A Contraction Theory-Based Analysis of the Stability of the Deterministic Extended Kalman Filter

The contraction properties of the extended Kalman filter, viewed as a deterministic observer for nonlinear systems, are analyzed. The approach relies on the study of an auxiliary “virtual” dynamical system. Some conditions under which exponential convergence of the state error can be guaranteed are derived. Moreover, contraction provides a simple formalism to study some robustness properties of the filter, especially with respect to measurement errors, as illustrated by a simplified inertial navigation example. This technical note sheds another light on the theoretical properties of this popular observer.

[1]  D. C. Lewis.  Metric Properties of Differential Equations , 1949 .

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

[3]  B. P. Demidovich.  THE DISSIPATIVITY OF A CERTAIN NONLINEAR SYSTEM OF DIFFERENTIAL EQUATIONS, 1 , 1961 .

[4]  P. Hartman.  On Stability in the Large for Systems of Ordinary Differential Equations , 1961, Canadian Journal of Mathematics.

[5]  J. Deyst,et al.  Conditions for asymptotic stability of the discrete minimum-variance linear estimator , 1968 .

[6]  D. Luenberger.  An introduction to observers , 1971 .

[7]  P. Krishnaprasad,et al.  Dynamic observers as asymptotic limits of recursive filters , 1982, CDC 1982.

[8]  J. Grizzle,et al.  The Extended Kalman Filter as a Local Asymptotic Observer for Nonlinear Discrete-Time Systems , 1992, 1992 American Control Conference.

[9]  J. Gauthier,et al.  High gain estimation for nonlinear systems , 1992 .

[10]  V. Fromion,et al.  An incremental framework for the extended Kalman filter , 1997, 1997 European Control Conference (ECC).

[11]  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..

[12]  Konrad Reif,et al.  An EKF-Based Nonlinear Observer with a Prescribed Degree of Stability , 1998, Autom..

[13]  Jean-Jacques E. Slotine,et al.  On Contraction Analysis for Non-linear Systems , 1998, Autom..

[14]  Jean-Jacques E. Slotine,et al.  On Contraction Analysis for Nonlinear Systems Analyzing stability differentially leads to a new perspective on nonlinear dynamic systems , 1999 .

[15]  A. Krener.  The Convergence of the Extended Kalman Filter , 2002, math/0212255.

[16]  Jean-Jacques E. Slotine,et al.  Methodological remarks on contraction theory , 2004, 2004 43rd IEEE Conference on Decision and Control (CDC) (IEEE Cat. No.04CH37601).

[17]  Jean-Jacques E. Slotine,et al.  Contraction Analysis of Nonlinear Distributed Systems , 2004 .

[18]  Jean-Jacques E. Slotine,et al.  On partial contraction analysis for coupled nonlinear oscillators , 2004, Biological Cybernetics.

[19]  J. Jouffroy.  Some ancestors of contraction analysis , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[20]  Domitilla Del Vecchio,et al.  A Contraction Theory Approach to Singularly Perturbed Systems , 2013, IEEE Transactions on Automatic Control.