On algebraic estimation and systems with graded polynomial structure

In the first half of this thesis the algebraic properties of a class of minimal, polynomial systems on IRn are considered. Of particular interest in the sequel are the results that (i) a tensor algebra generated by the observation space and strong accessibility algebra is equal to the Lie algebra of polynomial vector fields on IRn and (ii) the observation algebra of such a system is equal to the ring of polynomial functions on IRn. The former result is proved directly, but to establish the second we construct a canonical form for which the claim is trivial, the general case then following from the properties of the diffeomorphism relating the two realisations. It is also shown that, as a consequence of the structure of the observation space, any system in the class considered has a finite Volterra series solution, thereby showing that the canonical form developed is dual to that of Crouch. The second part of the work is devoted to the algebraic aspects of nonlinear filtering. The fundamental question that this 'algebraic estimation theory' seeks to answer is the existence of a homomorphism between a Lie algebra A of differential operators and a Lie algebra of vector fields. By restricting A to be finite dimensional we obtain a restrictive condition on the system generating A. Results of Ocone and Hijab are extended and connections with the work of Omori and de la Harpe established thus showing A seldom has a Banach structure. Finally, using an observability condition, we develop a further canonical form and thus define a class of systems for which A is isomorphic to the Weyl algebra on n-generators and hence cannot satisfy the above homomorphism principle