Recent results on classification of finite dimensional estimation algebras: dimension of state space ⩾2

All finite-dimensional estimation algebras with maximal rank can be classified if the dimension of the state space is at most two. A theorem is given which is characterized as the most important step towards the complete solution of the Brockett problem. The major difference between this theorem and the corresponding theorem of L.F. Tam et al. (1990) is that the exactness assumption in the theorem of Tam et al. has been removed. The novelty of the problem is that there is no assumption on the drift term of the nonlinear filtering system. However, in the course of the proof of the theorem, it is shown that the drift term cannot be very arbitrary if the estimation algebra is finite dimensional. In fact, it is shown that the drift term must be linear vector field plus gradient vector field