On Linear Systems Disturbed by Wide-Band Noise

On linear systems disturbed by wide-band noise

Agamirza E. Bashirov Department of Mathematics, Eastern Mediterranean University Gazimagusa, Mersin 10, Turkey [email protected] Institute of Cybernetics, Azerbaijan Academy of Sciences F. Agayev St. 9, Baku-370141, Azerbaijan

Keywords: Wide-band noise, white noise, stochastic sys- function £ is called an autocovariance function. The wide-

tem, ﬁltering. band noise process ³ is said to be stationary if the autoco-

× " variance function £ depends only on (see [1]). If is so small that it is normally assumed to be 0, then the wide-band

Abstract noise process ³ is transformed into a white noise process. As it was mentioned, in many fields such substitution gives rise An approach, based on a distributed delay, is presented to to tangible distortions. handle wide-band noise processes. This approach is closely There are different techniques of handling and working related with the semigroups of right translation and allows with the wide-band noise processes. For example, in [2] the a reduction of a wide-band noise driven linear system to a approximate approach is used. Another approach, based on white noise driven linear system. This reduction is applied a representation of wide-band noise as a distributed delay of to the Kalman-Bucy filtering in order to modify it to a wide- white noise, is suggested in [3]. band noise driven linear system. In [3, 4] the Kalman-Bucy estimation results are modified to a wide-band noise driven systems. The proofs of respective results are based on the duality principle and, technically, 1 Introduction they are routine being useless if one tries to modify them to nonlinear systems. It becomes important to develop a more The modern stochastic optimal control and filtering theories handle technique of working with the wide-band noise pro- use white noise driven systems. The results such as the sep- cesses represented through a distributed delay. aration principle and the Kalman-Bucy filtering are based on Some attempts in this way are made in [5] where a re- the white noise model. Indeed, white noise, being a math- duction of a wide-band nose driven system to a white noise ematical idealization, gives only an approximate description driven system is presented. The reduction from [5] requires to actual noise. In some fields the parameters of actual noise an unbounded observation system which does not allow to are near to the parameters of white noise and so the math- obtain a complete set of formulae for the respective optimal ematical methods of control and filtering for white noise filter. driven systems can be satisfactorily applied in them. But in In this paper we establish another reduction of a wide-band many fields white noise is a grude approximation to actual nose driven system to a white noise driven system which does noise. Consequently, the theoretical optimal controls and the not require an unbounded observation system. We apply this theoretical optimal estimators for white noise driven systems reduction to obtain a complete set of formulae for the re- become not optimal and, indeed, might be quite far from be- spective optimal filter. This reduction gives hope to push on ing optimal. It becomes important to develop the problems studying nonlinear systems disturbed by wide-band noise. of systems theory for a noise model that describe actual noise more adequately. 2 Wide-band noise The issue is that actual noise is marked with a property which ensures the correlation of its values within a small time Consider the random process

interval, i.e. if we denote such noise by ³, then

´× Øµ dÛ ´×µ; Ø ¼; ´Øµ=

³ (2)

Ø; ×µ ¼ ×<"

£´ if

ÑaÜ´¼;Ø "µ

Ú[³´Ø · ×µ;³´Øµ] =

cÓ (1) ¼

otherwise;

" > ¼ ¾ Ä ´ "; ¼µ Û

where , ¾ and is a stationary pro-

¼ × ¼ "> ¼ Û ´Øµ=Ø

where Ø and are time moments, is a small cess with orthogonal increments satisfying cov and

³ EÛ ´Øµ = ¼ Ø ¼ value and £ is a nonzero function. A random process with for all . One can easily verify that the

the property (1) is called a wide-band noise process. The random process ³, as deﬁned by (2), is a wide-band noise

´ µ

process which becomes stationary starting the time moment F is positive-deﬁnite, i.e. it is continuous at zero, it sat-

= " F ´ µ´ Ö µ=F ´ µ´Ö µ Ö

Ø with the autocovariance function isﬁes for all real numbers and

´Ö µ´Ö · ×µ dÖ; ¼ × ": ´×µ=

(3)

¼ F ´ µ´Ö Ö µ

i j i

i;j =½

Formally, the wide-band noise process (2) can be inter-

Ò ´Ö ;:::;Ö µ Ò Ò for all -tuples ½ of real numbers and for all -

preted as vibration. At moment Ø a vibration that is formed

´ ;:::; µ Ò

tuples ½ of complex numbers, where is the con-

by the action of white noise during the time between Ø

= ¾

jugate of . Since ½ for any pair of complex num-

½ ¾

Ø "

and Ø affects the system. Values of white noise until

F ´ µ ¾

do not take part in the formation of this vibration at moment bers ½ and , from the positive-deﬁniteness of , one

¼ ½=¾

´ µ can easily obtain that the function F is also positive-

Ø, because their weight is sufﬁciently small and we can ne- deﬁnite. Thus, again by the Bochner’s theorem, the inverse

glect them in the model (2). Consequently, the function

¼ ½ ¼ ½=¾

= F ´F ´ µ µ

stands for the coefﬁcient of relaxing the initial effect at dif- Fourier transformation is real-valued. ¼

ferent time moments. By this interpretation, the model (2) If is non-zero in some subset of the real line being in out-

;"]

corresponds to real cases when the vibration generated by side of [¼ and having nonzero Lebesgue measure, then

[¼;"]

white noise stands to affect a system starting with the initial the function does not vanish in outside of . Letting

´Ö µ= ´ Ö µ " Ö ¼

, , we obtain a desired solution

= ¼ time Ø (a reason for that may be the switching on the

system from resting state to dynamic state which changes its of the equation (3).

¼ <Ø <"

Ä ´¼;"µ sensitivity to random disturbances). Hence, when Thus, given a function in ¾ , there is a variety of

the wide-band noise proces (2) is formed by values of white wide-band noise processes in the form (2) which have the

[¼;Ø] " noise on . same autocovariance function starting the time moment . In practice wide-band noise is identiﬁed by autocovariance This variety is reached by different solutions of the equation

function. Therefore, to get the representation (2) one has to (3) and also by different processes Û .

solve the equation (3) in under given .

"> ¼ ¾ Ä ´¼;"µ

Theorem 1. Let and let ¾ . Then there 3 Notation

exists a ´non-unique in general solution of the equation

X Z À À À G

Ä ´ "; ¼µ ¾ Let , , , ½ , and be separable Hilbert spaces,

(3) in ¾ .

´X; Z µ

let Ä be the space of linear bounded operators from

¼ ¼

Proof. Deﬁne and by

X Z B ´a; b; Ä´X; Z µµ

to and let ¾ be the space of equiva-

lence classes of strongly Lebesgue measurable and square

´×µ ¼ × "

¼ if

´×µ=

a; b] Ä´X; Z µ integrable operator-valued functions from [ to . ¼ otherwise From the discussion given in Section 2, it is reasonable to

and deﬁne a Hilbert space version of the representation (2) and,

´ Ö µ ¼ Ö "

³ ³ X Z

¼ if ¾

hence, we let ½ and be - and -valued wide-band

´Ö µ=

¼ otherwise. noise processes in the form

¼ ¼

In terms of and the equation (3) has the form Ø

¨ ´× Øµ dÛ ´×µ; Ø ¼; ³ ´Øµ=

½ ½

½ (5)

ÑaÜ´¼;Ø "µ

¼ ¼ ¼

´Ö µ ´Ö ×µ dÖ = ´×µ; ½ <× < ½:

(4) Z

³ ´Øµ= ¨ ´× Øµ dÛ ´×µ; Ø ¼;

¾ ¾

¾ (6)

ÑaÜ´¼;Ø Æ µ

Since the left-hand side of (4) is the convolution of by

¼ ¼ ?

itself (we denote this convolution by ), a solution of the

" > ¼ Æ > ¼ Û Û À À

¾ ½ ¾ where , , ½ and are - and -

equation (4) can be derived by use of the direct and inverse

¨ ¨ ¾

valued Wiener processes, ½ and are strongly differen-

½ F

Fourier tramsforms F and as follows:

"; ¼] [ Æ; ¼]

tiable operator-valued functions from [ and to

Ä´À ;Xµ Ä´À ;Zµ

½ ¾

¼ ¼ ¼ ¼ ¼ ¼ and , respectively, with

? = µ F ´ ? µ=F ´ µ

¼ ¾ ¼

µ F ´ µ = F ´ µ

¨ ¾ B ´ "; ¼; Ä´À ;Xµµ; ¨ ´ "µ=¼;

½ ¾ ½ ½

¼ ¼ ½=¾

µ F ´ µ=F ´ µ

¼ ½=¾ ¼ ½

¨ ¾ B ´ Æ; ¼; Ä´À ;Zµµ; ¨ ´ Æ µ=¼:

: F ´ µ µ = F

¾ ¾ ¾ ¾

d«

¼ ½=¾

´ µ Note that the function F is multi-valued. To be Note that by the interpretation of the wide-band noise pro-

certain, we take the principal branch of the square root cess (2) given above, the function is the coefﬁcient of re-

¼ ½=¾ ¼

´ µ =

in F . It remains to show that the function laxing the initial effect at distinct time moments. Therefore,

½ ¼ ½=¾

F ´ µ

F is real-valued and it vanishes in outside of it should be a decreasing (hence, almost everywhere differ-

;"] F ´ µ ´ "µ=¼

[¼ . Since is a Fourier transformation of the real- entiable) function with . In view of this, the con-

¨ ¨ ¾ valued function , by the well-known Bochner’s theorem, ditions on ½ and are reasonable.

~ ~ Z

Consider the partially observable linear system to X and , respectively. We will use the notation

¾ ¿

dÜ´Øµ=´AÜ´Øµ·³ ´ØµµdØ ·¨dÛ ´Øµ;

½ A ¼

4 5

¼ d=d ¼

: A =

Ü´¼µ = Ü ; Ø ¼;

¼ (7)

¼ ¼ d=d«

dÞ ´Øµ=´CÜ´Øµ·³ ´ØµµdØ ·©aÚ ´Øµ;

´¼µ = ¼; Ø ¼;

Þ (8) One can easily verify that

¾ ¿

Þ A

where Ü is a state process, is an observation process,

A ¼ ¼

~ 5

is the inﬁnitesimal generator of a strongly continuous semi- 4

A = d=d ¼ ;

½ X

group Í of linear bounded operators on , for that we use

¼ ¼ d=d«

¾ Ë ´X µ C ¾ Ä´X; Z µ ¨ ¾ Ä´À; X µ

the symbol Í , , ,

A A d=d d=d« ¾

, ½ and are wide-band noise processes de- where is the adjoint of , and are differential

Ü X

ﬁned by (5) and (6), ¼ is an -valued Gaussian random operators from

Û Ú À G ©

variable, and are - and -valued Wiener processes so ¨

~ ~

D ´d=d µ= f ¾ X :´d=d µf ¾ X; f ´¼µ = ¼

Ü ´Û; Û ;Û µ Ú Û

½ ¾

that ¼ , and are mutually independent, and ,

Û Û ¾ ½ and are correlated. In integral form, the system (7)Ð

(8) can be represented as it is shown below: and

¨ ©

~ ~

D ´d=d«µ= g ¾ Z :´d=d«µg ¾ Z; g ´¼µ = ¼

Ü´Øµ=Í ´ØµÜ · Í ´Ø ×µ³ ´×µ d×

¼ ½

~ ~ ~

X Z A

Z to and , respectively. From [6], it follows that is the Ø

inﬁnitesimal generator of the strongly continuous semigroup

Í ´Ø ×µ¨ dÛ ´×µ; Ø ¼; ·

¾ ¿

Í ´Øµ E ´Øµ ¼

Þ ´Øµ= ´CÜ´×µ·³ ´×µµ d× ·©Ú ´Øµ; Ø ¼:

4 5

¼ Ì ´Øµ ¼

Í ´Øµ= ; Ø ¼;

¼ ¼ Ì ´Øµ ¾

In this paper we will reduce the system (7)Ð(8), disturbed

X; X :[¼; ½µ !Ä by wide-band and white noise processes, to the system which where E is the function deﬁned by

is disturbed by purely white noise. For this, we set some Z

special notations which will be used throughout this paper. ¼

E ´Øµh = Í ´Ø · Ö µh´Ö µ dÖ; Ø ¼; h ¾ X:

X = Ä ´ "; ¼; X µ

Let ¾ (the space of strongly measurable

ÑaÜ´ "; Øµ

"; ¼]

and square integrable vector-valued functions from [

X Z = Ä ´ Æ; ¼; Z µ ¨

to ) and let ¾ . We will consider the semi- We will denote by the operator

¡ ¡

~ ~

Ì ¾Ë Ì ¾Ë X Z ¾

groups of right translation ½ and de-

¿ ¾

¼ ¼

ﬁned by ¨

~ ~ ~

5 4

¼ ¨ ¼

¾Ä À ¢ À ¢ À ;X¢ X ¢ Z ; ¨=

½ ¾

f ´ Øµ; Ø "

¼ ¼ ¨

[Ì ´Øµf ]´ µ= ;

¼; Ø< "

~ ~

¨ ¨

¾ [ "; ¼]; Ø ¼; f ¾ X; ¾

a.e. where the operators ½ and are deﬁned by

g ´« Øµ; « Ø Æ

¢ £

[Ì ´Øµg ]´ µ= ;

¨ h ´ µ= ¨ ´ µh; a:e: ¾ [ "; ¼]; h ¾ À ;

¼; « Ø< Æ

½ ½ ½

£ ¢

« ¾ [ Æ; ¼]; Ø ¼; g ¾ Z;

a.e. d

¨ ´«µh; a:e:« ¾ [ Æ; ¼]; h ¾ À ; ¨ h ´«µ=

¾ ¾ ¾

d«

~ ~

X Z ¾

and the linear integral operators ½ and from and to

Û~ À ¢ À ¢ À

X Z ¾

and , respectively, deﬁned by and by the ½ -valued Wiener process

¿ ¾

Û ´Øµ

f = f ´ µ d ; f ¾ X;

5 4

Û ´Øµ

; Ø ¼: Û~ ´Øµ=

Û ´Øµ

g = g ´«µ d«; g ¾ Z:

¡ ¡

~ ~ ~ ~

¨ ¾Ä À ; X ¨ ¾Ä À ; Z

½ ¾ ¾

Clearly, ½ and . Ì

Ì Also, we will use the operators ¾

The inﬁnitesimal generators of ½ and are the differential

d=d d=d«

operators and operating from

~ ~ ~

C ¼

X ¢ Z; Z ] ¾Ä X ¢ C =[

¨ ©

~ ~

D ´ d=d µ= f ¾ X :´ d=d µf ¾ X; f ´ "µ=¼

and

~ ~ ~

Á ¼ ¼

=[ ] ¾Ä X ¢ X ¢ Z; X ;

and Á

¨ ©

~ ~

´ d=d«µ= g ¾ Z :´ d=d«µg ¾ Z; g ´ Æ µ=¼ Á X D where is the identity operator on .

Ü ³~ ³~ ¾ 4 Reduction By Lemmas 1 and 2, the random processes , ½ and ,

deﬁned by (7), (9) and (10), respectively, are mild solutions

~ ~

X Z ³~ ³~ ¾

Let ½ and be - and -valued random processes deﬁned of the linear stochastic differential equations

by 8

dÜ´Øµ = AÜ´Øµ· ³~ ´Øµ dØ ·¨dÛ ´Øµ;

> ½ ½

¢ £

d³~ ´Øµ=´ d=d µ~³ ´ØµdØ · ¨ dÛ ´Øµ;

½ ½ ½ ½

³~ ´Øµ ´ µ= ¨ ´× Ø · µ dÛ ´×µ;

½ ½ ½

d³~ ´Øµ=´ d=d«µ~³ ´ØµdØ · ¨ dÛ ´Øµ;

¾ ¾ ¾ ¾

ÑaÜ´¼;Ø " µ

Ü´¼µ = Ü ; ³~ ´¼µ = ¼; ³~ ´¼µ = ¼; Ø ¼;

¼ ½ ¾

:e: ¾ [ "; ¼]; Ø ¼;

a (9)

¢ £ Þ

d and the random process , deﬁned by (8), satisﬁes

³~ ´Øµ ´«µ= ¨ ´× Ø · «µ dÛ ´×µ;

¾ ¾ ¾

d«

ÑaÜ´¼;Ø Æ «µ

dÞ ´Øµ= CÜ´Øµ· ³~ ´Øµ dØ ·©dÚ ´Øµ; Þ ´¼µ = ¼; Ø ¼:

¾ ¾

:e:« ¾ [ Æ; ¼]; Ø ¼:

a (10) ¾

If ¿

Ü´Øµ

³~ ³~ ¾

Lemma 1. The random processes ½ and , deﬁned by

4 5

³~ ´Øµ

Ü~´Øµ= ; Ø ¼;

(9) and (10), have the representations ½

³~ ´Øµ

Z Ø

~ then using the notations from Section 3, for the random pro-

³~ ´Øµ= Ì ´Ø ×µ¨ dÛ ´×µ; Ø ¼;

½ ½ ½

½ (11)

Ü Þ

cesses ~ and , we obtain

~ ~ ~

³~ ´Øµ= Ì ´Ø ×µ¨ dÛ ´×µ; Ø ¼:

dÜ~ = AÜ~´ØµdØ · ¨dÛ~ ´Øµ; Ü~ ´¼µ = Á Ü ; Ø ¼;

¾ ¾ ¾ ¾ ¼

(12) Ø (14)

dÞ = C Ü~´ØµdØ ·©dÚ ´Øµ; Þ =¼; Ø ¼: ¼

Ø (15)

³~

Proof. Let us prove the representation in (11) for ½ .For

Thus, the system (7)Ð(8) disturbed by white and wide-band

h ¾ À

½ ,wehave

noise processes is reduced to the system (14)-(15) which is

¢ £

Ì ´Ø ×µ¨ h ´ µ ½

½ disturbed only by white noise. This reduction is signiﬁcant

¢ £ ~

as the operator C is bounded whereas in the reduction from

¨ h ´× Ø · µ; × Ø · " ½

= [5] the respective operator is unbounded.

¼; × Ø · < "

´d=d µ¨ ´× Ø · µh; × Ø · "

½ :

= 5 Application to ﬁltering

¼; × Ø · < "

Therefore, In this section we apply the reduction from Section 4 to the

ﬁltering problem for the partially observable system (7)Ð(8).

¢ £

d For, assume that the conditions and notations from Sections

³~ ´Øµ ´ µ= ¨ ´× Ø · µ dÛ ´×µ

½ ½ ½

Z = Ê k

d 3 and 4 hold and is the -dimensional Euclidean

ÑaÜ´¼;Ø " µ

cÓÚÜ = È EÜ = ¼

¼ ¼

space. Additionally, let ¼ and .We

Î =©Î © Î

Ì ´Ø ×µ¨ ´×µ dÛ ´×µ ´ µ: =

½ ½

½ denote , where is the covariance operator of

Î ©

the Wiener process Ú . Assume that and are invertible. Û

The covariance operator of ~ is

³~

The representation in (12) for ¾ can be proved in a similar ¾

way. ¿

Ï Ï Ï

¼½ ¼¾

4 5

Ï Ï Ï

Ï = ;

½½ ½¾

³ ³

¼½ ¾

Lemma 2. For ½ and , deﬁned by (5) and (6), and for

£ £

Ï Ï Ï

¾¾

³~ ³~

¼¾ ½¾ ¾

½ and , deﬁned by (9) and (10),

Ï Ï

where and ij are the respective covariance operators of

³ ´Øµ= ³~ ´Øµ; ³ ´Øµ= ³~ ´Øµ; Ø ¼:

½ ½ ¾ ¾ ¾

½ (13)

Û Û Û ¾

, ½ and .

Ü ´Øµ Ü~´Øµ

Proof. The ﬁrst equality in (13) follows from Then the best estimate ~ of based on the observa-

´×µ ¼ × Ø ´~Ü; Þ µ

tions Þ , , where is deﬁned by (14)Ð(15),

¼ £

¢ is a solution (in the mild sense) of the linear stochastic dif-

³~ ´Øµ= ³~ ´Øµ ´ µ d

½ ½

½ ferential equation (see [7])

Z Z

¼ Ø

£ ½

~ ~ ~ ~ ~

^ ^ ^

dÞ ´Øµ C Ü~ ´ØµdØ ; dÜ~ ´Øµ=A Ü~ ´ØµdØ · È ´ØµC Î

= ¨ ´× Ø · µ dÛ ´×µ d

½ ½

" ÑaÜ´¼;Ø " µ

Ü ´¼µ = ¼; Ø ¼;

~ (16)

Z Z

Ø ¼

= ¨ ´× Ø · µ d dÛ ´×µ

½ ½

where È is a solution (in the scalar product sense) of the

´¼;Ø "µ Ø × "

ÑaÜ Riccati equation

= ¨ ´× Øµ dÛ ´×µ=³ ´Øµ:

½ ½ ½

£ £

~ ~ ~ ~ ~ ~ ~ ~

È ´Øµ=È ´ØµA · AÈ ´Øµ·¨Ï ¨

ÑaÜ´¼;Ø "µ

dØ

£ ½ £

~ ~ ~ ~ ~ ~ ~ ~

È ´ØµC Î C È ´Øµ; È ´¼µ = Á È Á; Ø ¼:

The second equality in (13) can be proved in a similar way. ¼ (17)

£ £ ½ £

Ü^ ´Øµ Ü´Øµ È ´Ø; ¼µ·¨Ï ¨ Å ´ØµÎ Å ´Øµ;

Let be the best estimate of based on the observa- · (19)

¼½

Þ ´×µ ¼ × Ø ´Ü; Þ µ

@ tions , , where is deﬁned by (7)Ð(8) and @

· È ´Ø; µ=AÈ ´Ø; µ·È ´Ø; ¼;µ

¼½ ¼½ ½½

Ü ´Øµ = Á Ü~ ´Øµ Ü^´Øµ

(5)Ð(6). Then ^ . Hence, an equation for

@Ø @

can be deduced from the equations (16) and (17). For that,

£ ½ £

·¨Ï ¨ ´ µ Å ´ØµÎ Å ´Ø; µ; ¼½

~ (20)

½ ½

we decompose È in the form

@ @

¾ ¿

· È ´Ø; «µ=AÈ ´Ø; «µ·È ´Ø; ¼;«µ

~ ~

¼¾ ¼¾ ½¾

È ´Øµ È ´Øµ È ´Øµ

¼¼ ¼½ ¼¾

@Ø @«

~ ~ ~

4 5

È ´Øµ= ¾Ä X ¢ X ¢ Z :

È ´Øµ È ´Øµ È ´Øµ

½½ ½¾

£ ½ £

¼½

·¨Ï ¨ ´«µ Å ´ØµÎ Å ´Ø; «µ;

¼¾ (21)

¾ ¾

£ £

~ ~ ~

È ´Øµ È ´Øµ È ´Øµ

¾¾

¼¾ ½¾

with the initial and boundary conditions ~

Since the values of È are nuclear operators, the functions

~ ~ ~ ~ ~

È È È È È

¼¾ ½½ ½¾ ¾¾

¼½ , , , and can be represented in the follow-

È ´¼µ = È ;

¼¼ ¼ >

ing integral form <

È ´¼;µ=¼; " ¼;

¼½ (22)

È ´¼;«µ=¼ Æ « ¼;

¼¾

~ ~

È ´Ø; "µ=¼; È ´Ø; Æ µ=¼; Ø ¼:

È ´Ø; µf ´ µ d ; f ¾ X; È ´Øµf =

¼½ ¼¾

¼½ ¼½

" Z

¼ Sketch of proof. The derivation of the equations (19)Ð(21)

~ ~

È ´Øµg = È ´Ø; «µg ´«µ d«; g ¾ Z; ¼¾

¼¾ is routine and, therefore, here we give a brief sketch of that.

Z Writing the Riccati equation (17) in componentwise form,

¢ £

~ ~

È È È

¼½ ¼¾ ¼¼

È ´Ø; ; µf ´ µ d ; f ¾ X; ´Øµf ´ µ=

È we obtain the operator equations for , and and,

½½ ½½

using the fact that È is a scalar product solution of the Riccati

£ ¢ È

equation (17), we deduce the equations for the kernels ¼½

~ ~

È ´Ø; ; «µg ´«µ d«; g ¾ Z; È ´Øµg ´ µ=

½¾ ½¾

È È È È

¼¾ ¼½ ¼¾

and . Then, the equations (19)Ð(21) for ¼¼ , and

Æ Z

¼ will be seen. The conditions in (22) follow from the initial

¢ £

~ ~

È ´Øµg ´«µ= È ´Ø; «; µg ´ µ d; g ¾ Z;

condition in (17) and from the deﬁnition of the functions ¼½

¾¾ ¾¾

Æ È

and ¼¾ . The complete proof of this theorem can be found in

[8].

È È È È È

¼¾ ½½ ½¾ ¾¾

where ¼½ , , , and are the respective kernels

; ¾ [ "; ¼]

being square integrable functions in and in È

Theorem 3. Under the notation (18), the functions ½½ ,

«; ¾ [ Æ; ¼] Ø ¼

È ´ µ

under ﬁxed . Let È ¾¾ ½¾ , and are solutions in the ordinary sense of the

Z equations

È ´Ø; µ= È ´Ø; µ d ;

¼½ ¼½ ½ ½

@ @ @

· · È ´Ø; ; µ

½½

@Ø @ @

È ´Ø; « µ d« ; È ´Ø; «µ=

¼¾ ½ ½ ¼¾

£ ½ £

=¨ ´ µÏ ¨ ´ µ Å ´Ø; µÎ Å ´Ø; µ;

½ ½½

½ (23)

½ ½

Z Z

@ @ @

· · È ´Ø; ; «µ

È ´Ø; ; µ d d ; È ´Ø; ; µ=

½¾

½½ ½ ½ ½ ½ ½½

@Ø @ @«

" "

Z Z

£ ½ £

=¨ ´ µÏ ¨ ´«µ Å ´Ø; µÎ Å ´Ø; «µ;

½¾ ½

½ (24)

¾ ¾

È ´Ø; ; «µ= È ´Ø; ;« µ d« d ;

½¾ ½¾ ½ ½ ½ ½

@ @ @

" Æ

· · È ´Ø; «; µ

Z Z ¾¾

@Ø @« @

È ´Ø; « ; µ d d« ; È ´Ø; «; µ=

¾¾ ½ ½ ½ ½ ¾¾

£ ½ £

=¨ ´«µÏ ¨ ´ µ Å ´Ø; «µÎ Å ´Ø; µ

¾ ¾¾ ¾

Æ Æ

(25)

¾ ¾

¾ [ "; ¼] «; ¾ [ Æ; ¼] Ø ¼

where ; , and . We will with the initial and boundary conditions

Ü^´Øµ È È È È

¼½ ¼¾ ½½

derive an equation for in terms of ¼¼ , , , ,

È È

È ´¼;;µ=È ´Ø; "; µ=È ´Ø; ; "µ=¼;

½¾ ¾¾

½½ ½½

and . Therefore, at ﬁrst we present the equations for ½½

" ¼; " ¼; Ø ¼;

them. >

È ´¼;;«µ=È ´Ø; "; «µ=È ´Ø; ; Æ µ=¼;

½¾ ½¾

Theorem 2. Let ½¾ (26)

" ¼; Æ « ¼; Ø ¼;

Å ´Øµ=È ´ØµC · È ´Ø; ¼µ;

¼¼ ¼¾ È ´¼;«;µ=È ´Ø; Æ;µ=È ´Ø; «; Æ µ=¼;

¾¾ ¾¾ ¾¾

£ £

´Ø; µC · È ´Ø; ; ¼µ; Å ´Ø; µ=È

½¾ ½

Æ « ¼; Æ ¼; Ø ¼:

(18)

¼½

£ £

´Ø; «µC · È ´Ø; «; ¼µ; Å ´Ø; «µ=È

¾¾ ¾ ¼¾

Sketch of proof. The derivation of the equations (23)-(25)

¾ [ "; ¼] « ¾ [ Æ; ¼] Ø ¼

where , and . Then the functions is similar to the derivation of the equations (19)-(21). Writing

È È È ´ µ

¼½ ¼¾

¼¼ , , and are solutions in the scalar product sense the Riccati equation (17) in componentwise form, we obtain

~ ~

of the equations ~

È È È

½¾ ¾¾

the operator equations for ½½ , and and, using the ~

fact that È is a scalar product solution of (17), we deduce

È ´Øµ=È ´ØµA · AÈ ´Øµ·È ´Ø; ¼µ

¼¼ ¼¼ ¼¼ ¼½

È È È

½¾ ¾¾

the equations for the kernels ½½ , and . Then, the dØ

È È È ³~ ´Øµ=³^ ´Ø; ¼µ

½¾ ¾¾ ¾ ¾

equations (23)-(25) for ½½ , and will be seen. The In a similar way, . Therefore, ¾

conditions in (26) are easy to verify. For the complete proof

dÞ´Øµ=dÞ ´Øµ C Ü^ ´ØµdØ ³^ ´Ø; ¼µdØ

of Theorem 2, we again refer to [8]. ¾

= dÞ ´Øµ C Ü^ ´ØµdØ ³~ ´ØµdØ:

Ü ´Øµ Ü´Øµ Theorem 4. The best estimate ^ of based on the

Using this equality, we can write the equation (16) in com-

´×µ ¼ × Ø ´ observations Þ , , exists and it is a solution in ponentwise form as follows:

the mild senseµ of the equation

dÜ^´Øµ=´AÜ^ ´Øµ·³ ^ ´Ø; ¼µµdØ

£ £ ½

~ ~

· È ´ØµC · È ´Øµ Î dÞ´Øµ;

¼¼ ¼¾

£ ½

·´È ´ØµC · È ´Ø; ¼µÎ dÞ´Øµ;

¼¼ ¼¾

Ü´¼µ = ¼; Ø ¼;

^ (32)

Ü´¼µ = ¼; Ø ¼;

^ (27)

^ ^

³~ ´ØµdØ d³~ ´Øµ=

½ ½

Þ ´ µ

where the random process called an innovation process

£ £ £ ½

~ ~ ~

· È ´ØµC · È ´Øµ Î dÞ´Øµ;

½¾ ¾

satisﬁes ¼½

³ ´¼µ=¼; Ø ¼;

~ (33)

dÞ´Øµ=dÞ ´Øµ C Ü^ ´ØµdØ ³^ ´Ø; ¼µdØ;

^ ^

³~ ´ØµdØ d³~ ´Øµ=

¾ ¾

Þ´¼µ=¼; Ø ¼;

(28) d«

£ £ £ ½

~ ~ ~

· È ´ØµC · È ´Øµ Î dÞ´Øµ;

¾¾

¼¾ ¾

³^ ³^ ´ ¾

the random processes ½ and are solutions in the ordi-

³ ´¼µ=¼; Ø ¼:

~ (34) ¾

nary senseµ of the equations

From

@ @

· ³^ ´Ø; µdØ

£ £

@Ø @

È ´Øµh = È ´Ø; «µhd«

¼¾ ¼¾

£ £ ½

=´È ´Ø; µC · È ´Ø; ; ¼µµÎ dÞ´Øµ; ½¾

(29) Z

¼½

@ @

= ´Ø; «µhd« È

¼¾

· ³^ ´Ø; «µdØ

@«

@Ø @«

£ £

= È ´Ø; ¼µh È ´Ø; Æ µh

¼¾ ¼¾

£ £ ½

=´È ´Ø; «µC · È ´Ø; «; ¼µµÎ dÞ´Øµ;

¾¾ (30)

¼¾

= È ´Ø; ¼µh; h ¾ X;

¼¾ £

satisfying the initial and boundary conditions ~

È ´Øµ = È ´Ø; ¼µ ¼¾

it follows that ¼¾ . Using this equality in

8 Ü

(32), we obtain that ^ is a mild solution of the equation (27).

³^ ´¼;µ=¼; " ¼; <

½ Furthermore, from

³^ ´¼;«µ=¼; Æ « ¼;

¾ (31)

Z Z

¼ ¼

³^ ´Ø; "µ=¼; ³^ ´Ø; Æ µ=¼; Ø ¼;

£ £

½ ¾

È ´Øµf = È ´Ø; ; «µf ´ µ d d«

½¾ ½¾

Æ "

¼¼ ¼½ ¼¾ ½½ ½¾ ¾¾

Z Z

¼ ¼

È È È È È È

and , , , , , are solutions of the ¾

= È ´Ø; ; «µ f ´ µ d d«

equations (19)Ð(21) and (23)Ð(25) satisfying the initial and ½¾

@ @«

" Æ

boundary conditions in (22) and in (26), respectively.

È = ´Ø; ; ¼µ f ´ µ d ; f ¾ X;

Proof. Let ½¾

¿ ¾

Ü´Øµ

^ it follows that

5 4

³~ ´Øµ

; Ø ¼: Ü~ ´Øµ=

£ ¢

£ £

h ´ µ= È ´Øµ ´Ø; ; ¼µ h È

½¾

¾ ½¾

³~ ´Øµ

= ´Ø; ; ¼µh; h ¾ Z:

Denote È

½¾

h i

Also, we have

³^ ´Ø; µ= ³~ ´Øµ ´ µ d ; " ¼; Ø ¼;

¢ £

£ £

´Ø; µh ´Øµh ´ µ=È È

¼½ ¼½

h i

³^ ´Ø; «µ= ³~ ´Øµ ´ µ d; Æ « ¼; Ø ¼:

= È ´Ø; µh; h ¾ X:

¼½

Æ @

Then for this functions, the conditions in (31) hold. Further- Therefore,

¢ ¡ £

£ £ £ ~

more, we have ~

h ´ µ ´ØµC · È ´Øµ È

½¾

¾ ¼½

£ £

= ´Ø; µC · È ´Ø; ; ¼µµh ´È

½¾

¼½

³^ ´Ø; µ d ³~ ´Øµ=

½ ½

Å ´Ø; µh; h ¾ Z: =

= ³^ ´Ø; ¼µ ³^ ´Ø; "µ=³^ ´Ø; ¼µ:

½ ½ ½ @

This implies the linear observation system are perturbed by the sum of

¡ £

¢ white and wide-band noise processes.

£ £ £

~ ~

h ´ µ Ì ´Ø ×µ È ´×µC · È ´×µ

½ ½¾ ¾

¼½ The three major steps are needed to realize the formulae

@=@µÅ ´×; × Ø · µh; × · Ø "

´ for the optimal ﬁlter:

= ; h ¾ Z:

¼; × · <Ø "

¨ ¨ ¾

(a) Computation of the relaxing functions ½ and under

£ £ ¾

Using this equality in (33), we obtain given autocovariance functions ½ and of the wide-

³ ³

½ ¾ i

h band noise processes and acting to the state and

³ ´Øµ ´ µ

~ observation systems. This can be done by use of the

Z direct and inverse Fourier transformations as it is de-

£ £ £ ½

~ ~

~ scribed in Section 2.

= Ì ´Ø ×µ È ´×µC · È ´×µ Î dÞ´×µ ´ µ

½ ½¾

¼½ ¾

¼ Z

Ø (b) Numerical solution of the Riccati equation (19) and the

Å ´×; × Ø · µÎ dÞ´×µ: =

½ system of equations (20)Ð(21) and (23)Ð(25) under ini-

;Ø " µ ÑaÜ´¼ tial and boundary conditions in (22) and (26).

Hence, (c) Design the optimal ﬁlter in the linear feedback form by

i h

use of the formulae (27) and (28)Ð(31).

³~ ´Øµ ´ µ d ³^ ´Ø; µ=

Z Z

Ø References

= Å ´×; × Ø · µÎ dÞ´×µ d

" ;Ø " µ

ÑaÜ´¼ [1] Fleming W. H. and Rishel R. W. Deterministic and

Z Z

@ Stochastic Optimal Control. Springer, Berlin, 1975.

= Å ´×; × Ø · µÎ d dÞ´×µ

Ø × " ´¼;Ø " µ

ÑaÜ [2] Kushner H. J. and Runggaldier W. J. Filtering and con-

Ø ½

trol for wide-bandwidth noise driven systems. IEEE

= Å ´×; × Ø · µÎ dÞ´×µ

½ Transactions on Automatic Control, Vol.AC-32, No. 2,

ÑaÜ´¼;Ø " µ

Ø pp. 123Ð133, 1987.

= Ì ´Ø ×µÅ ´×; ¡µÎ dÞ´×µ ´ µ;

½ ½

¼ [3] Bashirov A. E. On linear ﬁltering under dependent

wide-band noises. Stochastics, Vol.23, No. 4, pp. 413Ð

³^ ³^ i.e. is a mild solution of the equation (29). To show that ½ 437, 1988.

is an ordinary solution of the equation (29), deﬁne the graph

´ d=d µ norm in D by [4] Bashirov A. E., Etikan H. and S¸emi N. Filtering,

smoothing and prediction of wide-band noise driven

¾ ¾ ¼ ¾

kf k = kf k · kf k :

~ ~

D ´ d=d µ X X linear systems. Journal of the Franklin Institute,

Vol.334B, pp. 667Ð683, 1997.

´ d=d µ

Then D becomes a Hilbert space and ¡

[5] Bashirov A. E. Representation of linear systems dis-

d=d ¾Ä D ´ d=d µ; X : turbed by wide-band noise. Preprint EMU AS 97-21

Using (submitted for publication in Appl. Math. Letters). £

£ [6] Bensoussan A., Da Prato G., Delfour M. S. and Mit-

´Ø; "µC · È ´Ø; "; ¼µ = ¼; Ø ¼; Å ´Ø; "µ= È

½¾ ½

¼½ ter S. K. Representation and Control of Inﬁnite Dimen-

´¡; ¡µh Å sional Systems 1,B¬urkhauser, Berlin, 1992.

one can easily verify that the restriction of ½ to the set

[¼;Ø] ¢ [ "; ¼] Ä ´¼;Ø; D ´ d=d µµ Ø>¼

belongs to ½ for all

[7] Curtain R. F. and Prichard A. J. Inﬁnite Dimensional

¾ Z ³^ and for all h . This sufﬁcies to conclude that is an ordinary solution of the equation (29). In a similar manner, it Linear Systems Theory, Lecture Notes on Control and

Information Sciences, Vol.8, 1978.

³^ can be proved that ¾ is a mild solution of the equation (30). [8] Bashirov A. E. Control and ﬁltering of linear systems 6 Conclusions driven by wide-band noise, Preprint EMU AS 98Ð07.

In this paper an approach to wide-band noise, based on a distributed delay of white noise, is developed. This approach allows to reduce the state-observation system disturbed by both white and wide-band noise processes to a linear system, disturbed by only white noise. This reduction is applied to linear ﬁltering and the complete set of formulae for the optimal ﬁlter is obtained when both the linear state system and