Structure theorem for five-dimensional estimation algebras

Abstract 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.

[1]  Amid Rasoulian,et al.  Finite dimensional filters with non-linear drift IX: construction of finite dimensional estimation algebras of non-maximal rank , 1997 .

[2]  Wing Shing Wong.  On a new class of finite dimensional estimation algebras , 1987 .

[3]  Xi Wu,et al.  Hessian matrix non-decomposition theorem , 1999 .

[4]  Elias M. Stein,et al.  Multilinear estimates and fractional integration , 1999 .

[5]  Xi Wu,et al.  Classification of Estimation Algebras with State Dimension 2 , 2006, SIAM J. Control. Optim..

[6]  Stephen S.-T. Yau,et al.  Classification of four-dimensional estimation algebras , 1999, IEEE Trans. Autom. Control..

[7]  Jie Chen,et al.  Finite-Dimensional Filters with Nonlinear Drift VII: Mitter Conjecture and Structure of $\eta$ , 1997 .

[8]  Jie Chen,et al.  Finite-dimensional filters with nonlinear drift, VI: Linear structure of Ω , 1996, Math. Control. Signals Syst..

[9]  W. Chiou.  A note on estimation algebras on nonlinear filtering theory , 1996 .

[10]  Daniel Ocone.  Finite Dimensional Estimation Algebras in Nonlinear Filtering , 1981 .

[11]  Wen-Lin Chiou,et al.  Finite-Dimensional Filters with Nonlinear Drift II: Brockett's Problem on Classification of Finite-Dimensional Estimation Algebras , 1994 .

[12]  Wen-Lin Chiou,et al.  Finite-dimensional filters with nonlinear drift XIII: Classification of finite-dimensional estimation algebras of maximal rank with state space dimension five , 2000 .

[13]  Stephen S.-T. Yau,et al.  Complete classification of finite-dimensional estimation algebras of maximal rank , 2003 .

[14]  Jie Chen,et al.  Finite-Dimensional Filters with Nonlinear Drift VIII: Classification of Finite-Dimensional Estimation Algebras of Maximal Rank with State-Space Dimension 4 , 1996 .

[15]  Wing Shing Wong,et al.  Structure and classification theorems of finite-dimensional exact estimation algebras , 1991 .

[16]  Wing Shing Wong.  Theorems on the structure of finite dimensional estimation algebras , 1987 .

[17]  S. S.-T. Yau *,et al.  Classification of finite-dimensional estimation algebras of maximal rank with arbitrary state–space dimension and mitter conjecture , 2005 .

[18]  Wing Shing Wong,et al.  On a necessary and sufficient condition for finite dimensionality of estimation , 1990 .