E l e c t r i c a l Engineering Department G i l m e r L. Blankenship

University of Maryland College Park, Maryland 20742

Several approximation techniques which have been developed in scattering theory are adapted to the t rea t - ment of a nonlinear filtering prcblem involving weak parametric noise. Formal asymptotic expansions are pre- sented for c e r t a i n averages of the un-normalized condi- tional density of the signal conditioned on the para- metric noise and the observations. These are validated by an analysis of the associated function space integral.

1. Introduction

Efficient representations for the conditional statis- t i c s of dependent randan processes have proven t o be very d i f f icu l t to cons t ruc t in a l l but a few special cases. For instance, in the case of conditional diff- usion processes,methods from algebraic and different ia l geometry, mathematical physics, par t ia l d i f fe ren t ia l equations, and probability theory have been used to understand the structure of the representations or t o construct systematic approximations to then-- this via an intensive s t u d y of the stochastic partial differen- t i a l equation, the Zakai - Mortenson equation, which i s central in this case. The work reported here is in the s p i r i t of t he l a t t e r i n t ha t scme classical perturbation methods are applied to the analysis of the evolution equations of nonlinear filtering.

We consider the Born approximation, a two-time method, the method of smoothing, and the nonlinear direct interaction methgd. All of these are applied to the problem: Estimate x (t) given y: = {p ( 8 ) ,%sLt]

dxE(t) = axE ( t ) d t + bdw(t) + en(t)x ' ( t )dt

d y E (t) = cx' ( t ) d t + dv(t) (1.1)

xE (0) = x, yE (0) = 0, 0 5 t 5 T.

Here a,b,c,x are constants, E > 0 is a parameter, w,v are independent standard W i e n e r processes, and n ( t ) is a randcm process independent of w,v which has trajec- tor ies i n D[O,T], zero m e a n , and stationary statistics

En(t)n(s) = N ( t - s ) (1.2)

distribution of (t) given yf . I f we compute the An "estimate" is an evaluation of the conditional

L

conditional density of x E ( t ) given (y:,nt),

pE(x,tlyf,nt), and integrate over the paths

the desired result follows. Arguments which 111, sec. 6.5, produce pE via

where the un-normalized conditional density

s aY of n, then

are standard

of Naval Research under contract No. N00014-79-C-0808. *This research w a s supported in par t by the Office

duE(x,t) = (~u ' (x , t ) - En(t)ax["uE(x, t ) ] )dt

UE (X,O) = n (x)

+ cxuE (x,t)dyE (t) (1.4)

D = %'a:(.) - aax(x-) (1.5)

Here II (x) is the initial density Of XE(0). Evidently, when E = 0 one recovers the Kalman - Bucy problem ( i f n is Gaussian). It is natural to consider an asymptotic analysis for (1.4) when E i s small.

Elsewhere, we have studied similar problems l i k e (1.1) i n which the coefficients were Markovian using a diffusion approximation and PDE methods (21. The work reported here e l e m a n t s [21 by investigating scme of

particular, our calculiltions should be compared with the the more classical methods of mathematical physics; i n

work of P.L. Chow [3][4] on wave propagation problems of the form

a w a t = (+A + en(x , t ) )u . (1.6)

In Chow's problem n(x, t ) corresponds to a fluctuation in the index of refraction or i n the sound speed prcf i le . Our recent survey [5] makes other connections be'.ween some perturbation methods for PDE's i n physics and con- t r o l theory. In [6] w e discussed some of e e s e meth& for stochastic ODE'S in the context of s tab i l i ty and wave propagation problems. The papers [ 7 ] [ 8 ] of Papanicolaou are also very helpful in these problems.

that the system (1.1) is typical of a class of es t i - It w i l l become evident in the course of the analysis

mationproblems which could be treated by the same methods (see [21 for indications of the range of such problems). The results reported here are: fo- and incomplete i n cmitting precise convergence estimates for the approxi- mations. We shall also pass over qaestions of existence, uniqueness and regularity. of the solutions of (1.4) since these require very different methods than those which occupy us here. We shall present some evidence for the validity of the approxi-ations i n tenns of function

Kac formula). The latter are also useful in existence space integrals (the Kallianpur - S t r i e b e l or Feynmann - arguments [91-[11].

2. Born Approximation

The method of regular wturbat ion, or the Born approximation, has a long history of successful use in scattering theory. I t is recognized to have a limited range of validity, because of the emergae of secular terms; however, i ts simplicity in computations has pro- mted its application.

Assume that uE i n (1.4) can be urpanded a8

u"(x,t) = 1 E un(x,t) + O ( E N n pH1 (2 .1)

IF0

:&ere N is an integer greater -one. Substituting (2 .1)

51 0191-2216/80/0000-0051$00.75 8 1980 IEEE

into (1.4) and collecting the coefficients of l ike powers of 5 , w e find

where u = 0 and 6 is the Kronecker delta. -1 nm

To solve the recursive system (2.2) consider the case n = 0,

duo = h o d t + cxuodyE (t)

UO(X,O) = %(X)

Now u(x,t) is a random process on the sample space of the three dimensiod process (v,w,n). Its (conditional) mcments are naturally of interest. Let<.> be the tation with respect to the process I n ( s ) , o -< s <- T L i.e., its marginal distribution. Since c n ( t b = 0,

c n ( t ) n ( s b = N ( t - s ) , we have

*ere and in all the expaasions which follow w e use the Landau symbol merely to indicate the next term. Convergence is not claimed a t th i s po in t .

52

. ( zk(z,q;s,p)aq(quo(q,p))~)~ + 0(E3) (2.9) R

( a f t e r an integration by parts). and th is is a closed form expression. mere kZ = a k.) Expansions for the

second cuE (xl,t)uE (x2,t)> and higher order moments may

be deteymined i n a similar fashion.

secular terms (higher order tenas in E which grow l ike

tk, k 1. 1). Hence, it is valid as an alpoximation.only

Tu see why, mnsider the E term and associate w i t h it for short intervals (O, t ) .pis is the case in (2 .9) .

(for the purpose of i l lustration only) the stationary integral

T ( t , s ) = Io N(6-p) ( I, K(x-zit-s)K(Z-q;s-p)

In other problem the Born woximation produces

t s

(2.10)

*K(q;P)dqdz)dpds

where K(x;t) = exp(-X2/2t)/(2nt)+. (This is the simplest

process.) The double convolution T can be analyzed by form of k, corresponding t o no observations of a Wiener

transform methods. Let

(2.11)

so lbiC) = @;<I (2.12)

and t h p has a double pole i n tkcomplex p-plane a t p = Y /2. This pole produces the secular ( i n t) tern in T(x;t).

The analysis of the aonstationary integral in (2.10) is much more complex, but it evidently exhibits the

ed expansion techniques usd in the l i t e ra ture were same flaw in the expansion. Most of the more sophisticat-

developed to circumvent t h i s problem.

3 . T!m - Time Method

This metbod, where applicable, is a wnvenient, sim- ple way of cbtaining asymptotic expansions uniformly valid on f ini te intervals . It was f i r s t adapted t o stochastic ODE'S by Papanicolaou and ~eller [121. It w a s used for the stochastic PDE'S of wave propagation by chow [3]. In [2 ] we used t h i s method on f i l t e r ing pro- blems by considering initial layer expansions i n singu- la r ly perturbed versions of the zakai - Mortenson equa- tion. The treatment which follows i s simpler and r a r e direct.

We seek a solutioq of (1.4) i n t he fom u'(x,t) = U" (x t t ,7 ) w i t h 'c = t. (The scaling arises because of the assuqt ion that n has zero m e a n . ) Assuming

uE(x;t,r) = z c"Un(x;t,T) + O(EW1) N

IF0 (3.1)

and substituting this into (1R) (note du' = dtuE = dtUE + c2dTUE 1 , gives the recursive system

dtUo - moat - =nodye (t) = 0 , Uo (x;O,O) = w (X) (3-2)

Uc (xl,t)U6 (x2,t). We will not carry this calculation

out here.

tics of the parametric process n(t) in a nontrival way. N o t e that (3.10) involves the second order statis-

It is a linear evolution equation of a type which we have not seen previously in estimation theory.

In wave propagation problans when the refractive index s hcsogeneous (stationary) in space and time, one can

which exploit the convolution structure of the integrals -olve the corresponding equation by transform methcds

because of the essential inhcsogeneity in the spatial in (3.10). In the present case this is not possible

(x,z) variables. One should, however, be able to replace k(x,z;t,s) by its stationary version, corres- ponding to the "steady state" Kalman filter, and achieve a certain simplification.

We are left with the problem of solving (3.10).

4. Method of Smoothing Perturbation

The systematic developnent of this method was

carried out by J.B. KelIer13 based on earlier work by Ament. BOurret,Tatarski, and others have applied variations of the method to wave propagation problems, see [Y] [ 6 ] for references. pollowing mller, consider a general stochastic equation of the form

L(W)U = f (4 .I) where f is a deterministic function and L(u) is a randan operator of the fonu L(w) = Lo + 6 V (w) with Lo

deterministic and V(w) randcan. If L is invertible almost surely, then

<* = <L%f (4.2)

a d so on. In addition, the Un are required to satisfy

certain boundedness conditions.

The solution of (3.2) is given by

U (x;t,r) = k(x,z;t,O) U(z;r\ dz 0 IR (3.5)

a r e U ( Z ; T ) is a slowly varying rand- function of to be determined, which must satisfy

ir (x; 0) = ll (x) (3-6)

Using (3.5) in (3.3) gives

ul(x;t,r) = - I: I, IR

k(x,z;t,s)n(s)

(3.7) 'a,(z k(X,Z;S,O)U(q;T)dq)dzdS

whose expectation <U > is assumed to be non-secular

(in t). using (3.7) (3.5) in (3.4) gives 1

The second term exhibits linear growth in t. For the mean solution to grow no more rapidly than t we require

lim CU >/t = 0 e- 2 (3.9)

This gives the equation

. az[z 1 kCz,q;s,u).a [q kCq,r;u,O) q ' I

R

<~(r,T)Xr]dq]dzduds (3.10)

<Y(x;O)> = n(x)

have for <%. If one can solve (3.101, then fran (3.11, we

<U'(x;t)>= k(x,z;t,O)<E(z,6 t)h32+0(6) (3.111 i 2

R

The second n n t may be expanded in a similar way

starting from the evolution equation for U:2 (x1,x2;t) =

or <L-5-1<* = f (4.3)

= Lo-'2<vL-b 0 + O ( 6 3 ) . Using this in (4.3) and dropping the O ( c ) term, the 3

equation of the smoothing perturbation is

Lo<**2<vL;b <* = f (4.5)

This result was applied to the wave propagation problem

(1.6) by Chow. In his case Lo4/at-$A, V=n(t,X,w),

and (4.5) is an integro-differential equation.

In our case the operators are

Lo = dt(.);p(.)dt-cx(.)dy6(t)

v(w) = + n(t,dax[x-1 (4.6)

Evidently, Lo is itself a random operator Since Y'iS a

random process. So long as we work on the product sample space and integrate with respect to the marginal distribution of n(t), this introduces no Special

difficulties. The inverse Lo1 is the integral Operator

with kernel k(x,z;t,s) and (4.5) becanes

~ u 6 ( x , t ) > - ~ u ' ( x , t ) > d t + ~ u 6 (x,t)>dy6 (t)

t (4.7)

+e2j N(t-s)ax[x k ( x , z ; t , s ) a Z [ ~ u 6 ( ( z , ~ ) > l d ~ l d s I 0 R

<U6(X,O)> = n(x) , CSksT

53

A similar equation for the second -t <ue > could be obtained by applying the smoothing equation (4.5) to the

evolution equation for uf2.

12

case when k is spatially a d -rally hamogeneous The integral in (4.7) is a convolution. In the

(this does not occur in our problem but does in wave propagation problems ) one can solve (4.7) by transform methods. If one applies the conventional, deteministic two-the method to (4.71, it can be swlified. Writing

<ue(x,t)> = <U'(x;t,?>, ? = e t, then the first term

the equation (3.10) derived using the stochastic two- in the expansion of as> in powers of s will satisfy

time method. This means that when both methods are applicable, the two-time method may be regarded as an asymptotic approximation to the smoothing perturbation

as t-, s+O with T = 6 C conatant.

3

2

2

5. Direct Interaction Approxiaation

This method, due originally to Kraichnan14, was interpreted and applied to wave propagation problems by It produces nonlinear approximations which

are know to be superior to the smoothing perturbation in scattering problems. The direct interaction approximation is derived by iterating the smoothing perturbation equations.

4

Let Klx,Z;t,s) be the randan Green's function for (1.4). That is,

dtK(tx,z;t,s)~K~x,z;t,s)dt+cxk6 (x,z;t,s)dye (t)

- sn(t)ax[se(x,z;t,s) I dt (5.1)

t&s lim Ks(x,Z;t,s) = ~(x-z) , SCW.

Applying smoothing (4.5) to (5.1), one finds that the mean Green's function satisfies

dtae>(x,z;t,s) = &K'>(x,z;t,s) dt

+ cdK'>(x,z;t,s)dys(t) (5.2)

+ e 2 rN(tq)ax[x I k(x,r;t,q) - <K'>(r,z;q,s)dr]dq + O ( S )

8 R 3

If we now average (5.1) and rewrite it as

k(x,z;t,s) = <XG>(x,Z;t,S)

4~~k(x,r;t,q)ar[*(q)Ke>(r.e;q,s) ldrdq (5.3)

s R

then substituting this in (5.2) leads to atas> = *ICs> dt + csKKs> dy' (t)

t + e 2 j !i(ttq)ax[x ~e>~x,r;t,q~<Ks~~r,z;q,s~dr~dq+O(E 1 I 3

S R (5.4)

lim &>(x,z;t,s) = ~(x-z) t&s

Dropping the O(s ) term in (5.41, one obtains the direct

Green's function. Clearly interaction approximation for the first maoent of the

3

<ue(x,t)> = as>(x,z;t,o)n(z)az I (4.5) R

with as> f r a (5.4) gives the approaimation of the first moment of the conditional density. Applying the same two step technique to the evolution equation for u6(xl,t)uU(x2,t) leads to approximations for the second

moinent.

since the nonlinear equation (5.4) is difficult to solve, its potential value in applications is unclear. There may, nevertheless, be S Q L ~ u s e m infomation implicit in the structure of this equation.

6. Slmmary and Validity of the Approximations

The techniques of asymptotic analysis from scatter- ing thaory that we have used produce a hierarchy of approximations for the original filtering prablem (1.1) in terms of its natural asymptotic limit. 'Ihe results range in complexity and accuracy fram the nonlinear di- rect interaction approximation to the simple Born ap- proximation with its limited validity. (See Figure 1.)

Direct Interaction Approximation Equ. (5.4)

A &

Smoothing Equ. (4.7)

a & 2

Two-Time Method Equ. (3.10) * *

Born Approximation mu. ( 2 . 9 )

filtering prcblem (1.1) - (1.4). Figure 1: Hierarchy of approximations to the

However, the analysis so far does not reveal fully

proceed further, we must estimate the error terms asso- the relationships among the several approximations. To

ciated with each. In 121 we indicated the use of PDE methods for this purpose. Here we follow Chow and use

Such integrals have been used recently by M. Davis and function space integrals as a basis for the estimates.

E. Kushner and others in connection with nonlineat E. Pardoux, awng others, and earlier by G. Kallianpur,

filtering. Their use here is in the spirit of [9] [15]

basis for approximations. Unfortunately, our analysis (and earlier work by Kushner) where they served as a

of the expansions is not ccsnpletely rigorous, and thus not definitive, sharing in this the chief defect of Chow's work [3] [4].

In this section we specialize (1.4) to

du'(x,t) = ax2 u"(x,t)dt - en(t)ax[x u'(x,t)]dt

UE(X,O) = n(x)

, +cxuC(x,t)dyE(t) (6.1)

with, of course, no loss of generality. In writing the solution of (6.1) as a Wiener integral, the term en(t)x has the effect of shifting the mean of the Wiener meas- ure. Since the two measures are equivalent, this creates no problem.

Let C [O,T] be the space of R-valued continuous func- tions'z on [0,t1 with z (o)=o. Let {so ,sl,. . . ,s I satis- fy O=s cs c.. .cs = t, and let zsx be a piecewige linear (or poPygAnal) A c t i o n on [~,t]satisfying zs.x=xi=z(si) where z(s) is a wiener path. For a smooth fdctional G ( z ) on Co the Wiener integral is defined by

'Ihe Wiener-Feynman integral is defined as follows:

54

The Wlianpur - Striebel formula backward equation [ll] -- formally, tering problem (6.1) must be written far

for the fil- the associated

J S

J S J S

.n(z(T-s)+x)}

Eere we have commuted and <- > which must be justi- fied. We use the assumption that n is Gaussian to eval- uate the inner average

J S J S

(6.6)

- f 1: ~ ~ c ~ n ~ ( t ) d t ] > .

This integral can be evaluated by using the character- istic function of the Gaussian distribufion and a limit- ing argument. Dropping the term of 0 (E ) and higher in the exponent, the resrilt is

J s J s

For convenience, we define

J S J s

so

must CcmQute the mean Green's function $(a,b;t,s) . Remce r(z(T-s)+x) in (6.10) by ~(z(T-s)+x), add x by a+, then the resulting integral is G.

To obtain the direct interaction approximation, we

We can rewrite (6.11) as

<KE(a,b;t,s)> = k(a,b;t,s) + Q(a,b;t,s)

where k(a,b;t,s) is the transition density in the Kalman - Bucy problem (recall (2.4)) and

(6.12)

(Here -- means the arguments in the previous line are repeated. )

we replace (s,r) In the exponential in Q by To repr uce the direct interaction approximation

55

2 E ELb(s,q) + EZH:-b(q,r) and all the resulting

expression Q,. A shple calculation shars that

<Kc> = k + Q, (6.14)

is the direct interaction approximation. The error in the approximation is thus Q - Ql *

The smoothing approximation results from discard- ing the upper-diagonal integral e2H2- (q,r) in the

expression for Q,. Call the result Qz; then a b

<KE> k +Q 2 (6.15)

is the smoothing approximation. he error is Q - Q2- The regular perturbation method is obtained by

expanding the exponential in (6.11) in powers of E and retaining the first two terms. The nature of the error is clear.

Further details of this analysis will be published elsewhere.

References

1. E. Wong, Stochastic Processes in Infomation and Dynamical Systems, McGraw-Hill, New York, 1971.

2. G. Blankenship, A. Haddad, "Asymptotic analysis of a class of nonlinear filtering problams," Proc. IPAC- IRIA Workshop S i n g u l a r Perturbations in Control, Paris, June, 1978.

3. P. Chow, "Perturbation methods in stochastic wave propaqation," SIAn Review, u(1975), pp. 57-81.

4. -, "On the exact and approximate solutions of a random parabolic equation," SIAM J Appl. Math., z(1974), pp. 376-397.

5. G. Blankenship, "Asymptotic analysis in mathematical physics and control theory: sane problems with conmwn features," Richerchi di Automatica, 1980.

6. -, "Perturbation theory for stchastic ordinary differential eauations with auulications to 0Dti-l - - waveguide analysis," in Applications of Lie Group Theory to Nonlinear Network PlOble~ls, C.A. Desoer, ed., Western Periodicals, North Hollywood, 1974, pp. 51-77.

7. G. Papanicolaou, "Introduction to the asymptotic analysis of stochastic equations," in A.M.S. book series Lectures in Applied Mathematics, ~ 1 . 16, Providence, R.I., 1977, pp. 109-148.

8. -, "Asymptotic Analysis of stochastic equations,"

M. Rosenblatt, ed., Math. Assoc. America, 1978, pp. 111- in M.A.A. studies No. 18, Studies in Probability Theory,

179,

9. J. Baras, G. Blankesnhip, "Nonlinear filtering Of diffusion processes: a generic example," to appear.

lo. S.K. Utter, to appear, Richerchi di Autmtica, 1980.

11. E. Pardoux, "Backward and forward stochastic part- ial differential equations associated with a non linear

56

Control, Ft. Lauderdale, 1979. filtering problam," Proc. Conference on Decision and

12. G. Papanicolaou, J. Keller, "Stochastic differen- tial equations with applications to random harmonic oscillators and wave propagation in random media, SIAM J. Appl. Math., %(1971), pp. 287-305.

tion in randan media," Proc. Sym.. -1. Math.,-?:. 16, 13. J. Keller, "Stochastic equations and wave propaga-

mer. mth. SOC., Providence, F.I., 1964, pp. 145-170.

14. R.H. xraichnan, "DYMULCS of nonlinear stochastic systgls," J. Math. Physics, 2(1961), p P * 124-148.

15. G. Blankenship, J. ~ a r a s , "Accurate evaluation of stochastic Wiener integrals," to appear.