Structure of finite dimensional exact estimation algebra on state dimension 3 and linear rank 2

The estimation algebra plays an important role in classification of finite dimensional filters. When finite dimensional estimation algebra has maximal rank, Yau et al. [Yau (2003). Complete classification of finite-dimensional estimation algebras of maximal rank. International Journal of Control, 76(7), 657-677; Yau & Hu (2005). Classification of finite-dimensional estimation algebras of maximal rank with arbitrary state-space dimension and Mitter conjecture. International Journal of Control, 78(10), 689-705.] have proved that eta must be a degree 2 polynomial. In this paper, we study the structure of finite dimensional exact estimation algebra with state dimension 3 and rank 2. We establish a sufficient and necessary condition for estimation algebra with nonmaximal rank to be finite dimensional. Importantly, in the new filtering system, eta needs not to be a degree 2 polynomial and can be of any degree 4n(1) +2, n(1) is an element of Z(+). It is the first time to systematically analyse nonmaximal rank exact estimation algebra in which eta is a polynomial of any degree 4n(1) + 2,n(1) is an element of Z(+). For Riccati-type equation, estimates have been done from the viewpoints of both classical solution and weak solution respectively. Finally, finite dimensional filters of Benes type are constructed successfully.