ,

t r

0, i ^E

N,

we the m e a s u r e Pt, ^{1 0}and time independent of the derivatives -

d Pt

have, in compliance with t h e analog of the Girsanov theorem on transformation of local process characteristics with absolutely continuous transformation' of probabilistic measures, t h a t the characteristics of process

t,

which is represented by equation ( 1 6 ) , do not change with t h e replacement of measure P by measure

P ,

i ^E

N.

The following equalities hold

P

-a.s.

whence follows presentation ( 1 7 ) .

The properties of process

t

a r e found to ensure equivalence between t h e probabilistic measures

Fk

and

Fi,

i # j . Let us formulate a n d prove this assertion.

Lemma

4. Measures

Fi

a n d

Fi

a r e e q u i v a l e n t o n m e a s u r a b l e space (

R , H ~

).

Proof. The boundedness of the coefficients A ( i , t ) , a ( i , t ) , b ( i , t ) , B ( t ) a n d non- singularity of the matrix B ( t ) B D ( t ) provide for any

t <

^mt h e inequality

where

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which is t r u e

P

and p - a . s . In accordance with the Liptzer-Shiryaev result

[ l a ]

(chapter 7) it yields the equivalence of pti a n d

P)

on

( o M , P J ) . I t

follows t h e n t h a t the restrictions of these measures to ( R,H( ) a r e equivalent, t h a t is

Fti - ^F/ ^.

Lemma 5.

7he processes

( m f ) t M , i E

N are Guussian o n the pro babiListic spaces ( R , f i , ? ) ,

k E N ; k is

not necessarily equal to

i .

Proof. I t is easy to see t h a t t h e processes ( m f ) t M , i E N a r e Gaussian on probabilistic spaces

( R , H ~ , P ) ,

i E

N.

Let us prove t h a t the processes

(m(i.t);,,

i E N a r e Gaussian on probabilistic spaces

( R , H ( , ~ ) ,

k #

i ,

k E

N.

Because the simultaneous initial distribution of t h e variables on any of the probabilistic spaces

(R,P.(,P),

ⁱ^E

^N

is Gaussian, t h e random variables

mi = E j ( g O 1 ~ 8 )

a r e linear functions of with any j E N and, consequently, simultaneous distributions of t h e variables

m & , g O ,

are also Gaussian on probabilistic spaces

( n , P . C . P )

where k E N, k # j . Substituting into equation (14) for t h e values of

mi.[t

from equation ( 1 ) we have

+ ( b 2 0 ,t ) ~ ' ( t ) + y j ( t ) A '( j ,t ) ( B ( t ) B ( t* ))-'B(t)dW,.t

The fact t.hat t h e joint distribution of

(ma,290,[o)

is Gaussian and t h e formula for 4, a n d

mi, ^j

^E^N,

^t

^r0 are linear, makes processes

(mi)tao. ^j

^E

^N

^and^([, Gaussian on probabilistic spaces

( Q , H ( , ~ ) ,

k E

N ,

k # j which was required. Consequently, the variables

( ~ ( i , t ) - ~ - ~ O , t ) m i ) ,

j E

N

a r e Gaussian for any of t h e m e a s u r e s

p.

k E

N. Lemma

6.

Singularity conditions for measures F(i and F/ m a y be w r i t t e n as fol-

l o w

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Proof. Using t h e main r e s u l t of paper [15] one can say t h a t measures and are singular if and only if

with F,-probability 1. Taking i n t o account t h a t processes rn/ are Gaussian on any probabilistic spaces ( R , H ~ . ? ), k E N and result [18], one can see t h a t this condition is equivalent to condition (18). Condition (19) is not very convenient for checking in a general case. For calculation of the mathematical expectation in (19) perform certain additional constructions.

Let @i.j(t) denote a block m a t r i x

and

q j ,

a block matrix

Using the well-known property of matrix multiplication [19] one can write

Direct verification shows t h a t t h e matrix

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can be represented in t e r m s of m a t r i c e s G i . , ( t ) and q m j ( t ) as follows :

Consequently

Let u s find now t h e r e c u r r e n c e equation for t h e matrix G i j ( t ) . Take u p a block m a t r i x f i j ( t ) which has t h e form

Using r e c u r r e n c e equation (14) for mj a n d

mi

a n d equation ( 1 ) for 19; we have for block e l e m e n t s of t h e m a t r i x GiJ ( t )

=

^~jF, J ( t )

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Introducing the matrices

the formula for a i i ( t ) can be rearranged into a matrix form

Consequently, the conditions for singularity of t h e measures and

P',

ⁱ# j , i , j E N are equivalent to (4) of the theorem.

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