Abstract
The error dynamics of the extended Kalman filter (EKF), employed as an observer for a general nonlinear, stochastic discrete time system, are analyzed. Sufficient conditions for the boundedness of the errors of the EKF are determined. An expression for the bound on the errors is given in terms of the size of the nonlinearities of the system and the error covariance matrices used in the design of the EKF. The results are applied to the design of a stable EKF frequency tracker for a signal with time-varying frequency.
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B. D. O. Anderson, R. R. Bitmead, C. R. Johnson, Jr, P. V. Kokotovic, R. L. Kosut, I. M. Y. Mareels, L. Praly, and B. D. Riedle.Stability of Adaptive Systems: Passivity and Averaging Analysis. M.I.T. Press, Cambridge, Massachusetts, 1986.
B. D. O. Anderson and J. B. Moore,Optimal Filtering. Prentice-Hall, Englewood Cliffs, New Jersey, 1979.
J. S. Baras, A. Bensoussan, and M. R. James. Dynamic observers as asymptotic limits of recursive filters: Special cases.SIAM J. Appl. Math.,48 (1988), 1147–1158.
S. W. Chan, G. C. Goodwin and K. S. Sin. Convergence properties of the Riccati difference equation in optimal filtering of nonstabilizable systems.IEEE Trans. Automat. Control,29 (1984), 110–118.
J. E. Dennis and R. B. Schnabel.Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Prentice-Hall, Englewood Cliffs, New Jersey, 1983.
J. J. Deyst, Jr and C. R. Price. Conditions for asymptotic stability of the discrete minimum-variance linear estimator.IEEE Trans. Automat. Control,13 (1968), 702–705.
B. James.Approaches to Multiharmonic Frequency Tracking and Estimation. Ph.D. thesis, Australian National University, Canberra, Australia, 1992.
C. N. Kelly and S. C. Gupta. The digital phase-locked loop as a near-optimum FM demodulator.IEEE Trans. Comm.,20 (1972), 406–411.
A. Nehorai and B. Porat. Adaptive comb filtering for harmonic signal enhancement.IEEE Trans. Acoustics, Speech Signal Processing,34 (1980), 1124–1138.
H. Nijmeijer. Observability of autonomous discrete time non-linear systems: A geometric approach.Internat. J. Control,36 (1982), 867–874.
P. J. Parker and B. D. O. Anderson. Frequency tracking of nonsinusoidal periodic signals in noise.Signal Processing,20 (1990), 127–152.
B. G. Quinn and J. M. Fernandes. A fast efficient technique for the estimation of frequency.Biometrika,78 (1991), 489–497.
D. L. Snyder.The State-Variable Approach to Continuous Estimation with Applications to Analog Communications Theory. M.I.T. Press, Boston, Massachusetts, 1969.
Y. Song and J. W. Grizzle. The extended Kalman filter as a local asymptotic observer for nonlinear discrete-time systems.J. Math. Systems, Estimation Control,5 (1995), 59–78.
R. L. Streit and R. F. Barrett. Frequency line tracking using Hidden Markov Models.IEEE Trans. Acoustics, Speech Signal Processing,38 (1990), 586–598.
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This research was supported by the Co-operative Research Centre for Robust and Adaptive Systems ((CR)2 ASys). The authors wish to acknowledge the funding of the activities of (CR)2 ASys by the Australian Commonwealth Government under the Co-operative Research Centre Program.
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La Scala, B.F., Bitmead, R.R. & James, M.R. Conditions for stability of the extended Kalman filter and their application to the frequency tracking problem. Math. Control Signal Systems 8, 1–26 (1995). https://doi.org/10.1007/BF01212364
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DOI: https://doi.org/10.1007/BF01212364