On lie algebras and finite dimensional filtering

Abstract

A Lie algebra L(:E) can be associated with each nonlinear filtering problem, and the realizability or, better, the representability of L(:E) or quotients of L(L:) by means of vector fields on a finite dimensional manifold is related to the existence of finite dimensional recursive filters. In this paper, the structure and representability properties of L(L:) are analyzed for several interesting and/or well known classes of problems. It is shown that, for certain nonlinear filtering problems, L(:E) is given by the Wey! algebra It is proved that neither W. nor any quotient of W. can be realized with C"' or analytic vector fields on a finite dimensional manifold, thus suggesting that for these problems, no statistic of the conditional density can be computed with a finite dimensional recursive filter. For another class of problems (including bilinear systems with linear observations), it is shown that L(:E) is a certain type of filtered Lie algebra. The algebras of this class are of a type which suggest that "sufficiently many" statistics are exactly computable. Other examples are presented, and the structure of their Lie algebras is discussed.

Lectures on Linear and Nonlinear Filtering

“…The original 12 page proof of this result, [25], was long and computational. Another much shorter proof has more recently been given by Toby Stafford.…”

“…The generators are 2 -2 +1x?2-x 1 -a--2x~ and x2, and (again) [25] with the extra conditions that f, G and h are real analytic and that f (0) = 0, G (0) = 0, cf [25]. Another example is 9.31.…”

Current algebras and the identification problem

“…By the estimation algebra of the identification problem we mean the operator Lie algebra G generated by (d0 -86'6) and 86' 0 . For more general nonlinear filtering problems, estimation algebras analogous to G have been emphasized by Brockett and Clark [7], , Mitter [12,2,5], Hazewinkel and Marcus [13] and others (see [14]) as being objects of central interest. In the papers [24,15]) we give a classification theorem for identification problems in terms of G. See Theorem 1 below.…”

Nongaussian linear filtering, identification of linear systems, and the symplectic group

“…Thus the Lie Algebra generated by the operators£, h 1(x), ... ,hp (x) occuring in (2.l) clearly has much to say about how difficult the filtering problem is. This Lie algebra is called the estimation lie algebra of the system (I.I)-( 1.2) and it can be used to prove a variety of positive and negative results about the filtering problem [4,5,9].…”