Structure theorem for five-dimensional estimation algebras

The problem of classification of finite-dimensional estimation algebras was formally proposed by Brockett in his lecture at International Congress of Mathematicians in 1983. Due to the difficulty of the problem, in the early 1990s Brockett suggested that one should understand the low-dimensional estimation algebras first. In this paper we give classification of estimation algebras of dimension at most five. Although the classification of finite-dimensional estimation algebra of maximal rank was completed by Yau and his coworkers Chen, Chiou, Hu, Wong and Wu; the problem of classification of non-maximal rank finite-dimensional estimation algebra is still wide open except for the case of state space dimension 2. Hopefully, the result of this paper will shed some light on the non-maximal rank estimation algebras.