Classification Study Of Finite Dimensional Estimation Algebra In Nonlinear Filtering

Estimation theory is about determining the values of parameters of a model based on empirical data or measurements.Specifically,in control theory the problem is determining the state and parameters of a control system by measured system outputs.In practical,the state of a control system is a stochastic process due to various external disturbance or noise.As to the observation of a control system,on the one hand only partial state can be directly observed.On the other hand,the observations usually have stochastic error or subject to noise as well.Hence,from a mathematical point of view,the estimation problem is to determine the state distribution of a stochastic differential equations given certain observation history.Based on the state needed to be estimated is in the past,present or future,the estimation problem is defined as smoothing,filtering or prediction respectively.According to the system is linear or nonlinear,filtering problem can be divided into linear filtering or nonlinear filtering.For general nonlinear filtering problem,the unnormalized state conditional density function ?(t,x)satisfies Duncan-Mortensen-Zakai(DMZ)equation which is a stochastic partial differential equation and intractable.However,a finite-dimensional filter can compute ?(t,x)efficiently by only computing finite sufficient statistics which is very attractive,and it has been a research focus since the late 1970s.Motivated by the Wei-Norman method in solving time varying linear operator differ-ential equation,Brockett,Mitter et al.proposed the idea of constructing finite dimensional filters by classify corresponding finite dimensional estimation algebras generated by differ-ential operators in DMZ equation.In the 1983 International Congress of Mathematicians,Brockett proposed the problem of classifying all finite dimensional estimation algebras.Since the 1980s,the fundamental way to find new classes of finite dimensional system is classifying finite-dimensional estimation algebras.In the beginning of this century,the classification of finite dimensional maximal rank estimation algebra was completed by Yau and his coworkers.However,for non-maximal rank case there are only a few results due to the difficulty of the problem.In this thesis,the main work is to classify finite-dimensional non-maximal rank estimation algebra with state dimension n=3.By the theories of Euler operator and under-determined partial differential equations combined with the strategies of constructing infinite-dimensional sequences in finite-dimensional estimation algebra,we proved the following results under the rank 2 assumption:(i)All the entries of Wong ? are affine functions;(ii)Mitter conjecture holds,i.e.all the func-tions in the underlying estimation algebra are affine functions.Moreover,concrete finite dimensional filtering systems have been constructed corresponding to rank 2 and 1 case,respectively,and the construction of universal robust finite-dimensional filter has been shown via Wei-Norman method in rank 1 case.The advantages of Lie algebra method lie in:(1)It is the systematic way to construct finite-dimensional filters,and we do not need to make any assumptions on the initial condition;(2)The sufficient statistics of Lie algebra approach is O(n);(3)The filter constructed by Lie algebra approach is universal under the sense of Chaleyat-Maurel-Michel.