Event-based state estimation of linear dynamic systems with unknown exogenous inputs

In this work, an event-based optimal state estimation problem for linear-time varying systems with unknown inputs is investigated. By treating the unknown input as a process with a non-informative prior, the event-based minimum mean square error (MMSE) estimator is obtained in a recursive form. It is shown that for the general time-varying case, the closed-loop matrix of the optimal event-based estimator is exponentially stable and the estimation error covariance matrix is asymptotically bounded for each sample path of the event-triggering process. The results are also extended to the multiple sensor scenario, where each sensor is allowed to have its own event-triggering condition. The efficiency of the proposed results is illustrated by a numerical example and comparative simulation with the MMSE estimators obtained based on time-triggered measurements. The results are potentially applicable to event-based secure state estimation of cyber-physical systems.

[1]  M. Lundberg,et al.  On posterior distributions for signals in Gaussian noise with unknown covariance matrix , 2005, IEEE Transactions on Signal Processing.

[2]  Mircea Lazar,et al.  Event Based State Estimation With Time Synchronous Updates , 2012, IEEE Transactions on Automatic Control.

[3]  Ling Shi,et al.  Event-Based Sensor Data Scheduling: Trade-Off Between Communication Rate and Estimation Quality , 2013, IEEE Transactions on Automatic Control.

[4]  T. Floquet,et al.  On Sliding Mode Observers for Systems with Unknown Inputs , 2006, International Workshop on Variable Structure Systems, 2006. VSS'06..

[5]  Bernard C. Levy,et al.  Principles of Signal Detection and Parameter Estimation , 2008 .

[6]  John S. Baras,et al.  Adaptive Sampling for Linear State Estimation , 2009, SIAM J. Control. Optim..

[7]  Jingang Yi,et al.  On stable simultaneous input and state estimation for discrete‐time linear systems , 2011 .

[8]  John B. Moore,et al.  Hidden Markov Models: Estimation and Control , 1994 .

[9]  K. Åström,et al.  Comparison of Riemann and Lebesgue sampling for first order stochastic systems , 2002, Proceedings of the 41st IEEE Conference on Decision and Control, 2002..

[10]  H. Jeffreys.  An invariant form for the prior probability in estimation problems , 1946, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[11]  Marek Miskowicz,et al.  Send-On-Delta Concept: An Event-Based Data Reporting Strategy , 2006, Sensors (Basel, Switzerland).

[12]  W. P. M. H. Heemels,et al.  Periodic Event-Triggered Control for Linear Systems , 2013, IEEE Trans. Autom. Control..

[13]  Weiyi Liu,et al.  Markov Chain Approximation Algorithm for Event-Based State Estimation , 2015, IEEE Transactions on Control Systems Technology.

[14]  Ling Shi,et al.  An event-triggered approach to state estimation with multiple point- and set-valued measurements , 2014, Autom..

[15]  M. Lemmon,et al.  Event-triggered state estimation in vector linear processes , 2010, Proceedings of the 2010 American Control Conference.

[16]  Mohamed Darouach,et al.  Unbiased minimum variance estimation for systems with unknown exogenous inputs , 1997, Autom..

[17]  Edward R. Dougherty,et al.  Bayesian Minimum Mean-Square Error Estimation for Classification Error—Part I: Definition and the Bayesian MMSE Error Estimator for Discrete Classification , 2011, IEEE Transactions on Signal Processing.

[18]  J.S. Baras,et al.  Multiple Sampling for Estimation on a Finite Horizon , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[19]  Peter K. Kitanidis,et al.  Unbiased minimum-variance linear state estimation , 1987, Autom..

[20]  Tor Arne Johansen,et al.  Fault tolerant control allocation using unknown input observers , 2014, Autom..

[21]  Jerry L. Prince,et al.  On the optimality of recursive unbiased state estimation with unknown inputs , 2000, Autom..

[22]  Robert D. Nowak,et al.  Wavelet-based image estimation: an empirical Bayes approach using Jeffrey's noninformative prior , 2001, IEEE Trans. Image Process..

[23]  Raffaello D'Andrea,et al.  An Experimental Demonstration of a Distributed and Event-Based State Estimation Algorithm , 2011 .

[24]  Nandit Soparkar,et al.  Trading computation for bandwidth: reducing communication in distributed control systems using state estimators , 2002, IEEE Trans. Control. Syst. Technol..

[25]  George J. Pappas,et al.  Stochastic game approach for replay attack detection , 2013, 52nd IEEE Conference on Decision and Control.

[26]  Benjamin Noack,et al.  A study on event triggering criteria for estimation , 2014, 17th International Conference on Information Fusion (FUSION).

[27]  Bruno Sinopoli,et al.  Secure Estimation in the Presence of Integrity Attacks , 2013, IEEE Transactions on Automatic Control.

[28]  Ling Shi,et al.  On Set-Valued Kalman Filtering and Its Application to Event-Based State Estimation , 2015, IEEE Transactions on Automatic Control.

[29]  Robert J. Elliott,et al.  Event-based state estimation of discrete-state hidden Markov models , 2016, Autom..

[30]  Qing-Long Han,et al.  Event-based H∞ filtering for sampled-data systems , 2015, Autom..

[31]  Pablo A. Iglesias,et al.  A Spectral Test for Observability and Reachability of Time-Varying Systems , 1999 .

[32]  M. Darouach,et al.  Full-order observers for linear systems with unknown inputs , 1994, IEEE Trans. Autom. Control..

[33]  Donghua Zhou,et al.  Unbiased minimum-variance state estimation for linear systems with unknown input , 2009, Autom..

[34]  Paulo Tabuada,et al.  Secure Estimation and Control for Cyber-Physical Systems Under Adversarial Attacks , 2012, IEEE Transactions on Automatic Control.

[35]  Arthur G. O. Mutambara,et al.  Decentralized Estimation and Control for Multisensor Systems , 2019 .

[36]  Baibing Li,et al.  State estimation with partially observed inputs: A unified Kalman filtering approach , 2013, Autom..

[37]  Michael D. Lemmon,et al.  Performance and average sampling period of sub-optimal triggering event in event triggered state estimation , 2011, IEEE Conference on Decision and Control and European Control Conference.

[38]  Sebastian Trimpe,et al.  Event-Based State Estimation With Variance-Based Triggering , 2012, IEEE Transactions on Automatic Control.

[39]  Huazhen Fang,et al.  On the asymptotic stability of minimum-variance unbiased input and state estimation , 2012, Autom..

[40]  Florian Dörfler,et al.  Attack Detection and Identification in Cyber-Physical Systems -- Part II: Centralized and Distributed Monitor Design , 2012, ArXiv.

[41]  Ling Shi,et al.  Stochastic event-triggered sensor scheduling for remote state estimation , 2013, 52nd IEEE Conference on Decision and Control.

[42]  B. Anderson,et al.  Optimal Filtering , 1979, IEEE Transactions on Systems, Man, and Cybernetics.

[43]  Sandra Hirche,et al.  An Iterative Algorithm for Optimal Event-Triggered Estimation , 2012, ADHS.

[44]  T. Başar,et al.  Optimal Estimation with Limited Measurements , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[45]  Joris Sijs,et al.  Relevant Sampling Applied to Event-Based State-Estimation , 2010, 2010 Fourth International Conference on Sensor Technologies and Applications.

[46]  Tongwen Chen,et al.  Event detection and control co-design of sampled-data systems , 2014, Int. J. Control.

[47]  Sebastian Trimpe,et al.  Stability analysis of distributed event-based state estimation , 2014, 53rd IEEE Conference on Decision and Control.

[48]  Thor I. Fossen,et al.  On estimation of wind velocity, angle-of-attack and sideslip angle of small UAVs using standard sensors , 2015, 2015 International Conference on Unmanned Aircraft Systems (ICUAS).

[49]  Mohamed Darouach,et al.  Extension of minimum variance estimation for systems with unknown inputs , 2003, Autom..

[50]  Jinya Su,et al.  On existence, optimality and asymptotic stability of the Kalman filter with partially observed inputs , 2015, Autom..

[51]  Huijun Gao,et al.  Event-Triggered State Estimation for Complex Networks With Mixed Time Delays via Sampled Data Information: The Continuous-Time Case , 2015, IEEE Transactions on Cybernetics.

[52]  Bart De Moor,et al.  Unbiased minimum-variance input and state estimation for linear discrete-time systems , 2007, Autom..

[53]  Karl Henrik Johansson,et al.  Distributed Event-Triggered Estimation in Networked Systems , 2012, ADHS.

[54]  Giuseppe de Nicolao.  On the time-varying Riccati difference equation of optimal filtering , 1992 .

[55]  Ling Shi,et al.  Event-triggered maximum likelihood state estimation , 2014, Autom..

[56]  Uwe D. Hanebeck,et al.  Event-based state estimation with negative information , 2013, Proceedings of the 16th International Conference on Information Fusion.

[57]  Ling Shi,et al.  Time and Event-based Sensor Scheduling for Networks with Limited Communication Resources , 2011 .

[58]  Tongwen Chen,et al.  Optimal Sampling and Performance Comparison of Periodic and Event Based Impulse Control , 2012, IEEE Transactions on Automatic Control.

[59]  Siva Sivaganesan,et al.  Bayes factors for a test about the drift of the Brownian motion under noninformative priors , 2000 .

[60]  B. Anderson,et al.  Detectability and Stabilizability of Time-Varying Discrete-Time Linear Systems , 1981 .

[61]  Bingchang Wang,et al.  Comparison of Periodic and Event Based Sampling for Linear State Estimation , 2014 .

[62]  Holger Dette,et al.  Optimal designs for nonlinear regression models with respect to non-informative priors , 2013, 1311.0835.