A stochastic realization approach to the smoothing problem

Abstract

formatique et d'Automatiquein France, where he worked on the time delays), linear time-varying systems, linear two-dimensional sysmathematical theory of infinite-dimensional systems. At Georgia Tech, tems, and bilinear systems.he has been the principal investigator for several research grants dealing Dr. Kamen is a member of Sigma Xi, Eta Kappa Nu, Tau Beta Pi; with the theory of infinite-dimensional systems (including systems with and Phi Kappa Phi.Abstract--'Ihe pmpose of this paper is to develop a theory of smoothing for linite dimensional linear stochastic systems in the context of storhastic realization theory. "le basic idea is to embed the given stochastic system ina~ofsimllarsystemsallhavlagthesameootputprocesfandthe same Kalman-Bucy filter. 'Ibis class bas a lafflce shucture with a smallest and a largest element; these two elements completely determine the smoothing estimates.approach enables rn to obtain stochastic interpretations of many important smoothing formulas and to explain the relattomhip between t hem.the representation S is minimal in the sense that there is recommended by A. Ephremides, Chairman of the Estimation Com-Manuscript received April 2, 1979; revised August 17, 1979. Paper mittee. This work was supported by the Air Force Office of Scientific 78-3519. Research, Air Force Systems Command, USAF, under Grant AFOSR-( Q > 0) means that the symmetric matrix Q is positive (nonnegative) definite. Lemma 2.2: Let P be the state covariance function of the linear stochastic system S defined in Section I . litren, for any e > 0, P -' exists and is analytic on the interval [ E , r]. If ll> 0, the same holds for the compIete interval [0, TI. Proof: Integrating (1.2) yields P( t ) = @( t, O)II@( t, 0)' + ~r~( t y 7 ) B (~) B (~) '~( t ,~) ' (2.6) 'Some of these shortcomings have been pointed out in a recent thesis which is positive definite if I I > O ; hence, since A and B by Wall [&I, brought to OUT attention after the submission of this paper. are analytic an [o, TI, SO is ?' -'. NOW assume that n $0. 'For example, the Moore-Penrase pseudo-inverse can be used, Since S is minimal , ( A , B ) must be completely controlla-

On the fixed-interval smoothing problem

“…These smoothers are obtained from simple, first principles arguments using reversed-time realizations of the state process. Badawi et al [5] have very recently derived similar smoothing formulas based on stochastic realizations. It is to be hoped that the analysis and discussion in Sections I11 and IV will enable the reader to obtain a clear understanding of how future observations are incorporated in the FI smoothing problem.…”

Markovianity of a subset of components of a Markov process

“…In general, since k < n, the process (Y t ) t≥0 is not necessarily a Markov one. However, as is known from the stochastic realization theory [7] and [2], under some proper conditions on matrices A, C, and Q = Cov(X 0 , X 0 ), the process (Y t ) t≥0 is nevertheless a Markov itself.…”

“…In this paper necessary and sufficient conditions for the (Y t ) t≥0 to possess a Markov property in terms of matrices A, C and Q are formulated (not explicitly using [7] and [2]). The Itô linear differential equation for Y t is also given.…”

Multiscale smoothing error models

“…Several such applications have been developed in the context of 1-D GaussMarkov models by exploiting relatively recent results which show that the smoothing error processes associated with Gauss-Markov models are themselves Gauss-Markov processes (7,8,10,11] 4 . In this paper, we derive a dynamic model for the smoothing error process associated with multiscale stochastic models.…”