Least-squares filtering and smoothing for linear distributed parameter systems

Automatica, Vol. 7, pp. 315-322. Pergamon Press, 1971. Printed in Great Britain.

Least-Squares Filtering and Smoothing for Linear Distributed Parameter Systems" Filtrage et lissage par la m6thode des moi dres carr6s our les syst~mes lin6aires ~t param~tres repartis Filterung u d Gl~ittu g nach der Methode der kleinsten Quadrate ftir lineare Systeme mit verteilten Parametern MmmMa~bHO-rBa~paTrlqecroe qbri~bTprlpoBamle 14 cr~a~rIBaHne ~ I I H H e H H b I X C H C T e M C pacr[pe)2eJIeHHblMri ImpaMeTpaMri J. S. M E D I T C H t

Utilizing the least-squares estirnation viewpoint and the calculus of variations, a KalmanBucy type filter may be formally derived for distributed parameter systems. Summary--The problem of estimating the state of a class of linear distributed parameter systems from noisy measurements is considered from the viewpoint of weighted leastsquares estimation over the spatial domain of the system and the time interval of the measurement data. The problem is reduced to a two-point boundary-value problem via the calculus of variations. The two-point boundary-value problem is then solved in closed form via the sweep method to obtain a Kalman-Bucy type filter. Solution of the smoothing problem then follows directly. Cases are considered where measurement data are obtained over the entire spatial domain of the system or at discrete points in this domain, and where the system is subject to internal and external disturbances as well as measurement errors. Some resulting problems for future study are discussed.

squares e s t i m a t i o n via a p p l i c a t i o n o f the calculus o f variations. In section 3, we o b t a i n a s o l u t i o n o f the resulting t w o - p o i n t b o u n d a r y - v a l u e p r o b l e m in the f o r m o f a K a l m a n - B u c y filter a n d indicate a p r o c e d u r e for h a n d l i n g also the i n t e r p o l a t i o n o r s m o o t h i n g p r o b l e m . A n e x a m p l e is included in this section to illustrate the results. W e extend o u r results to treat the case where m e a s u r e m e n t s are m a d e at discrete spatial l o c a t i o n s in section 4. W e conclude in section 5 with a discussion o f o u r work, an extension t h e r e o f to systems with disturbances, a n d some p r o b l e m s f o r future study. W e also indicate there the relation o f o u r w o r k to previous studies in this p r o b l e m area. W e emphasize t h a t o u r theoretical d e v e l o p m e n t here is purely f o r m a l a n d no claim to m a t h e m a t i c a l rigor is m a d e .

1. INTRODUCTION WE CONSIDER the p r o b l e m o f estimating the state o f a class o f linear d i s t r i b u t e d p a r a m e t e r systems f r o m noisy m e a s u r e m e n t s . T h e e s t i m a t i o n viewp o i n t a d o p t e d is t h a t o f weighted least-squares over the spatial d o m a i n o f the system a n d the t i m e interval o f the m e a s u r e m e n t data. W e t r e a t the two cases where m e a s u r e m e n t s are m a d e either cont i n u o u s l y o r at discrete l o c a t i o n s in the system's spatial d o m a i n . W e give the p r o b l e m f o r m u l a t i o n in section 2 a n d d e v e l o p there the c a n o n i c e q u a t i o n s for least-

2. PROBLEM FORMULATION AND CANONIC EQUATIONS Let f l be a b o u n d e d o p e n set in euclidean N - s p a c e E N with piecewise s m o o t h b o u n d a r y F , a n d let x = (xl . . . . , x s ) d e n o t e a generic p o i n t in fl. L e t t denote time, defined on the fixed interval T = [to, tl], to < tl, a n d define Z = f l x T. C o n s i d e r the h o m o g e n e o u s linear distributed parameter systemt

* Received 3 April 1970; revised 20 October 1970. The original version of this paper was presented at the 2nd IFAC Symposium on Identification and Process Parameter Estimation which was held in Prague, Czechoslovakia during June 1970. It was recommended for publication in revised form by associate editor B. Morgan. This work was supported in full by the Boeing Company, Seattle, Washington 98124. I" Department of Electrical Engineering, University of California, Irvine, California 92664.

au(x, t)_ ~(x)u(x, t)

(1)

Ot t The case where an additive system disturbance is present is considered in section 5 as a simple extension of the results in sections 3 and 4. 315

316

J.S. MEDIrCH

where (x, t)sY. In equation (1), u(x, t) is the . dimensional state function of the system, viz.

ul(x, t) u(x, 0 =

Itere, z(x, t) is an m-vector, H(x, t) is an m × n matrix whose elements arc continuous, bounded functions of x and t, and v(x, t) is an m-vector of unknown measurement errors. We assume that v(x, t)eL"(E). In the sequel, we shall also consider measurement processes which can be modeled as

z(xj, t)=H(xj, t)u(xj, t)+ v(xj, t)

(4)

u.(x, 0 We assume that each u~(x, t), i = 1. . . . . n, is an element in L2(~'~) for each teT. The state function space S(f~) of the system is then defined as a subset++ of the n product space L~(f~) = L2(f~) x . . . x L2(O ). Also in equation (1), 5e(x) is an n x n matrix, linear, spatial, differential, or integro-differential operator whose parameters are continuous and bounded functions of x and/or t for all (x, t)eY.. The adjoint operator ~ * ( x ) of £Z(x) is then defined, formally, by the relation

= < ~*(x)a(x, t), u(x, t)> for all a(x, t) and u(x, t)eS(f~) for each < . , • > denotes the inner product

teT. Here

where the x j, j = 1. . . . . M, are distinct points in f2 and teT. We define the estimation problem to be that of determining u(x, t) for all (x, t)eZ such that the functional f~ J = ½~Jn [u(x, to)--Uo(X)]'Ax, ~[u(a, to) - Uo(a)]dx +

[z(x, t) to

-H(x, t)u(x, t)]'R-l(x, t)[z(x, t) --H(x, t)u(x, t)]dxdt t

(5)

is minimized subject to the constraint in equation (1) and the boundary condition in equation (2). In equation (5), Ax, ~ is the operator

< a(x, t), b(x, t)> Afna'(x, t)b(x, t)dx

o,( for all a(x, t) and b(x, t)eS(O) for each teT. We take the boundary conditions for the system of equation (1) to be inhomogeneous Dirichlet conditions given by the relation u(e, t ) = f ( ~ , t)

(2)

for all (~, t)sF x T where f(ce, t) is a continuous, bounded, n-vector on the same domain. This is a matter of convenience; cases where we have Neumann or mixed conditions on the boundary can also be handled. We assume that the initial state u(x, to) of the system in equation (1) is unknown, but that we have available an a priori estimate Uo(X) of it. We assume further that the solution of equation (1) subject to the boundary conditions in equation (2) is unique for a given u(x, to)eS(f D. Further, we assume that this solution depends continuously on

u(x, to). We suppose now that measurement data z(x, t), (x, t)eY~, are given, and that the measurement process is governed by the relation

z(x, t) = H(x, t)u(x, t) + v(x, t).

(3)

*+It may be the entire space rather than a subset of it, i.e. S(II) ~ L~(~).

,dq

where [ ]-1 denotes the inverse of the indicated operator and Po(x, ~) is an n ×n, symmetric, positive definite matrix kernel which is continuous and bounded for all (x, a)ef~×fL We assume further that Po(x, a)=Po(a, x). This particular choice is made to keep the formulation parallel to that for the corresponding lumped parameter system estimation problem. In the latter case, the spatial dependence is absent and the first term in equation (5) reduces to [ u ( t 0 ) - Uo]'Po l[U(to)- Uo] with Po 1 n x n, symmetric and positive definite. In a similar way, we choose R(x, t) to be m x m, symmetric, and positive definite with its elements continuous and bounded for all (x, t)eE. Here, as in the case of the first term in equation (5), the matrix inverse is used to imitate the formulation of the lumped parameter problem. The choice of the above two matrices is arbitrary. It does, however, allow relative weighting between the a priori estimation error u(x, to)-Uo(X) and the a posteriori measurement residual

z(x, t ) - U ( x , t)u(x, O, as well as a relative weighting of individual components within each of the two telms, over fl in the first term and Z in the second.

Least-squares filtering and smoothing for linear distributed parameter systems The estimation problem here is readily treated via the calculus of variations [1] by introducing a set of Lagrange multipliers --

m

O~(x, 0

present procedure, however, differs considerably from theirs in a number of significant and important details. Since the system in equations (6) and (7) is linear, but inhomogeneous, we propose a trial solution of the form

O(x, 0 =

f

U(X, t)= / P(x, o., t)O(o., t)do.+~(x, t) (10) J

¢.(x, 0 defined for all (x, t)e~, and considering the augmented functional

j=j+fljnO,(x ' ,(au t)

~(x),(x,

t)]dxdt.

au(x, t)- L#(x)u(x, t) at

(6)

and

(x, o., 0ef~ xlC and assumed to be such that and

-H'(x, OR- (x, t)[z(x, t) -H(x, t)u(x, t)]

~k(x, to)= Ax,~, [u(,~ t o ) - Uo(o-)]

(7)

o=f

, P(x, o., t) a~b(o''att) do"

+f n aP(x,ato., t)0(o.,

t)do.4

O~(x, at t)

(11)

From equations (7) and (10), we have

(8)

dO(o., O_

and

at O(x, t , ) = 0

x, t)

Also, ¢(x, t)eS(f~) for all teT. Our problem xs that of determining P(x, o., 0 and ~(x, t). Differentiating in equation (10) with respect to t, we get

au(x,at

for all (x, t)elg. The transversality condition leads to the two relations

P(x, o., t)=P(o.,

f e(x, o., t)~(o., t)do.~S(~).

.~*(x)O(x, t)

at

fl

for all (x, t) and (o., t)eE. Here P(x, o., t) is an n x n, symmetric matrix which is defined for all

The relevant Euler-Lagrange equations are then easily shown to be

dO(x, t)_

317

(9)

for all xe~. We defer treatment of the boundary conditions to the next section. The estimation problem has thus been reduced to a two-point boundary-value problem. We note that solution of equations (6-9) for u(x, t) where t o < t < t, yields the smoothed estimate of the system's state, i.e. the estimate which is based on all of the measurement data over the interval [to, q]. On the other hand, solution for u(x, q) gives the filtered estimate, i.e. the estimate of the state at time tx given measurement data up to and including that at t~. 3. FILTERING AND SMOOTHING We now obtain a solution of the above two-point boundary-value problem by utilizing the sweep method [1]. The same technique was employed by BaYSON and FghZIER [2] to solve the corresponding problem for linear lumped parameter systems. The

~*(a)~k(o., t)

-H'(o., 0R-'(o., 0[z(o., t)-H(o., 0~(o-, O]+H'(o., t)R-'(o., 0 H(o., tl I P(o., q, t)~b(q, t)dq J i'l

where qel2 and d q = d q l . . , dr/,. Substituting this result into equation (11), it follows that

au(x,at t ) _ l a P ( x ,

o., t)~*(o.)0(o., t)do.

- f a P(x, o., t)H'(o., t)R- 1(o., t)" [z(o., t) daja

(o., t)R-'(o., OH(O-, t)P(o., t/, t)

#/(q, t)drldo. + a~(x, at

t)

(12)

318

J.S. MEDITCH

From equations (6) and (10), we have also that

Ou(x, t)__f ..g2(x)P(x, a, t)~(a, t)da ~t .J n + ~(x)~(x, t).

of braces in equation (14). By virtue of the definition of the adjoint operator, we have

t)~(x, t), c~a*(a)~(a, t)> = <[.~P(a)P(x, a, t)]O(x, t), ~9(a, t)>

I =
(13)

--<¢(x, t), [~(a)P(x, o', t)J*~k(a, t)> Equating the right-hand sides of equations (12) and (13), interchanging the roles of u and q in the third term on the right in equation (12) and then interchanging the order of integration there, and finally grouping terms, we obtain the result

=f d

¢'(x, t)[.~(a)P(x, a, t)]*O(a, t)da.

Hence, equation (14) becomes

fa~,(x ' t){O~(_~t t) £Z(x)~(x, t) Ot

.~(x)¢(x, t ) -

P(x, a, t)

- f uP(X' a, t)H'(a, t)R-l(a, t).[z(a, t)

H'(a, t)g-l(a, t)[z(tL t)

-~(x)P(x, a, t)-P(x, a, t)~*(u)

OP(x_,a, t) .~(x)P(x, a, t) Ot

+f~ P(x, 11, t)H'(11, t)R-1(11, t)H(11, t)

-[£P(u)P(x, a,

P(11, a, t)d11}O(e, t)da=O

t)], +fta P(x, tl, t)

H'(11, t)R-1(~1, t)H(11, t)P(11, a, t)d11} for all (x, t)eY. and (x, u, t)ef~ x Y,. Premultiplying this relation by ~O'(x, t) and integrating over x, we have

~,'(x, t){~(;[

.~(x)~(x,t)

(15

@(o', t)dadx = 0 .

Since ~(x, t) is necessarily the nontrivial solution of equation (7), a sufficient condition for the satisfaction of equation (14) is that each term in braces vanish almost everywhere on its domain of definition. We have, therefore, that

- f n P(x, a, t)H'(a, t)R-l(a, t) O~(x, t)_.~(x)~(x, t)+f P(x, a, t)H'(a, t) Ot

• [z(a, t)-H(u, t)~(a, t)]da}dx

Jn

R-l(a, t)[z(a, t)-H(a, t)~(a, t)]da [ @'(x, t)~OP(x' u' t)-.~(x)P(x, a, t) dndn ( at

+[

-P(x, ~,

t)~*(,7)

for all

(x, OeZ and

OP(x, a, t)_.~p(x)P(x, a, t)+ [.LP(a)P(x, tr, t)]* Ot

+f l i P(x, 11, t)H'(11, t)R-l(11, t)H(11, t) 3 P(11, tr, t)d11}~b(a, t)dadx=O

(16)

- f II P(x, q, t)n'(q, t)R-l(q, t) 3 H(11, t)P(11, tr, t)d11 (17)

(14)

for all teT. Let us consider the integral

I =~ ~b'(x, t)P(x, a, t).oq'*(a)~(a, t)da J J~ which arises in the third term within the second set

for all (x, a, t)ef~ x Z. Considering next the transversality condition, first at t=tl, we have from equations (9) and (10) that u(x,

tl)=fnP(x,

a,

=~(x, tl).

tl)~b(a,tl)da+ ~(x, tO

Least-squares filtering and smoothing for linear distributed parameter systems This means that ~(x, t~) is the filtered estimate of the system's state at t I and further that ~(x, t) is the filtered estimate of the state for all t~T. Further, ¢(x, t~) is the "missing" terminal condition for the two-point boundary-value problem which is defined by equations (6) and (7). Thus, once ~(x, tO is determined, the smoothed estimate of the system's state, and, in particular, the smoothed estimate of u(x, to), can be determined by backward integration, in time, of equation (6). The implicit assumption here is that ~ ( x ) is such that the problem of backward integration is well-posed. Secondly, at t = to, we have from equations (10) and (8), respectively, and the definition of the operator Ax, ~ that

u(x, to)--¢(x, to)=fa P(x, a, to)~/(a , to)da and

u(x, to)- Uo(X)= A2 ~¢(o-, to)

Equations (20) and (21) are the boundary conditions for equations (16) and (17), respectively. The algorithm for obtaining the filtered estimate of the state of the system in equation (1) is thus defined by equations (16) and (17) subject to the initial conditions in equations (18) and (19), and the boundary conditions in equations (20) and (21). This algorithm is obviously sequential in time and has the same structure as the Kalman-Bucy filter for continuous-time lumped parameter systems. Equation (17) is of the Riccati type, but considerably more involved than its lumped parameter analog. We remark that it is no surprise that the above results reduce trivially to the Kalman-Bucy filter if the spatial dependence is removed. The study of numerical methods for implementation of the above algorithm is a research topic unto itself and beyond the scope of this paper. However, we can illustrate the algorithm with a simple analytical example involving the scalar heat equation. We let

=fn,] Po(x, a)O(a, to)do'.

Ou(x, t)

Comparing these two equations, we see that sufficient conditions for satisfaction of the transversality condition at t = to are

~(x, to)=Uo(X )

Ot

1 82u(x, t) --n 2

c]x 2

and

z(x, t)= u(x, t)+ v(x, t)

(18)

and

319

and 0~
p(x, a, to)=Po(x,~).

(19)

Equations (18) and (19) provide the initial conditions for equations (16) and (17), respectively. We have thus reduced the two-point boundaryvalue problem of equations (6) and (7) to an initialvalue problem. To treat the boundary conditions, we require that

3 fl for all (a, t)eF x T. Then, from equation (2),

This relation can be satisfied by setting (20)

for all (~, t)eF × T, and

for all (a, a, t)eF x Z.

For this case, equation (17) becomes

-e s

f fl P(a, a, t)¢(a, t)da + ~(~, t)=f(a, t).

P(~t, o', t)=0

O<~t <~q.

OP(x, a, t) 1 02p(x, a, t) 1 02p(x, a, t) t~t -re 2 OX 2 "'q-'~ (~ff2

u(a, t ) = f P(cz, a, t)~b(tr, t)dtr+~(a, t)

~(ct, t)=f(ct, t)

data are of equal accuracy spatially, but improve exponentially with time. The boundary conditions are taken to be homogeneous, viz. u(0, t) = u(1, t) = O,

(21)

f

l

P(x, q, t)P(q, cr, t)dq 0

which is subject, according to the algorithm, to the boundary conditions P(x, 0, t) = P(0, x, t) -- P(x, 1, t) =P(1, x, t)=0 for all 0 < x < 1 and 0~< t~< t x. This integro-partial differential equation can be solved by means of separation of variables. The result is

P(x, a, t) = 6e- s, sin nxsin na for 0~< t~< tl, 0 < x < 1, and 0 < a < 1.

320

J.S. MEDITCH

The optimal filter, equation (16), is then given by ~ ( x , t) I ~2~(x, t) ~t "= ~ 2 OX 2 + 6 sin g

4'

[z(a, t)- ~(a, t)]da

0

for O<~t<<.t~ and O < x < l . 4. DISCRETE LOCATION MEASUREMENTS In many practical problems of estimation for distributed parameter systems, it is more realistic to consider the situation where sensors such as thermocouples, pressure transducers, and accelerometers are located at discrete points in f~ rather than throughout all of ~. In such cases, the measurement process is modeled by equation (4). To handle this situation, we weight the measurement residuals only at the x j, j = 1. . . . . M, in equation (5) by setting M

R-t(x, t)= ~ R-t(xi, t)6(x-xj) j=l

where 6(.) is the Dirac delta function. Substitution of this relation into equations (16) and (17) gives M

O~(x, t)_.Lp(x)~(x, t)+ ~,, P(x, a t, t) Ot j=l H'(aj, t)R-a(trj, t)[z(aj, t)-H(aj, t)~(trj, t)] (22)

is also in S(~) for each t~T when ~b(o-, t)~S(f~) tbr each t~T. Since equation (7) is also linear, the latter holds so that our hypotheses on the assumed solution of equation (10) are satisfied. Equations (6-9) are only necessary conditions for J to be an extremum. However, since the system of equations (1) and (3) is linear and J is quadratic, we conjecture, but do not prove, that the four equations are both necessary and sufficient for J to be a minimum. In the same sense, equations (16) and (17) are sufficient, but not necessary conditions, for the satisfaction of equation (15). The situation is the same for the initial conditions of equations (18) and (19), and the boundary conditions of equations (20) and (2l). In our work, we have assumed that measurements are made in f~, but not on the boundary F. An excellent treatment of the latter case is given in [3] where certain difficulties are noted. We have omitted here any treatment of the question of observability. Implicit in our development is the assumption that the system defined by equations (1), (2) and (3), the latter with v(x, t ) = 0 for all (x, t)~E, is observable over T. A well-known study of observability for distributed parameter systems is that of WANG [4]. His results can be adapted here if ~ ( x ) is the infinitesimal generator of a semigroup {~(t, z)}, t, z~T. In this case, the noise-free measurement can be expressed in terms of Uo(X) as

z(x, t)= H(x, t)~(t, to)Uo(X).

and

OP(x, or, t)_~(x)P(x, ~, t) + [Le(~r)P(x, ~r, t)]* dt M

-- ~ P(x, rlj, t)H'(qi, t) j=l

Premultiplying this result by [H(x, t)O(t, to)]* =~*(t, to)H'(x, t) and integrating over T, we get f"q)*(t, to)It'(x, t)H(x, t)~(t, to)dtluo(X) alto

=

R-X(qj, t)n(tl~, t)P(qj, a, t), (23)

~*(t, to)H'(x, t)z(x, t)dt. to

respectively. The initial and boundary conditions are unchanged. 5. CONCLUSION Since equation (16) is linear, it follows that ¢(x, t)eS(~) for each t~T. The same is true in the case of equation (22). Further, since

f P(x, tr, t)(')do" is a linear operator, the term

f P(x, a, t)lp(a, t)da

It is then straightforward to show that a necessary and sufficient condition for our system to be observable is that the linear, self-adjoint operator

f

t, t~*(t, to)H'(x , t)H(x, t)O(t, to)dt to

have a bounded inverse for some finite t~ _>t o. The question of observability for the case where the noise-free measurements are of the form

z(xj, t)=H(xj, t)u(xj, t) j = 1. . . . , M, is somewhat more involved and is left for future study.

Least-squares filtering and smoothing for linear distributed parameter systems There remain also the questions of the stability of the filter relations, equations (16) and (22), and the qualitative properties of the corresponding Riccati equations, equations (17) and (23), respectively. Also, the computational problems which arise here are considerably more complex than those associated with the lumped parameter estimation problem. The results in sections 3 and 4 are readily extended to the estimation problem where a system disturbance n-vector w(x, t) is present. In this case, equation (1) becomes

Ot

with the assumption that w(x, t)eL"2(f~) for each

teT. The functional J in equation (5) is modified to include the dynamic error term

-t t)JBx'°k -

o

~(~r)u(a, t)/dcrdt.

(24)

Here, Bx,, is the operator

where Q(x, ~, t) is an n x n, symmetric, positive definite matrix kernel which is continuous and bounded for all (x, a, t) in Q x IL Also, Q(x, ~, 0 = Q(cr, x, t) is assumed. In view of the constraint

Ou(x, t)_ ~(x)u(x, t) + w(x, t) dt

the above dynamic error term becomes

½~t,[ W'(X, t)Bx, oW(cr, /)dxd' {tojn

so t h a t n o w

j = ½{f~ [u(x, to)-Uo(X)]Ax,~[u(tr, to) -Uo(tr)]dx+

f'f to

[z(x, t)-H(x, t)u(x, t)] fl

R-l(x, t)[z(x, t)-H(x, t)u(x, t)]dxdt + [ t , [ w'(x, t)Bx,~w(tr, t)dxdt} j,ojn

+f,'of°¢(x' L - w(x, tQdxdt.

t) at

The corresponding Euler-Lagrange equations are then

Ou(x, t) = .~(x)u(x, t) + B~. l~k(a, t) Ot

and 0@(x, t)

~t

= -~*(x)q/(x, t) -H'(x, t)R-'(x, t)[z(x, t) - H ( x , t)u(x, t)].

(26)

By virtue of the definition of B~, ,, equation (25) becomes

gu(x, t) = ~(x)u(x, t) + w(x, t)

ff"f IP u(x, '> ~,o,] n k at

321

t)

~u(x, t______)_ L#(x)u(x, t)+f Q(x, a, t)~k(er, t)der. (27) Ot .I The transversality condition leads to the relations already given in equations (8) and (9). The boundary conditions are, of course, unchanged. Repetition of the development in section 3 leads to equations (16) and (17) with the exception of the presence of the term Q(x, or, t) on the right in equation (17). The initial and boundary conditions are the same as before. The extension to the situation where the measurement locations are discrete is obvious. In this case, once ~(x, tl) is determined, solution for the smoothed estimate of the system's state is obtained by backward integration, in time, of both equations (26) and (27) subject to the terminal conditions @(x, q ) = 0 and u(x, q ) = ~(x, q) for all x~f~. This is in contrast to backward integration of only the single relation, equation (6), when the system disturbance is absent. We conclude with some remarks relating the present results with those for lumped parameter systems and a brief discussion of some previous work on parameter estimation distributed. First, the weighting matrix R(x, t) can be viewed, loosely speaking, as the space-time "covariance" matrix of the measurement noise v(x, t). The residual term in the performance functional J is then weighted by the inverse of this matrix to reflect measurement accuracy. Secondly, the analogy of the operator Ax, ~ in the present case to the weighting matrix Pc ~ in the lumped parameter problem has already been noted. In the latter case, Pc is taken to be the covariance matrix of the initial state assuming this matrix is known. Again by analogy, A~., -1 here can be viewed as the "covariance operator" of the system disturbance w(x, t). Previous work on estimation for distributed parameter systems based on the least-squares formulation has been carried out by PHILLIPSON

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and MITTER [5, 6] for measurement data on the boundary. TZAFESTASand NIGHTINGALE[7-9] have utilized the maximum likelihood viewpoint and have developed estimation algorithms for both linear and non-linear systems. THAU [10] approached the problem from the minimum variance standpoint and considered both discrete and continuous-time measurements. The most rigorous treatment, to date, is that of KUSHNER[1 l] who has adopted the conditional expectation viewpoint. Some interesting results involving both filtering and identification are given by BENSOUSSAN[12]. REFERENCES [1] I. M. G~LrAND and S. V. FOMIN: Calculus of Variations, Prentice-Hall, Englewood Cliffs, N.J. (1963). [2] A. E. BaYSON, JR. and M. FRAZIER: Smoothing for linear and nonlinear dynamic systems. TDR-63-119, Aerospace Systems Division, Wright-Patterson A.F.B., Ohio, pp. 353-364, February (1963). [3l A. V. BALAKmSHNANand J. L. LIONS: State estimation for infinite-dimensional systems. J. Comp. Syst. Sci. 1 (1967), 391-403 (1967). [4] P. K. C. WANG: Control of distributed parameter systems. In Advances in Control Systems--Theory and Applications, Vol. 1, pp. 75-172 (Ed. C. T. LEONDES). Academic Press, New York (1963). [5] G. A. PrIILLIVSON and S. K. Mn'rER: State identification of a class of linear distributed systems. Proc. Fourth IFAC Congress, Warsaw, Poland, June (1969). [6] S. K. MrrrER: Optimal control of distributed parameter systems. In Control of Distributed Parameter Systems. A.S.M.E., New York (1969). [7] S. G. TZAFESTAS and J. M. NIGHTINGALE: Optimal filtering, smoothing and prediction in linear distributedparameter systems. Proc. lEE 115, 1207-1212 (1968). [8] S. G. TZArESTAS and J. M. NIGHTINGALE: Concerning optimal filtering theory of linear distributed-parameter systems. Proc. IEE 115, 1737-1742 (1968). [9] S. G. TZAFESTAS and J. M. NIGHTINGALE: Maximumlikelihood approach to optimal filtering of distributedparameter systems. Proc. lEE 116, 1085-1093 (1969). [10] F. E. THAu: On optimum filtering for a class of linear distributed-parameter systems. Proc. 1968 Joint Automatic Control Conf., University of Michigan, Ann Arbor, Michigan, pp. 610-618, June (1968). [11] H. J. KUSHNER: Filtering for linear distributed parameter systems. Center for Dynamical Systems, Brown University, Providence, Rhode Island (1969). [12] A. BEr~SOUSSAN: L'Identification et L'Filtrage. Cahiers No. 1, IRIA, February (1969).

R~sumr--L'article considrre le probl~me d'evaluation de l'rtat d'une catrgorie de syst~mes li@aires a param~trcs repartis & partir de mesures ent'~chres de bruit, du point de vue de l'estimation par la methode des moindres carrrs pondrrrs dans le domaine spatial du systSme ct dans l'intervalle de temps des donnres de mesure. Le probl~mc est reduit A une probl~me bi-ponctuel des valeurs-limites par I'intermrdiare du calcul des variations. Le probl~me biponctuel des valeurs-limites est ensuite resolu par Fintermrdiare de la mrthode de balayage pour obtenir un filtre du type Kalman-Bucy. La solution du probl~me de lissage en decoule directement. L'article consid~re les cas lorsque les donnres de mesure sont obtenues sur l'ensemble du domainc spatial du syst~me ou lorsqu'elles sont obtenues en des points discrets de ce domaine, le syst~me 6rant soumis '~ des perturbations internes et externes, de m~me qu'h des erreurs de mesure. I1 discute certains probl~mes en rrsultant et h 6tudier ultrrieurement. Zusammenfassung--Das Problem der Sch/itzung des Zustandes einer Klasse yon linearen Systemen mittverteilten Parametern aus verrauschten Messungen wird vom Gesichtspunkt der Sch~itzung mit einer Methode gewichteter kleinster Quadrate fiber dem r/iumlichen Gebiet des Systems und dem Zeitintervall der Mel3daten betrachtet. Das Problem wird unter Benutzung der Variationsrechnung auf ein Zweipunkt-Randwertproblem reduziert. Das Zweipunkt-Randwertproblem wird in geschlossener Form gel6st, um ein Kalman-Bucy-Filter zu erhalten. Die Lrsung des Gl~ittungsproblems folgt dann direkt. Betrachtet werden Ffille, wo die MeBdaten tiber das gesamte r/iumliche Gebiet oder in diskreten Punkten dieses Gebietes gewonnen werden und wo das System sowohl yon inneren und /~uBeren Strrungen als auch von MeBfehlern beeinfluBt wird. Einige fiJr zukfinftige Studien resultierende Probleme werden diskutiert. Pe3IOMe---CTaTbn paccMaTpaBaeT npor.qeMy otl.~rlKri COCTOnHI,m O~IHoro :~Jmcca .rlHHe~HblXCHCTeM C pacnpe~e:l~ HHIalMH riapaMeTpaMH H c x o ~ ~3 H3Mepea~t~ C IIIyMOM, C ToqKtt 3peHu~l MIIHHMa~IbrtOKBa~lpaTHqecKo~ oIIeHKH C I~COBblMM KogqbqbHnJ,IeHTaM.q, B IIpocTpaHCTBe CI,ICTeMbI H B HHTepBaJle BpeMettll jIaHrlr~ix II3Me.pcHH~I. 3aaaaa CBO~H'rca K ]IByXTOqeqHol~ n p o r s I e M e rIpe,IlenbHblX 3Haqen~Ii~ q e p e 3 nocpe)lCTBO BapltallrlOHHOrO I-ICqlICJ~eHHSt. ,~ByXToqeqHa~l npo6cieMa

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