An interval Kalman filter enhanced by lowering the covariance matrix upper bound

Tran, Tuan Anh ; Jauberthie, Carine ; Trave-Massuyés, Louise ; Lu, Quoc Hung

International Journal of Applied Mathematics and Computer Science, Tome 31 (2021) no. 2, pp. 259-269.

Résumé

This paper proposes a variance upper bound based interval Kalman filter that enhances the interval Kalman filter based on the same principle proposed by Tran et al. (2017) for uncertain discrete time linear models. The systems under consideration are subject to bounded parameter uncertainties not only in the state and observation matrices, but also in the covariance matrices of the Gaussian noises. By using the spectral decomposition of a symmetric matrix and by optimizing the gain matrix of the proposed filter, we lower the minimal upper bound on the state estimation error covariance for all admissible uncertainties. This paper contributes with an improved algorithm that provides a less conservative error covariance upper bound than the approach proposed by Tran et al. (2017). The state estimates are determined using interval analysis in order to enclose the set of all possible solutions of the classical Kalman filter consistent with the uncertainties.

Détail

Keywords: uncertain linear systems, Kalman filter, interval analysis, estimation, covariance matrix
Mots-clés : układ liniowy, filtr Kalmana, analiza interwałowa, macierz kowariancji

Bibliographie

[1] [1] Cayero, J., Rotondo, D., Morcego, B. and Puig, V. (2019). Optimal state observation using quadratic boundedness: Application to UAV disturbance estimation, International Journal of Applied Mathematics and Computer Science 29(1): 99–109, DOI: 10.2478/amcs-2019-0008.

[2] [2] Chabir, K., Rhouma, T., Keller, J.Y. and Sauter, D. (2018). State filtering for networked control systems subject to switching disturbances, International Journal of Applied Mathematics and Computer Science 28(3): 473–482, DOI: 10.2478/amcs-2018-0036.

[3] [3] Chen, G., Wang, J. and Shieh, L.S. (1997). Interval Kalman filtering, IEEE Transactions on Aerospace and Electronic Systems 33(1): 250–259.

[4] [4] Combastel, C. (2015). Merging Kalman filtering and zonotopic state bounding for robust fault detection under noisy environment, Proceedings of the 9th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes, Paris, France, pp. 289–295.

[5] [5] Combastel, C. (2016). An Extended Zonotopic and Gaussian Kalman Filter (EZGKF) merging set-membership and stochastic paradigms: Toward non-linear filtering and fault detection, Annual Reviews in Control 42: 232–243.

[6] [6] Ingimundarson, A., Manuel Bravo, J., Puig, V., Alamo, T. and Guerra, P. (2009). Robust fault detection using zonotope-based set-membership consistency test, International Journal of Adaptive Control And Signal Processing 23(4): 311–330.

[7] [7] Jaulin, L., Braems, I., Kieffer, M. and Walter, E. (2001a). Nonlinear state estimation using forward-backward propagation of intervals in an algorithm, in W. Krämer and J.W. von Gudenberg (Eds), Scientific Computing, Validated Numerics, Interval Methods, Springer US, New York, pp. 191–204.

[8] [8] Jaulin, L., Kieffer, M., Didrit, O. and Walter, E. (2001b). Applied Interval Analysis with Examples in Parameter and State Estimation, Robust Control and Robotics, Springer, London.

[9] [9] Kalman, R.E. (1960). A new approach to linear filtering and prediction problems, Journal of Basic Engineering 82(Series D): 35–45.

[10] [10] Kieffer, M., Jaulin, L., Walter, E. and Meizel, D. (1999). Guaranteed mobile robot tracking using interval analysis, Proceedings of the Workshop on Application of Interval Analysis to System and Control, Girona, Spain, pp. 347–360.

[11] [11] Lesecq, S., Barraud, A. and Dinh, K. (2003). Numerical accurate computations for ellipsoidal state bounding, Proceedings of the Mediterranean Conference on Control and Automation (MED’03), Rhodes, Greece.

[12] [12] Masreliez, C. and Martin, R. (1977). Robust Bayesian estimation for the linear model and robustifying the Kalman filter, IEEE Transactions on Automatic Control 22(3): 361–371.

[13] [13] Moore, R.E. (1959). Automatic error analysis in digital computation, Technical report LMSD-48421, Lockheed Missiles and Space Co, Palo Alto.

[14] [14] Moore, R.E. (1966). Interval Analysis, Prentice-Hall, Englewood Cliffs.

[15] [15] Moore, R., Kearfott, R. and Cloud, M. (2009). Introduction to Interval Analysis, Society for Industrial and Applied Mathematics, Philadelphia.

[16] [16] Raka, S.-A. and Combastel, C. (2013). Fault detection based on robust adaptive thresholds: A dynamic interval approach, Annual Reviews in Control 37(1): 119–128.

[17] [17] Ribot, P., Jauberthie, C. and Trave-Massuyés, L. (2007). State estimation by interval analysis for a nonlinear differential aerospace model, Proceedings of the European Control Conference, Kos, Greece, pp. 4839–4844.

[18] [18] Rump, S.M. (1999). INTLAB—INTerval LABoratory, in T. Csendes (Ed.), Developments in Reliable Computing, Kluwer Academic Publishers, Dordrecht, pp. 77–104.

[19] [19] Tran, T.A. (2017). Cadre unifié pour la modélisation des incertitudes tatistiques et bornés: application à la détection et isolation de défauts dans les systémes dynamiques incertains par estimation PhD thesis, Université Toulouse 3—Paul Sabatier, Toulouse, www.theses.fr/2017TO U30292.

[20] [20] Tran, T.A., Jauberthie, C., Le Gall, F. and Travé-Massuyès, L. (2017). Interval Kalman filter enhanced by positive definite upper bounds, Proceedings of the 20th IFAC World Congress, Toulouse, France, pp. 1595–1600.

[21] [21] Tran, T. A., Le Gall, F., Jauberthie, C. and Travé-Massuyès, L. (2016). Two stochastic filters and their interval extensions, Proceedings of the 4th IFAC International Conference on Intelligent Control and Automation Sciences, Reims, France, pp. 49–54.

[22] [22] Welch, G. and Bishop, G. (2001). An introduction to the Kalman filter, SIGGRAPH, Los Angeles, USA, Course 8.

[23] [23] Xiong, J., Jauberthie, C., Travé-Massuyès, L. and Le Gall, F. (2013). Fault detection using interval Kalman filtering enhanced by constraint propagation, Proceedings of the IEEE 52nd Annual Conference on Decision and Control (CDC), Florence, Italy, pp. 490–495.