Journal ArticleDOI
Finite-Dimensional Filters with Nonlinear Drift II: Brockett's Problem on Classification of Finite-Dimensional Estimation Algebras
Wen-Lin Chiou,Stephen S,T. Yau +2 more
TLDR
In this article, all finite-dimensional algebras with maximal rank are classified if the dimension of the state space is less than or equal to two and therefore, from the Lie algebraic point of view, all finite dimensional nonlinear filters are understood generically in the case of state space dimension less than three.Abstract:
The idea of using estimation algebras to construct finite- dimensional nonlinear filters was first proposed by Brockett and Mitter independently. It turns out that the concept of estimation algebra plays a crucial role in the investigation of finite-dimensional nonlinear filters. In his talk at the International Congress of Mathematics in 1983, Brockett proposed classifying all finite-dimensional estimation algebras. In this paper, all finite-dimensional algebras with maximal rank are classified if the dimension of the state space is less than or equal to two. Therefore, from the Lie algebraic point of view, all finite-dimensional filters are understood generically in the case where the dimension of state space is less than three.read more
Citations
More filters
Journal ArticleDOI
Finite-Dimensional Filters with Nonlinear Drift VIII: Classification of Finite-Dimensional Estimation Algebras of Maximal Rank with State-Space Dimension 4
TL;DR: In this article, Chiou and Yau classified all finite-dimensional estimation algebras of maximal rank with dimension of the state space less than or equal to two, and then they succeeded in classifying all finite dimensional estimation algebra with state-space dimension equal to three.
Journal ArticleDOI
Real time solution of nonlinear filtering problem without memory I
Shing-Tung Yau,Stephen S.-T. Yau +1 more
TL;DR: The nonlinear filtering problem as mentioned in this paper involves the estimation of a stochastic process x = {xt} (called the signal or state process) that cannot be directly observed directly.
Journal ArticleDOI
Finite-dimensional filters with nonlinear drift, VI: Linear structure of Ω
Jie Chen,Stephen S.-T. Yau +1 more
TL;DR: It is proved that if the estimation algebra is finite dimensional and of maximal rank, then the Ω=(∂fj/ ∂xi−∂fi/∂xj)matrix is a linear matrix in the sense that all the entries in Ω are degree one polynomials.
Journal ArticleDOI
Real Time Solution of the Nonlinear Filtering Problem without Memory II
Shing-Tung Yau,Stephen S.-T. Yau +1 more
TL;DR: The purpose of this paper is to show that, under very mild conditions, theDMZ equation admits a unique nonnegative weak solution u which can be approximated by a solution of the DMZ equation on the ball $B_R$ with $u_R|_{\pat B_R}=0$.
Journal ArticleDOI
Finite-dimensional filters with nonlinear drift. III: Duncan-Mortensen-Zakai equation with arbitrary initial condition for the linear filtering system and the Benes filtering system
Shing-Tung Yau,S.S.-T. Yau +1 more
TL;DR: In this paper, the Duncan-Mortensen-Zakai (DMZ) equation for the Kalman-Bucy filtering system and Benes filtering system is considered and it is shown that this equation can be solved explicitly with an arbitrary initial condition by solving a system of ordinary differential equations and a Kolmogorov-type equation, Let n be the dimension of state space.