lEEE TRANSACTIONS ON AUTQMATTC CONTROL, VOL AC-24, NO. 2, APRIL 1979 335

Theorem Assume that f ( t ) is a compound Poisson process whose IV. CONCLUSIONS amplitudes Q possess a moment generating function -and that the intensity function X( e) is positive on a set of positive Lebesgue measure. Also, assume that P( /3U, > 0) >O. Then (IO) is not first moment asymptotically stable.

Notice that if the support of the distribution of U, includes both positive and negative numbers, then P { /3Ul > 0) > 0.

Example 3: Now consider the following nth-order system:

where A , and A , are upper triangular n X n matrices and x ( t ) is an n-dimensional vector. Notice that the ith term of the diagonal of the transition matrix @At,O) will be of the form

where and Pi are the ith terms of the diagonals of A , and A I , respectively. If the initial state x(0) is a constant vector with a one in the ith position and zeros elsewhere, then the ith component of x( t ) will be given by (13). This results in the following corollary.

Corollary: If at least one diagonal element p of A , is nonzero and if f(r) satisfies the conditions of the theorem, then (12) is not first moment asymptotically stable.

Example 4: Consider the system (IO) and let the noise process be given by (3) where

h ( t , T ) = e x p [ - y ( t - ~ ) ] u ( t - - ) , y>O.

Let M(.) be the moment generating function of UI. We find that

Assuming that X(-) is integrable, we see that this quantity can be easily upper bounded. In this case we see that (IO) is asymptotically stable if a<O.

Example 5: Consider the scalar equation (10). Letf(t) be given by (3) and assume that h is a function of the difference of its arguments, say

h(t,T)’h(f-7).

Assume that U, possesses the moment generating function M(.) . Also, assume that

Then from (4) we see that

From (14) we conclude that

M[ BJcof- ‘ i (s)h] <Mo<co.

Assume that the intensity function A(-) is such that

t- tm lim +(T)&=o. 0

Then we have that

We have presented a method for analyzhg the pth moment asymp- totic stability of a class of linear systems with multiplicative state noise. The class of systems considered was required to satisfy a certain Lie algebraic condition. The noise processes were taken to be a form of filtered Poisson noise. The utility of the method was illustrated with several examples. We note that these methods can be extended in a straightforward fashion to include noise processes having the form considered in this paper plus an additive independent Gaussian compo- nent, as well as any other noise processes for which we can calculate the appropriate characteristic functional.

REFERENCES

[I] A. S. Wiusky, S. I. M a m q and D. N. Martin, “On the stochastic stability of linear

AC-20, pp. 711-713, Oct. 1975. systems containing colored multiplicative noise,” IEEE Trans. A u l ~ . Contr., vol.

[2] F. KO- “A survey of stability of stochastic systems,” Aufomatica, vol. 5, pp. 95-112, 1969.

131 W. Wedig, ‘‘Regions of instability for a linear system with random parametric excitation,” in Prm. Symp. on Srmharric D y m i c a l SysI. (Math. Lecture Now

[4] W. M. Wonbam, “Random differential equations in control theory,” in Probabilisfic 294). New York: Springer-Verlag. 1972.

M e h & in Applied Mathematics, voL 2, A. T. Bharucha-Reid, Ed. New York: Academic, I%&

[SI D. L. Kleinman, “On the stabiity of linear stochastic systems,” IEEE Tram Auromar. Contr, voL AG14, pp. 429-430, Aug. 1969.

[q R W. Brwkett, “Lie theory and control systems on spheres,” SIAM J. Appl. Marh..

[7l R. W. Brockets “Lie algebras and Lie groups in control theory,” in Geometric voL 2.5. pp. 213-225, Sept. 1973.

Merhodr in @ s t e m %y, D. Q. Mayne and R. W. Brockets Eds. The Nether-

[Sj J. L. Winems, “Stability of higher order moments for linear stochastic system$” lands: Reidel, 1973.

[9] A. S. WiUsky and S. 1. Marcus, “Analysis of bilinear noise models in circuits and Ingenieur-Archjo, vol. 44, pp. 123-129. 1975.

[IO] J. L. Willems, “Stability criteria for stochastic systems with colored multiplicative devices,”J. Franklin Insf., vol. 301, pp. 103-122, Jan. 1976.

[II] G. L Blankemhip, “Lie theory and the moment stability problem in stochastic noise,” Acta Mechaniw, voL 23, pp. 171-178, 1975.

[I21 M. B. Nevelson and R Z. Khas’miaskii, “Stability of a linear system with random differential equations,” in Prqrinrs IFAC 75, Cambridge, M A , Aug. 1975, no. 362.

[I31 J. L. Willems and D. Aeyels “‘An equivalence result for moment stability criteria disturbances of its parameters,” AppL Math Me&, vol. 30, pp. 487493, 1966.

for parametric stochastic systems and It6 equations,” Int. J. @st. Sci., voL 7, pp. 577-590. 1976.

1141 -, “Moment stability of hear stochastic systems ~ ’ t h solvable Lie algebras,” IEEE T m . Aulomm Corn., vol. AG21, p. 285, Apr. 1976.

[IS] G. L Blankenship, “Stability of linear differential equations with random coefficients,” IEEE Trans. Aulomat. Gnu. , vol. AC-22, pp. 834-838, OCL 1977.

[ I q J. Morrison and J. McKenna, “Analysis of some stochastic ordinary differential equations,” in Stochastic Differential Eqwrrion$ SIAM-AMs Prm, vol. 6. hvi-

[I71 R. W. Brockett and G. L. Blaakenship, -A representation theorem for linear dence, RI: her. Math. Soc.. 1973. pp. 97-162.

differential equations with Markovian coefficients.” in Roc ISlfi Annu. Allenon Con$ on Gmmun., Comr.., andcOmput., Sept 28-30, 1977, pp. 671-679.

[IS] D. L. Snyder, Random Point Procwes. New York: Wiley, 1975.

Stable Continuous-Time Fixed-Lag Smoothem for Stationary Random Processes

JAMES S. MEDITCH

Abstrucj-We present two hfhbdhensiond dynamic representations of figed-lag smoothem for rontinuaus-time stationary random pmceses. The representations are stable and provide a natural framework for the study of this elass of fixed-lag moothers from both the theoretical and device technology ViewpOiJtts.

of Scientific Research, Air Force Systems Command, under Grant AFOSR-78-3546. Manuscript received August 8,1978. This work was supported by the Air Force Office

The author is with the Department of Electrical Engineering, University of Washing- Thus, we see that if a < 0, the system is asymptotically stable. ton, Seattle, W A 98195.

Oa18-9286/79/0400-0335$00.75 01979 IEEE

336 IEEE TRANSACnONS ON AUTOMATTC CONTFtOL, VOL. AC-24, NO. 2, APRU 1979

I. I~TRODUCTION

In this note, we present two dynamic representations of fixed-lag smoothers for continuous-time stationary random processes. Their de- rivation is direct and follows from the stable transfer matrix representa- tion of fied-lag smoothers developed in [l]. The representations are in the form of partial differential equations thus providing the infinite- dimensional state-space characterization for this class of smoothers. Our results are in contrast to the earlier finite-dimensional representations in [2], [3] which are known to possess an internal, unstable block which is uncontrollable [4]. In the representations here, that block has been removed as shown in [ 11.

In our first representation, the innovations process from the Kalman filter [5] is the input; in the second, the optimal filtered state estimate is the input. We believe that both representations are new.

We remark that interest in fixed-lag smoothing stems from the fact that it can provide significantly smaller estimation error variances than filtering [6]-[8].

11. MODEL AND SMOOTHER TRANSFER MArrux

We consider the class of systems

i(r)=Fx(r)+w(t) z ( f ) = H x ( t ) + u ( r )

where the dot denotes the derivative with respect to time t >ro, the specified initial time. The vectors x and w are n-dimensional, and z and u are m-dimensional. The matrices F and H are n X n and m x n, respectively, with all eigenvalues of F in the open left-half plane. AU vectors and matrices are real.

We assume that (w(.)) and (u(.)) are independent zero-mean Gaus- sian random processes with

E [ w ( ~ ) w ' ( u ) ] = Q ~ ( ~ - u ) , E[u(Z)U'(U)]=R~(~-U) (3)

for all r, u > to with R positive definite. In (3), E denotes the expected value, prime the transpose and 6(.) and Dirac delta function. The initial condition x(ra) is zero mean and Gaussian with covariance matrix P(r,,), and x(ro) is independent of {w(-)) and (o(.)}. We also assume that [F,H,Q,R] satisfy the sufficient conditions of [9] for asymptotic stability of the Kalman filter for the system in (1) and (2).

Letting r o d - co, we have for all r the Kalman filter

$ t l t ) = ( F - K H ) i ( r l r ) + K t ( t )

=Fi(r l t )+K~(r) (4)

v( r )=z ( t ) - Hi(rlr) ( 5 )

K= PH'R-' (6)

FP+PF'-PH'R-'HP+Q=O (7)

for the stationary process ( x ( . ) } . In these relations, is the optimal filtered estimate of x(r) given {z(a) , -co<a<t) , ( . ) - I denotes the matrix inverse, P is the covariance matrix of the filtering error x ( r l r ) x( r ) - i ( t l t ) , and (4.)) is the innovations process [lo]. We consider only cases where P is the unique positive definite solution of (7). Situations where P is nonnegative definite are of no interest since the filter then provides an exact estimate of one or more linear combinations of the components of x , and smoothing is not needed to estimate them.

We let $2- TIr) denote for d r the optimal fixed-lag smoothed estimate of x(t- 11 given (z(a), - 00 <n<t) where T >O is the fixed lag. We also let X(s) denote the double-sided Laplace transform of ;(rlr), Z(S) that of z(t), and Q(s,T) that of e(t,.r) ~ ( Z - T ~ Z ) for each T E[O, TI. Then from [I]

where Z is the n X n identity matrix and

Equation (8) was arrived at in [ 11 by determining the transfer matrix of the finite-dimensional fixed-lag smoothers in [2], [3] and demonstrating the cancellation of a factor ( S I - F - QP-')-' by its inverse. This served to remove the internal stabhty problem without affecting the input- output map. We remark that since ( F - KH)'= - P-'(F+QP-')P, the eigenvalues of F + QP - I are the negatives of the eigenvalues of ( F - KH), the latter being those of the Kalman filter. Hence, the internal instability existed under exady those conditions for which the Kalman filter was asymptotically stable.

The poles of the transfer matrix in (8) are identically those of the Kalman filter. Recognizing that (SI- F+KH)-'K is the transfer matrix of the Kalman filter, we have

for each T E[O, TI as an alternate to (8). The transfer matrix in (IO) has no poles in the finite comple? plane and is thus an all-zero system. Finally, we note that Q(s,O)=X(s).

111. DYNAMIC REPRESENTATIONS

We begin by differentiating in (10) with respect to T :

Introducing the change of variable k = j + 1 in the second term in (1 I), we get

=eATBX(S)-(sz+A)[8(S,T)--ATX(S)]

= - A ~ ( s , T ) - - @ ( s , T ) + B ~ ~ ( s ~ + A + B ) J ~ ( s ) = - A @ ( S , T ) - S @ ( S , T ) + e A T ( S z - F ) X ( S )

where we have made use of the commutativity of A and eA', and (9) and (10). Substituting this result into (ll), we s e e that

= -s@(s,7)+eA'(sZ-F)~(s). aT Inverse Laplace transformation gives

which, as a consequence of (4), becomes

For the stationary K h a n filter, it can be shown [ I ] that

e-(F+QP-')r=pe(F-Knyrp-l

where r > - co and O<T < T. The boundary condition for (13) is evi- dently O ( t , O ) = i ( r l t ) , r > - 03. The desired output is obviously

i ( t - ~ I t ) = e ( t , ~ ) . (15)

Equations (14) and (15) specify our first representation, the one for which the input is the innovations process. The Kalman filter is, of course, needed to obtain both v( r ) and i ( r l r ) which are required in (14).

We observe from (12) and the definition of c?(t,T) that

IEEE TRANSACTIONS ON AUTOMATIC C O ~ O L , VOL. AC-24, NO. 2, APRIL 1979 337

Hence, for zero lag, (14) reduces to the Kalman filter as it should.

(10) and (15) that We develop our second representation by first noting with the aid of

i ( t - TIr)=B(t,T)

=eATi(t lr)+A(t,T) (16)

where A(t, 2‘) is the inverse Laplace transform of

evaluated at T = T. Clearly, A(t,O)=O for all t .

tion Proceediig exactly as above, we obtain the partial differential equa-

where t > - 03, O < T < T, and the boundary condition is A(t,O)=O for all t . Substitution for A in (16) from (9) gives

i ( t - TI1)=e(F+QP-’)Ti(rlr)+h(t,T) ( 18)

which completes specification of the second dynamic representation for which the input is i ( t l t ) . There is apparently no way to avoid the presence of P - I in this representation.

For either representation, the covariance matrix S of the fixed-lag smoothing error i ( t - Tit) x ( t ) - i ( t - rlt) is the solution of the linear system 11

(F+QP-’)S+S(F+QP-’)’=P~(T)H’R-’H~(T)P+Q

where

a( T ) = e(F-Kf’)T.

N. CONCLUSION

We have given two infinite-dimensional state-space representations of continuous-time fixed-lag smoothen for stationary random processes. The representations are stable and provide the natural framework for the study of optimal fixed-lag smoothing in the time domain. While the study of suitable approximation of these representations is worth pursu- ing, a more important technological issue may be that of distributed- parameter devices for implementing these representations for signal processing purposes.

On State Estimation for Linear DiscreteTune Systems with Unknown Noise Covariances

J. K. TUGNAIT AND A. H. HADDAD

Abstract-A combined detectionestimation scheme for the estimation of the state of a linear discretetime system with unknown noise covari- ances is amsidered, It is a heuristic extension of the standard minimax scheme to the case when multiple born& on the anknown parameters are available. The convergence of the scheme is diseussed and an example is comidered for illust&ion.

I. INTRODUCTION

The problem of state estimation in linear system driven by and observed in the presence of white Gaussian noise, has been treated extensively in the literature. When the signal and the noise models are completely known, well-known solutions exist [ 11. The problem is consid- erably more difficult when uncertainty exists regarding the noise statis- tics. Several approaches have been developed for solving such problems. These can be roughly classified into two major categories: minimax estimation [4], [IO], and adaptive estimation [2], [3], 181, [Ill. The mini- max schemes yield estimates which are too conservative when large observation records are available but are robust for small samples. Adaptive estimation schemes have good large sample properties in that the state estimate converges to the optimal estimate (corresponding to the completely known model case) as length of the observation record tends to infinity. However, for small samples, adaptive schemes usually yield state estimates having large error variances. Finally, a combined detection-estimation approach has also been developed (51, [13]-[15].

In the next few sections we present a detection-estimation scheme for state estimation in uncertain linear dynamical systems with unknown noise covariances. It is based, to an extent, on the approach given in [5]. It is a heuristic extension of the standard minimax scheme to the case when multiple bounds (disjoint or nested) on the unknown parameters are available. The proposed approach is an attempt to alleviate the pessimistic performance of the standard minimax estimator for large observation records and large uncertainties, while retaining its desirable small-sample properties. A Bayesian approach would require specifica- tion of a priori probabilities of occurrence of all possible parametric values. The small-sample performance of a given scheme is particularly important for the time-varying parameter case when not much is known regarding the time-variation of the parameter. The given scheme is expected to be useful mainly in the time-varying parameter case; the present work limited to constant parameters is intended as an illustration of the concepts involved.

11. PROBLEM STATEMENT

Consider a discrete-time linear dynamic system modeled by the state equation

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propriate dimensions. The sequences { u(k)) and { ~ ( k ) ) are indepen- algorithms,” 3. Franklin Inrf., voL 291, pp. 271-281, Apr. 1971. dent, white, zero-mean Gaussian sequences with covariances Q and R, theory:’ Tram. ASME. Ser. D, J. Basic Eng.. voL 83, pp. 95-108, Mar. 1961. R. E. Kalman and R. S. Bucy, “New results in l i n e a r filtering and prediction

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m a r . Contr,vol. AC-14, pp. 114-115, Feb. 1969. B. D. 0. Anderson, “Roperties of optimal linear smoothing,” IEEE Tmns. Awe and in part by the Joint Services Electronics Program ( U S . Army, U.S. Navy, and US.

B. D. 0. Anderson and S. Chirarauananon, ‘Smoothing as an improvement on J. IC. Tugnait was with the Coordinated Science Laboratory and the Departmat of filtering: A universal bound,” Electron. L e f f . , vol. 7, pp. 524-525. Sepr 9, 1971. Electrical Engineering University of Illinois at UrbanaChampaign, Urbana, IL 61801.

voL 291. pp. 137-144, Feb. 1971. B. D. 0. Andenon, ‘Stability properties of Kalman-Bucy fiiters,” J. FrmJdin Inrt., He is now with the Division of Information Engineering, University of Iowa, Iowa City.

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646-655, Der 1968. NIering in additive while noise,” IEEE Tram. Auromru. Contr, vol. AC-13, pp. ELectrical Engineering, University of Illinois at Urbana-Champaign. Urbana, IL 61801.

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Air Force) under Contract DAAE-07-72-C-0259.

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He is now with the Dynamics Research Corporation, WiLnington, MA 01887.

0018-9286/79/0400-0337$00.75 01979 IEEE