This paper presents modifications to the stochastic stability lemma which is then used to estimate the convergence rate and persistent error of the linear Kalman filter online without using knowledge of the true state. Unlike previous uses of the stochastic stability lemma for stability proof, this new convergence analysis technique considers time-varying parameters, which can be calculated online in real-time to monitor the performance of the filter. Through simulation of an example problem, the new method was shown to be effective in determining a bound on the estimation error that closely follows the actual estimation error. Different cases of assumed process and measurement noise covariance matrices were considered in order to study their effects on the convergence and persistent error of the Kalman filter. 1. Introduction Since its introduction in 1960, the linear Kalman filter (LKF) [1] has been used widely in industry. When the LKF is implemented in real-time applications, it is often difficult to quantify the performance of the filter without access to some reference “truth.” Offline simulations can provide some indication of the filter performance; however accurate mathematical models are not always available. For the LKF, there are two primary sources of error in the estimation: initialization error and stochastic errors due to the process and measurement noise. In the early stages of the filter, the initialization error is dominant, and it takes some amount of time for the estimated state to converge to the true state from this incorrect initial state. After the initial error convergence, the errors due to the noise terms remain, resulting in “persistent” errors. Because of these types of error, there is a need to analyze the performance of the LKF online by quantifying the convergence rate and persistent error bounds of the real system. Such a tool could benefit many safety or performance critical systems, such as the aircraft health management system. Existing techniques for online performance analysis of the LKF include outlier detection [2], performance reliability prediction [3], and confidence bounds from the covariance matrix; for example, see [4]. Confidence bounds can also be established through use of the Chebyshev inequality [5], although these bounds tend to be too large for practical use [6]. Some other investigations for confidence bounds on the Kalman filter consider the non-Gaussian case using enhancements to the Chebyshev inequality [6] or the Kantorovich inequality [7]. The work presented herein offers a novel online method
References
[1]
R. E. Kalman, “A new approach to linear filtering and prediction problems,” Journal of Basic Engineering, vol. 82, pp. 35–45, 1960.
[2]
H. Liu, S. Shah, and W. Jiang, “On-line outlier detection and data cleaning,” Computers and Chemical Engineering, vol. 28, no. 9, pp. 1635–1647, 2004.
[3]
S. Lu, H. Lu, and W. J. Kolarik, “Multivariate performance reliability prediction in real-time,” Reliability Engineering and System Safety, vol. 72, no. 1, pp. 39–45, 2001.
[4]
D.-J. Jwo and T.-S. Cho, “A practical note on evaluating Kalman filter performance optimality and degradation,” Applied Mathematics and Computation, vol. 193, no. 2, pp. 482–505, 2007.
[5]
J. G. Saw, M. C. K. Yang, and T. C. Mo, “Chebyshev inequality with estimated mean and variance,” The American Statistician, vol. 38, no. 2, pp. 130–132, 1984.
[6]
J. L. Maryak, J. C. Spall, and B. D. Heydon, “Use of the Kalman filter for inference in state-space models with unknown noise distributions,” IEEE Transactions on Automatic Control, vol. 49, no. 1, pp. 87–90, 2004.
[7]
J. C. Spall, “The kantorovich inequality for error analysis of the Kalman filter with unknown noise distributions,” Automatica, vol. 31, no. 10, pp. 1513–1517, 1995.
[8]
R. E. Kalman, “New methods in Wiener filtering theory,” in Proceedings of the 1st Symposium on Engineering Applications of Random Function Theory and Probability, J. L. Bogdanoff and F. Kozin, Eds., Wiley, New York, NY, USA, 1963.
[9]
R. J. Fitzgerald, “Divergence of the Kalman filter,” IEEE Transactions on Automatic Control, vol. 16, no. 6, pp. 736–747, 1971.
[10]
H. W. Sorenson, “On the error behavior in linear minimum variance estimation problems,” IEEE Transactions on Automatic Control, vol. 12, no. 5, pp. 557–562, 1967.
[11]
J. J. Deyst Jr. and C. F. Price, “Conditions for asymptotic stability of the discrete minimum-variance linear estimator,” IEEE Transactions on Automatic Control, vol. 13, no. 6, pp. 702–705, 1968.
[12]
A. H. Jazwinski, Stochastic Processes and Filtering Theory, Academic Press, New York, NY, USA, 1970.
[13]
K. L. Hitz, T. E. Fortmann, and B. D. O. Anderson, “A note on bounds on solutions of the Riccati equation,” IEEE Transactions on Automatic Control, vol. 17, no. 1, p. 178, 1972.
[14]
E. Tse, “Observer-estimators for discrete-time systems,” IEEE Transactions on Automatic Control, vol. 18, no. 1, pp. 10–16, 1973.
[15]
J. J. Deyst Jr., “Correction to “Conditions for asymptotic stability of the discrete minimum-variance linear estimator”,” IEEE Transactions on Automatic Control, vol. 18, no. 5, pp. 562–563, 1973.
[16]
J. B. Moore and B. D. O. Anderson, “Coping with singular transition matrices in estimation and control stability theory,” International Journal of Control, vol. 31, no. 3, pp. 571–586, 1980.
[17]
S. W. Chan, G. C. Goodwin, and K. S. Sin, “Convergence properties of the Riccati difference equation in optimal filtering of nonstabilizable systems,” IEEE Transactions on Automatic Control, vol. 29, no. 2, pp. 110–118, 1984.
[18]
L. Guo, “Estimating time-varying parameters by the Kalman filter based algorithm: stability and convergence,” IEEE Transactions on Automatic Control, vol. 35, no. 2, pp. 141–147, 1990.
[19]
J. L. Crassidis and J. L. Junkins, Optimal Estimation of Dynamic Systems, chapter 5, CRC Press, Boca Raton, Fla, USA, 2004.
[20]
E. F. Costa and A. Astolfi, “On the stability of the recursive Kalman filter for linear time-invariant systems,” in Proceedings of the American Control Conference (ACC '08), pp. 1286–1291, Seattle, Wash, USA, June 2008.
[21]
R. G. Agniel and E. I. Jury, “Almost sure boundedness of randomly sampled systems,” SIAM Journal on Control, vol. 9, no. 3, pp. 372–384, 1971.
[22]
T.-J. Tarn and Y. Rasis, “Observers for nonlinear stochastic systems,” IEEE Transactions on Automatic Control, vol. 21, no. 4, pp. 441–448, 1976.
[23]
K. Reif, S. Günther, E. Yaz, and R. Unbehauen, “Stochastic stability of the discrete-time extended Kalman filter,” IEEE Transactions on Automatic Control, vol. 44, no. 4, pp. 714–728, 1999.
[24]
K. Xiong, H. Y. Zhang, and C. W. Chan, “Performance evaluation of UKF-based nonlinear filtering,” Automatica, vol. 42, no. 2, pp. 261–270, 2006.
[25]
Y. Wu, D. Hu, and X. Hu, “Comments on “Performance evaluation of UKF-based nonlinear filtering”,” Automatica, vol. 43, no. 3, pp. 567–568, 2007.
[26]
M. Rhudy, Y. Gu, and M. R. Napolitano, “Relaxation of initial error and noise bounds for stability of GPS/INS attitude estimation,” in AIAA Guidance, Navigation, and Control Conference, Minneapolis, Minn, USA, August 2012.
[27]
D. Simon, Optimal State Estimation, Wiley, New York, NY, USA, 2006.
[28]
J. C. Spall, “The distribution of nonstationary autoregressive processes under general noise conditions,” Journal of Time Series Analysis, vol. 14, no. 3, pp. 317–320, 1993, (correction: vol. 14, p. 550, 1993).
[29]
J. L. Maryak, J. C. Spall, and G. L. Silberman, “Uncertainties for recursive estimators in nonlinear state-space models, with applications to epidemiology,” Automatica, vol. 31, no. 12, pp. 1889–1892, 1995.
[30]
R. V. Hogg, J. W. McKean, and A. T. Craig, Introduction to Mathematical Statistics, Pearson Education, 2005.
[31]
E. Kreyszig, Advanced Engineering Mathematics, Wiley, New York, NY, USA, 9th edition, 2006.
[32]
F. L. Lewis, Optimal Estimation, Wiley, New York, NY, USA, 1986.
