The Study Of Classifying Finite-dimensional Nonlinear Filters And Developing Real-time Filtering Methods Without Memory

The information in the real world can not be accurately measured due to the disturbances or noises.Filtering is how to extract useful information from the noisy process in discrete or continuous system,or how to obtain the "best" estimate of the useful information in the noisy signals.As one of the research focus in estimation theory,the filtering theory is widely applicable in the fields of communication,tracking,images processing and so on.According to the linearity of the system,the filtering problem can be split up linear and nonliner filtering.For the general nonlinear filtering problem,as long as the evolution equation about the posterior conditional dentity of the states in the system is precisely described,the optimal nonlinear filtering can be solved.It can be realized by taking normalizing technique to the unnormalized posterior conditional dentity of the states p(t,x),which satisfies the Duncan-Mortensen-Zakai(DMZ)equation.The DMZ equation is a second-order stochastic partial differential equation,and usually intractable.Since the late 1970s,one of the research forcus is to construct the finite dimensional filters,which makes it possible and acctractive to compute p(t,x)efficiently by only computing finite sufficient statistics.Motivated by the Wei-Norman approach in solving time-varying linear operator differential equation,Brockett,Clark,Mit-ter have proposed to firstly classify the corresponding finite dimensional estimation algebras generated by differential operators in DMZ equation,and then construct finite dimensional filters for nonlinear filtering systems by utilizing the Wei-Norman approach.In the early of this century,the classification of finite dimensional esti-mation algebras with maximal rank was completed by Yau and his collaborators.However,due to the difficulty of the problem,the classification of finite dimensional estimation algebras with non-maximal rank are only a few results.In this thesis,we mainly give the construction of finite dimensional estimation algebra with state space dimension 4 and linear rank equal to 1,and further obtain a new class of nonlinear filtering systems with finite dimensional filters.Importantly,we show that there is a class of polynomial finite-dimensional filtering system in state space dimension 4 with linear rank one,but the coefficients in Wong's ?-matrix are poly-nomials of degree two,or higher.Furthermore,we write down several easily satisfied sufficient conditions for the construction of more special classes of finite dimensional filters.Additionally,we derive the finite dimensional filters for the proposed non-linear filtering systems by using the Wei-Norman approach.Another research thought is how to efficiently design nonlinear filters in real time based on solving partial differential equation.The DMZ equation is a second-order stochastic partial differential equation,which is usually unable to be solved in closed form,therefore it is possible to numerically solve the DMZ equation and further layout the sub-optimal nonlinear filters.In this dissertation,we shall follow the socalled Yau-Yau's filtering approach to split the solution of the DMZ equation into on-and off-line part,where the off-line part is to solve the forward Kolmogorov equation(FKE),which is a second-order partial differential equation without obser-vation term.We mainly forcus on exploring the generalized Legendre polynomials and using the Galerkin spectral method to numerically solve the FKE.Under certain conditions,the convergence rate of Legendre Galerkin spectral method is given out.Two 2-d numerical experiments of NLF problems(time-invariant and time-varying cases)have been numerically solved to illustrate the feasibility of our algorithm.Our algorithm outperforms the extended Kalman filter and particle filter in both real-time manner and accuracy.