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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 &amp; 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  η  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,  η  needs not to be a degree 2 polynomial and can be of any degree. It is the first time to systematically analyse nonmaximal rank exact estimation algebra in which  η  is a polynomial of any degree. For Riccati-type equation, estimates have been done from the viewpoints of both classical solution and weak solution respectively. Finally, finite dimensional filters of Benés type are constructed successfully.

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