A Kalman filter with intermittent observations and reconstruction of data losses

Rhouma, Taouba ; Keller, Jean-Yves ; Abdelkrim, Mohamed Naceur

International Journal of Applied Mathematics and Computer Science, Tome 32 (2022) no. 2, pp. 241-253.

Résumé

This paper deals with the problem of joint state and unknown input estimation for stochastic discrete-time linear systems subject to intermittent unknown inputs on measurements. A Kalman filter approach is proposed for state prediction and intermittent unknown input reconstruction. The filter design is based on the minimization of the trace of the state estimation error covariance matrix under the constraint that the state prediction error is decoupled from active unknown inputs corrupting measurements at the current time. When the system is not strongly detectable, a sufficient stochastic stability condition on the mathematical expectation of the random state prediction errors covariance matrix is established in the case where the arrival binary sequences of unknown inputs follow independent random Bernoulli processes. When the intermittent unknown inputs on measurements represent intermittent observations, an illustrative example shows that the proposed filter corresponds to a Kalman filter with intermittent observations having the ability to generate a minimum variance unbiased prediction of measurement losses.

Détail

Keywords: Kalman filter, intermittent unknown inputs, linear system, intermittent observation
Mots-clés : filtr Kalmana, układ liniowy, obserwacja okresowa

Bibliographie

[1] [1] Alouani, A., Rice, T. and Blair, W. (1992). A two-stage filter for state estimation in the presence of dynamical stochastic bias, American Control Conference, Chicago, USA, pp. 1784–1788.

[2] [2] Blanke, M., Kinnaert, M., Lunze, J., Staroswiecki, M. and Schröder, J. (2006). Diagnosis and Fault-Tolerant Control, Vol. 2, Springer, Berlin.

[3] [3] Censi, A. (2010). Kalman filtering with intermittent observations: Convergence for semi-Markov chains and an intrinsic performance measure, IEEE Transactions on Automatic Control 56(2): 376–381.

[4] [4] 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.

[5] [5] Chang, Y. H., Hu, Q. and Tomlin, C.J. (2018). Secure estimation based Kalman filter for cyber-physical systems against sensor attacks, Automatica 95(14): 399–412.

[6] [6] Chen, J. and Patton, R.J. (1996). Optimal filtering and robust fault diagnosis of stochastic systems with unknown disturbances, IEE Proceedings—Control Theory and Applications 143(1): 31–36.

[7] [7] Dán, G. and Sandberg, H. (2010). Stealth attacks and protection schemes for state estimators in power systems, 1st IEEE International Conference on Smart Grid Communications, Gaithersburg, USA, pp. 214–219.

[8] [8] Darouach, M. and Zasadzinski, M. (1997). Unbiased minimum variance estimation for systems with unknown exogenous inputs, Automatica 33(4): 717–719.

[9] [9] Darouach, M., Zasadzinski, M. and Keller, J.Y. (1992). State estimation for discrete systems with unknown inputs using state estimation of singular systems, American Control Conference, Chicago, USA, pp. 3014–3015.

[10] [10] De Sá, A.O., da Costa Carmo, L.F.R. and Machado, R.C. (2017). Covert attacks in cyber-physical control systems, IEEE Transactions on Industrial Informatics 13(4): 1641–1651.

[11] [11] Ding, B. and Fang, H. (2018). Fault estimation and prediction for nonlinear stochastic system with intermittent observations, International Journal of Robust and Nonlinear Control 28(4): 1165–1181.

[12] [12] Fang, H., Shi, Y. and Yi, J. (2011). On stable simultaneous input and state estimation for discrete-time linear systems, International Journal of Adaptive Control and Signal Processing 25(8): 671–686.

[13] [13] Fletcher, A.K., Rangan, S. and Goyal, V.K. (2004). Estimation from lossy sensor data: Jump linear modeling and Kalman filtering, 3rd International Symposium on Information Processing in Sensor Networks, Berkeley, USA, pp. 251–258.

[14] [14] Friedland, B. (1969). Treatment of bias in recursive filtering, IEEE Transactions on Automatic Control 14(4): 359–367.

[15] [15] Gillijns, S. and De Moor, B. (2007). Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough, Automatica 43(5): 934–937.

[16] [16] Hespanha, J.P., Naghshtabrizi, P. and Xu, Y. (2007). A survey of recent results in networked control systems, Proceedings of the IEEE 95(1): 138–162.

[17] [17] Hmida, F.B., Khemiri, K., Ragot, J. and Gossa, M. (2010). Robust filtering for state and fault estimation of linear stochastic systems with unknown disturbance, Mathematical Problems in Engineering 2010, Article ID: 591639.

[18] [18] Hou, M. and Patton, R.J. (1998). Optimal filtering for systems with unknown inputs, IEEE Transactions on Automatic Control 43(3): 445–449.

[19] [19] Hsieh, C.-S. and Chen, F.-C. (1999). Optimal solution of the two-stage Kalman estimator, IEEE Transactions on Automatic Control 44(1): 194–199.

[20] [20] Huang, H., Ahmed, N. and Karthik, P. (2011). On a new type of denial of service attack in wireless networks: The distributed jammer network, IEEE Transactions on Wireless Communications 10(7): 2316–2324.

[21] [21] Huang, M. and Dey, S. (2007). Stability of Kalman filtering with Markovian packet losses, Automatica 43(4): 598–607.

[22] [22] Ignagni, M. (2000). Optimal and suboptimal separate-bias Kalman estimators for a stochastic bias, IEEE Transactions on Automatic Control 45(3): 547–551.

[23] [23] Jie, S., Guoqing, Q., Yinya, L. and Andong, S. (2018). Stochastic convergence analysis of cubature Kalman filter with intermittent observations, Journal of Systems Engineering and Electronics 29(4): 823–833.

[24] [24] Kailath, T., Sayed, A.H. and Hassibi, B. (2000). Linear Estimation, Prentice Hall, Englewood Cliffs.

[25] [25] Kalman, R.E. (1960). A new approach to linear filtering and prediction problems, ASME Journal of Basic Engineering 82(1): 23–45.

[26] [26] Keller, J.Y. and Darouach, M. (1997). Optimal two-stage Kalman filter in the presence of random bias, Automatica 33(9): 1745–1748.

[27] [27] Keller, J.Y. and Sauter, D. (2013). Kalman filter for discrete-time stochastic linear systems subject to intermittent unknown inputs, IEEE Transactions on Automatic Control 58(7): 1882–1887.

[28] [28] Kim, K.H., Lee, J.G. and Park, C.G. (2006). Adaptive two-stage kalman filter in the presence of unknown random bias, International Journal of Adaptive Control and Signal Processing 20(7): 305–319.

[29] [29] Kitanidis, P.K. (1987). Unbiased minimum-variance linear state estimation, Automatica 23(6): 775–778.

[30] [30] Li, W., Jia, Y. and Du, J. (2015). Distributed Kalman consensus filter with intermittent observations, Journal of the Franklin Institute 352(9): 3764–3781.

[31] [31] Liang, J., Sankar, L. and Kosut, O. (2015). Vulnerability analysis and consequences of false data injection attack on power system state estimation, IEEE Transactions on Power Systems 31(5): 3864–3872.

[32] [32] Nosrati, K. and Shafiee, M. (2018). Kalman filtering for discrete-time linear fractional-order singular systems, IET Control Theory Applications 12(9): 1254–1266.

[33] [33] Rhouma, T., Chabir, K. and Abdelkrim, M.N. (2018). Resilient control for networked control systems subject to cyber/physical attacks, International Journal of Automation and Computing 15(3): 345–354.

[34] [34] Rhouma, T., Keller, J.Y., Sauter, D., Chabir, K. and Abdelkrim, M. (2015). Active GLR detector for resilient LQG controller in networked control systems, IFAC-PapersOnLine 48(21): 754–759.

[35] [35] Schenato, L., Sinopoli, B., Franceschetti, M., Poolla, K. and Sastry, S.S. (2007). Foundations of control and estimation over lossy networks, Proceedings of the IEEE 95(1): 163–187.

[36] [36] Shi, L., Xie, L. and Murray, R.M. (2009). Kalman filtering over a packet-delaying network: A probabilistic approach, Automatica 45(9): 2134–2140.

[37] [37] Simon, D. (2006). Optimal State Estimation: Kalman, H∞, and Nonlinear Approaches, Wiley, Hoboken.

[38] [38] Sinopoli, B., Schenato, L., Franceschetti, M., Poolla, K., Jordan, M.I. and Sastry, S.S. (2004). Kalman filtering with intermittent observations, IEEE Transactions on Automatic Control 49(9): 1453–1464.

[39] [39] Sumithra, S. and Vadivel, R. (2021). An optimal innovation based adaptive estimation Kalman filter for accurate positioning in a vehicular ad-hoc network, International Journal of Applied Mathematics and Computer Science 31(1): 45–57, DOI: 10.34768/amcs-2021-0004.

[40] [40] Sun, S. and Ma, J. (2014). Linear estimation for networked control systems with random transmission delays and packet dropouts, Information Sciences 269(2014): 349–365.

[41] [41] Tran, T.A., Jauberthie, C., Travé-Massuyès, L. and Lu, Q.H. (2021). An interval Kalman filter enhanced by lowering the covariance matrix upper bound, International Journal of Applied Mathematics and Computer Science 31(2): 259–269, DOI: 10.34768/amcs-2021-0018.

[42] [42] Varshney, D., Bhushan, M. and Patwardhan, S.C. (2019). State and parameter estimation using extended Kitanidis Kalman filter, Journal of Process Control 76(2019): 98–111.

[43] [43] Wang, H., Tan, S., Zhu, Y. and Li, M. (2020). Deterministic scheduling with optimization of average transmission delays in industrial wireless sensor networks, IEEE Access 8: 18852–18862.

[44] [44] Wang, Q. and Yang, H. (2019). A survey on the recent development of securing the networked control systems, Systems Science Control Engineering 7(1): 54–64.

[45] [45] Yuan, Y. and Sun, F. (2015). Data fusion-based resilient control system under DoS attacks: A game theoretic approach, International Journal of Control, Automation and Systems 13(3): 513–520.

[46] [46] Yuan, Y., Wang, Z. and Guo, L. (2017). Event-triggered strategy design for discrete-time nonlinear quadratic games with disturbance compensations: The noncooperative case, IEEE Transactions on Systems, Man, and Cybernetics: Systems 48(11): 1885–1896.

[47] [47] Zhang, D., Shi, P., Wang, Q.-G. and Yu, L. (2017). Analysis and synthesis of networked control systems: A survey of recent advances and challenges, ISA Transactions 66(1): 376–392.

[48] [48] Zhang, H., Song, X. and Shi, L. (2012). Convergence and mean square stability of suboptimal estimator for systems with measurement packet dropping, IEEE Transactions on Automatic Control 57(5): 1248–1253.

[49] [49] Zhang, X. and Ding, F. (2020). Adaptive parameter estimation for a general dynamical system with unknown states, International Journal of Robust and Nonlinear Control 30(4): 1351–1372.

[50] [50] Zhu, M. and Martinez, S. (2013). On the performance analysis of resilient networked control systems under replay attacks, IEEE Transactions on Automatic Control 59(3): 804–808.