New Algorithms in Real Time Solution of the Nonlinear Filtering Problem

It is well known that the filtering theory has important applications in both military and commercial industries. The Kalman-Bucy filter has been used in many areas such as navigational and guidance systems, radar tracking, solar mapping, and satellite orbit determination. However, the Kalman-Bucy filter has limited applicability because of the linearity assumptions of the drift term and observation term as well as the Gaussian assumption of the initial value. Therefore there has been an intensive interest in solving the nonlinear filtering problem. The central problem of nonlinear filtering theory is to solve the DMZ equation in real time and memoryless way. In this paper, we shall describe three methods to solve the DMZ equation: Brockett-Mitter estimation algebra method, direct method, and new algorithm method. The first two methods are relatively easy to implement in hardware and can solve a large class of nonlinear filtering problems. We shall present the recent advance in the third method which solves all the nonlinear filtering problems in a real-time manner in theory. 1. Introduction. It is well known that the filtering theory has important ap- plications in both military and commercial industries. The Kalman-Bucy filter has been used in many areas such as navigational and guidance systems, radar tracking, solar mapping, and satellite orbit determination. However, the Kalman-Bucy filter has limited applicability because of the linearity assumptions of the drift term and observation term as well as the Gaussian assumption of the initial value. Therefore there has been an intensive interest in solving the nonlinear filtering problem. The nonlinear filtering problem involves the estimation of a stochastic process x = {xt} (called the signal or state process) that cannot be observed directly. Information con- taining x is obtained from observations of a related process y = {yt} (the observation process). The goal of nonlinear filtering is to determine the conditional density �(t,x) of xt given the observation history of {ys: 0 6 s 6 t}. In the late 1960s, Duncan (Du), Mortensen (Mo) and Zakai (Za) independently derived the Duncan-Mortensen-Zakai (DMZ) equation for the nonlinear filtering theory which the conditional probability density �(t,x) must satisfy. The central problem of nonlinear filtering theory is to solve the DMZ equation in real time and memoryless way. In this paper, we shall describe three methods to solve the DMZ equation: Brockett-Mitter estimation al- gebra method, direct method, and new algorithm method. The first two methods are relatively easy to implement in hardware and can solve a large class of nonlinear filtering problems. We shall present the recent advance in the third method which solves all the nonlinear filtering problems in a real-time manner theoretically.

[1]  Stephen S.-T. Yau,et al.  Structure theorem for five-dimensional estimation algebras , 2006, Syst. Control. Lett..

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

[3]  R. Bucy,et al.  A priori error bounds for the cubic sensor problem , 1978 .

[4]  Lei Guo,et al.  Brockett’s problem on nonlinear filtering theory , 2000 .

[5]  S. Marcus.  Algebraic and Geometric Methods in Nonlinear Filtering , 1984 .

[6]  Stephen S.-T. Yau,et al.  New direct method for Kalman-Bucy filtering system with arbitrary initial condition , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

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

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

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

[10]  Shing-Tung Yau,et al.  Real Time Solution of the Nonlinear Filtering Problem without Memory II , 2008, SIAM J. Control. Optim..

[11]  G.-G. Hu.  Finite-dimensional filters with nonlinear drift. , 1997 .

[12]  T. Başar,et al.  A New Approach to Linear Filtering and Prediction Problems , 2001 .

[13]  M. Zakai.  On the optimal filtering of diffusion processes , 1969 .

[14]  Steven I. Marcus,et al.  An Introduction to Nonlinear Filtering , 1981 .

[15]  Stephen S.-T. Yau,et al.  Real time solution of nonlinear filtering problem without memory I , 2000 .

[16]  R. E. Kalman,et al.  New Results in Linear Filtering and Prediction Theory , 1961 .

[17]  S. Marcus,et al.  Nonexistence of finite dimensional filters for conditional statistics of the cubic sensor problem , 1983 .

[18]  R. E. Kalman,et al.  A New Approach to Linear Filtering and Prediction Problems , 2002 .

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

[20]  Mitter.  On the analogy between mathematical problems of non-linear filtering and quantum physics. Interim report , 1980 .

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

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

[23]  Stephen S.-T. Yau.  Recent results on nonlinear filtering: new class of finite dimensional filters , 1990, 29th IEEE Conference on Decision and Control.

[24]  T. Duncan.  PROBABILITY DENSITIES FOR DIFFUSION PROCESSES WITH APPLICATIONS TO NONLINEAR FILTERING THEORY AND DETECTION THEORY , 1967 .

[25]  Roger W. Brockett,et al.  Nonlinear Systems and Nonlinear Estimation Theory , 1981 .

[26]  E. Norman,et al.  ON GLOBAL REPRESENTATIONS OF THE SOLUTIONS OF LINEAR DIFFERENTIAL EQUATIONS AS A PRODUCT OF EXPONENTIALS , 1964 .

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

[28]  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 .

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

[30]  D. Ocone.  Topics in Nonlinear Filtering Theory. , 1980 .

[31]  Stephen S.-T. Yau,et al.  Finite-dimensional filters with nonlinear drift. XV. New direct method for construction of universal finite-dimensional filter , 2002 .

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

[33]  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 .

[34]  Stephen S.-T. Yau,et al.  Finite-dimensional filters with nonlinear drift X: explicit solution of DMZ equation , 2001, IEEE Trans. Autom. Control..

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

[36]  Mark H. A. Davis.  On a multiplicative functional transformation arising in nonlinear filtering theory , 1980 .

[37]  W. Wong,et al.  The estimation algebra of nonlinear filtering systems , 1998 .

[38]  S. Yau,et al.  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 , 1997, IEEE Transactions on Aerospace and Electronic Systems.

[39]  D. Michel,et al.  Des resultats de non existence de filtre de dimension finie , 1984 .

[40]  Stephen S.-T. Yau,et al.  New direct method for Yau filtering system with arbitrary initial conditions , 1996, Proceedings of 35th IEEE Conference on Decision and Control.

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

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