IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 4, APRIL 2018 1003 Kalman Filtering Over Gilbert–Elliott Channels: Stability Conditions and Critical Curve Junfeng Wu , Guodong Shi , Brian D. O. Anderson , Life Fellow, IEEE, and Karl Henrik Johansson , Fellow, IEEE Abstract—This paper investigates the stability of Kalman tal question has inspired various significant results focusing on filtering over Gilbert–Elliott channels where random packet the interface of control and communication and has become drops follow a time-homogeneous two-state Markov chain a central theme in the study of networked sensor and control whose state transition is determined by a pair of failure and recovery rates. First of all, we establish a relaxed condition systems [2]–[4]. guaranteeing peak-covariance stability described by an in- Early works on networked control systems assumed that sen- equality in terms of the spectral radius of the system matrix sors, controllers, actuators, and estimators communicate with and transition probabilities of the Markov chain. We further each other over a finite-capacity digital channel, e.g., [2] and show that the condition can be interpreted using a linear [5]–[13], with the majority of contributions focused on one or matrix inequality feasibility problem. Next, we prove that the peak-covariance stability implies mean-square stability, both finding the minimum channel capacity or data rate needed if the system matrix has no defective eigenvalues on the for stabilizing the closed-loop system, and constructing optimal unit circle. This connection between the two stability no- encoder–decoder pairs to improve system performance. At the tions holds for any random packet drop process. We prove same time, motivated by the fact that packets are the funda- that there exists a critical curve in the failure-recovery rate mental information carrier in most modern data networks [3], plane, below which the Kalman filter is mean-square stable and no longer mean-square stable above. Finally, a lower many results on control or filtering with random packet dropouts bound for this critical failure rate is obtained making use appeared. of the relationship we establish between the two stability State estimation, based on collecting measurements of the criteria, based on an approximate relaxation of the system system output from sensors deployed in the field, is embedded in matrix. many networked control applications and is often implemented Index Terms—Estimation, Kalman filtering, Markov pro- recursively using a Kalman filter [14], [15]. Clearly, channel cesses, stability, stochastic systems. randomness leads to that the characterization of performance is not straightforward. A burst of interest in the problem of the stability of Kalman filtering with intermittent measurements has I. INTRODUCTION arisen after the pioneering work [16], where Sinopoli et al. mod- A. Background and Related Works eled the statistics of intermittent observations by an independent IRELESS communications are being widely used nowa- and identically distributed (i.i.d.) Bernoulli random process and W days in sensor networks and networked control systems. studied how packet losses affect the state estimation. Tremen- New challenges accompany the considerable advantages wire- dous research has been devoted to stability analysis of Kalman less communications offer in these applications, one of which is filtering or the closed-loop control systems over i.i.d. packet how channel fading and congestion influence the performance lossy packet networks in [17]–[22]. of estimation and control. In the past decade, this fundamen- To capture the temporal correlation of realistic commu- nication channels, the Markovian packet loss model has Manuscript received May 14, 2017; accepted July 14, 2017. Date of been introduced to partially address this problem. Since the publication July 27, 2017; date of current version March 27, 2018. This Gilbert–Elliott channel model [23], [24], a classical two-state work was supported in part by the Knut and Alice Wallenberg Foundation, time-homogeneous Markov channel model, has been widely in part by the Swedish Research Council, and in part by the National Natural Science Foundation of China under Grant 61120106011. This applied to represent wireless channels and networks in indus- paper was presented in part at the 34th Chinese Control Conference, trial applications [25]–[27], the problem of networked control Hangzhou, China, July 2015. Recommended by Associate Editor S. over Gilbert–Elliott channels has drawn considerable attention. Andersson. J. Wu and K. H. Johansson are with the ACCESS Linnaeus Center, Huang and Dey [28], [29] considered the stability of Kalman School of Electrical Engineering, Royal Institute of Technology, Stock- filtering with Markovian packet losses. To aid the analysis, holm SE-100 44, Sweden (e-mail: [email protected]; [email protected]). they introduced the notation of peak covariance, defined by G. Shi and B. D. O. Anderson are with the Research School of Engineering, The Australian National University, Canberra, ACT 0200, the expected prediction error covariance at the time instances Australia (e-mail: [email protected]; brian.anderson@anu. when the channel just recovers from failed transmissions, as edu.au). an evaluation of estimation performance deterioration. The Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. peak-covariance stability (PCS) can be studied by lifting the Digital Object Identifier 10.1109/TAC.2017.2732821 original systems at stopping times. Sufficient conditions for the 0018-9286 © 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications standards/publications/rights/index.html for more information. 1004 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 4, APRIL 2018 PCS were proposed for general vector systems, and a necessary be recast as a linear matrix inequality (LMI) feasibility and sufficient condition for scalar systems. The existing problem. These conditions are theoretically and numer- literature [28]–[31] has however made restrictive assumptions ically shown to be less conservative than those in the on the plant dynamics and the communication channel. We literature. show by numerical examples that existing conditions for PCS 2) We prove that PCS implies MSS if the system matrix has only apply to relative reliable channels with low failure rate. no defective eigenvalues on the unit circle. Remarkably Moreover, existing results rely on calculating an infinite sum enough this implication holds for any random packet drop of matrix norms for checking stability conditions. process that allows PCS to be defined. This result bridges The PCS describes covariance stability at random stopping two stability criteria in the literature and offers a tool for times, whereas the mean-square stability (MSS) describes co- studying MSS of the Kalman filter through its PCS. Note variance stability at deterministic times. In the literature, the that MSS was previously studied using quite different main focus is on MSS rather than PCS. This is partly because methods such as analyzing the boundness of the expec- the definition of the former is more natural and practically useful tation of a kind of randomized observability Gramians than that of the latter. However, it is difficult to analyze MSS over a stationary random packet loss process to establish over Gilbert–Elliott channels directly through the random Ric- the equivalence between stability in stopping times and cati equation, due to temporal correlation between the packet stability in sampling times [32], and characterizing the losses. If we could build clearer connections relating these two decay rate of the prediction covariance’s tail distribution stability notions, the MSS can be conveniently studied through for so-called nondegenerate systems [34]. PCS. The relationship between the MSS and the PCS was pre- 3) We prove that there is a critical p − q curve, with p be- liminarily discussed [29]. Improvements to these results can be ing the failure rate and q being the recovery rate of the found in [30] and [31]. Particularly in [32], by investigating the Gilbert–Elliott channel, below which the expected pre- estimation error covariance matrices at each packet reception, diction error covariance matrices are uniformly bounded necessary and sufficient conditions for the MSS were derived for and unbounded above. The existence of a critical curve second-order systems and some certain classes of higher order for Markovian packet losses is an extension of that of systems. Although it was proved that with i.i.d. packet losses the critical packet loss rate subject to i.i.d. packet losses the PCS is equivalent to the MSS for scalar systems and sys- in [16]. However, the proof method of [16] does not ap- tems that are one-step observable [29], [31], for vector systems ply to Markovian packet losses. In this paper, the critical with more general packet drop processes, this relationship is curve is proved via a novel coupling argument, and to the unclear. best of our knowledge, this is the first time phase transi- There are some other works studying distribution of error co- tion is established for Kalman filtering over Markovian variance matrices. Essentially, the probabilistic characteristics channels. Finally, we present a lower bound for the crit- of the prediction error covariance are fully captured by its prob- ical