Stability of the Extended Kalman Filter When the States are Constrained

In this note, stability of the projection-based constrained discrete time extended Kalman filter (EKF) when applied to deterministic nonlinear systems has been studied. It is proved that, like the unconstrained case, under certain assumptions, the EKF with state equality constraints is an exponential observer, i.e., it keeps the dynamics of its estimation error exponentially stable. Also, it has been shown that a simple modification to the general definition of the EKF with exponential weighting increases the filter's degree of stability and convergence speed with or without state constraints.

[1]  L. Ozbek,et al.  Comments on "Adaptive fading Kalman filter with an application" , 1998 .

[2]  Murat Efe,et al.  Application of Topographic State Constraints in Ground Target Tracking , 2007, 2007 IEEE 15th Signal Processing and Communications Applications.

[3]  Petros G. Voulgaris,et al.  On optimal ℓ∞ to ℓ∞ filtering , 1995, Autom..

[4]  Won Chang Lee,et al.  Observers for Nonautonomous Discrete-Time Nonlinear Systems , 1991 .

[5]  Xuemin Shen,et al.  Adaptive fading Kalman filter with an application , 1994, Autom..

[6]  Alfredo Medio,et al.  Nonlinear Dynamics: Contents , 2001 .

[7]  Matthew Botur,et al.  Non-linear dynamics , 2008 .

[8]  A. E. Nordsjo,et al.  A constrained extended Kalman filter for target tracking , 2004, Proceedings of the 2004 IEEE Radar Conference (IEEE Cat. No.04CH37509).

[9]  Mónica F. Bugallo,et al.  Performance comparison of EKF and particle filtering methods for maneuvering targets , 2007, Digit. Signal Process..

[10]  Konrad Reif,et al.  The extended Kalman filter as an exponential observer for nonlinear systems , 1999, IEEE Trans. Signal Process..

[11]  MingQing Xiao,et al.  A direct method for the construction of nonlinear discrete-time observer with linearizable error dynamics , 2006, IEEE Transactions on Automatic Control.

[12]  Tzyh Jong Tarn,et al.  Exponential Observers for Nonlinear Dynamic Systems , 1975, Inf. Control..

[13]  Thia Kirubarajan,et al.  Estimation with Applications to Tracking and Navigation: Theory, Algorithms and Software , 2001 .

[14]  Alfredo Medio,et al.  Nonlinear Dynamics: Subject index , 2001 .

[15]  M. Boutayeb,et al.  Convergence analysis of the extended Kalman filter used as an observer for nonlinear deterministic discrete-time systems , 1997, IEEE Trans. Autom. Control..

[16]  M. Efe,et al.  An Adaptive Extended Kalman Filter with Application to Compartment Models , 2004 .

[17]  Donato Trigiante,et al.  THEORY OF DIFFERENCE EQUATIONS Numerical Methods and Applications (Second Edition) , 2002 .

[18]  D. Simon,et al.  Kalman filtering with state equality constraints , 2002 .

[19]  Wei Lin,et al.  Remarks on linearization of discrete-time autonomous systems and nonlinear observer design , 1995 .

[20]  Arthur Gelb,et al.  Applied Optimal Estimation , 1974 .

[21]  V. Sundarapandian,et al.  General observers for discrete-time nonlinear systems , 2004 .

[22]  Levent Özbek,et al.  Employing the extended Kalman filter in measuring the output gap , 2005 .

[23]  Robert Dewar,et al.  Non-linear dynamics , 2000 .