Automatica 62 (2015) 32–38 Contents lists available at ScienceDirect Automatica journal homepage: www.elsevier.com/locate/automatica Brief paper An improved stability condition for Kalman filtering with bounded Markovian packet lossesI Junfeng Wu a, Ling Shi b, Lihua Xie c, Karl Henrik Johansson a a ACCESS Linnaeus Center, School of Electrical Engineering, Royal Institute of Technology, Stockholm, Sweden b Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong c School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore article info a b s t r a c t Article history: In this paper, we consider the peak-covariance stability of Kalman filtering subject to packet losses. The Received 22 January 2015 length of consecutive packet losses is governed by a time-homogeneous finite-state Markov chain. We Received in revised form establish a sufficient condition for peak-covariance stability and show that this stability check can be 21 May 2015 recast as a linear matrix inequality (LMI) feasibility problem. Compared with the literature, the stability Accepted 21 August 2015 condition given in this paper is invariant with respect to similarity state transformations; moreover, our Available online 8 October 2015 condition is proved to be less conservative than the existing results. Numerical examples are provided to demonstrate the effectiveness of our result. Keywords: Networked control systems ' 2015 Elsevier Ltd. All rights reserved. Kalman filtering Estimation theory Packet losses Stability 1. Introduction works assume the packet drops, due to the Gilbert–Elliott channel (Gilbert, 1960; Elliott, 1963), are governed by a time-homogeneous Networked control systems are closed-loop systems, wherein Markov chain. Huang and Dey(2007) introduced the notion of peak sensors, controllers and actuators are interconnected through a covariance, which describes an upper envelope of the sequence of communication network. In the last decade, advances of modern error covariance matrices for the case of an unstable scalar system. control, micro-electronics, wireless communication and network- They focused on its stability with Markovian packet losses and ing technologies have given birth to a considerable number of net- gave a sufficient stability condition. The stability condition was worked control applications. further improved in Xie and Xie(2007) and Xie and Xie(2008). In networked control systems, state estimation such as using a In Wu, Shi, Anderson, and Johansson(0000), the authors proved Kalman filter is necessary whenever precise measurement of the that the peak-covariance stability implies mean-square stability system state cannot be obtained. When a Kalman filter is running for general random packet drop processes, if the system matrix has no defective eigenvalues on the unit circle. In addition to the peak- subject to intermittent observations, the stability of the estimation covariance stability, the mean-square stability was considered for error is affected by not only the system dynamics but also by some classes of linear systems in Mo and Sinopoli(2012), You, the statistics of the packet loss process. The stability of Kalman Fu, and Xie(2011), and weak convergence of the estimation error filtering with packet drops has been intensively studied in the covariance was studied in Xie(2012). literature. In Sinopoli et al.(2004), Plarre and Bullo(2009), Mo and In the aforementioned packet loss models, the length of consec- Sinopol(2010), Shi, Epstein, and Murray(2010) and Kar, Sinopoli, utive packet losses can be infinitely large. In contrast, some works and Moura(2012), independently and identically distributed (i.i.d.) also considered bounded packet loss model, whereby the length of Bernoulli packet losses have been considered. Some other research consecutive packet losses is restricted to be less than a finite inte- ger. A real example of bounded packet losses is the WirelessHART (Wireless Highway Addressable Remote Transducer) protocol, the state-of-the-art wireless communication solution for process au- I The material in this paper was not presented at any conference. This paper was tomation applications. In WirelessHART, there are two types of recommended for publication in revised form by Associate Editor Tongwen Chen under the direction of Editor Ian R. Petersen. time slots: one is the dedicated time slot allocated to a specific E-mail addresses: [email protected] (J. Wu), [email protected] (L. Shi), field device for time-division multiple-access (TDMA) based trans- [email protected] (L. Xie), [email protected] (K.H. Johansson). mission and the other is the shared time slot for contention-based http://dx.doi.org/10.1016/j.automatica.2015.09.005 0005-1098/' 2015 Elsevier Ltd. All rights reserved. J. Wu et al. / Automatica 62 (2015) 32–38 33 communication. A contiguous group of time slots during a con- the Kronecker delta function with δkj D 1 if k D j and δkj D 0 stant period of time forms a superframe, within which every node otherwise. The initial state x0 is a zero-mean Gaussian random is guaranteed at least one time slot for data communication. Vari- vector that is uncorrelated to wk and vk, with covariance Σ0 ≥ 0. ous networked control problems with bounded packet loss models It can be seen that, by applying a similarity transformation, the have been studied, e.g., Wu and Chen(2007) and Xiong and Lam unstable and stable modes of the LTI system can be decoupled. (2007); while the stability of Kalman filtering for this kind of mod- An open-loop prediction of the stable mode always has a bounded els was rarely discussed. In Xiao, Xie, and Fu(2009), the authors estimation error covariance, therefore, this mode does not play gave a first attempt to the stability issue related to the Kalman fil- any key role in the problem considered below. Without loss of tering with bounded Markovian losses. They provided a sufficient generality, all eigenvalues of A are assumed to have magnitudes not condition for peak-covariance stability, the stability notion stud- less than 1. We also assume that .A; C/ is observable and .A; Q 1=2/ ied in Huang and Dey(2007), Xie and Xie(2007) and Xie and Xie is controllable. We introduce the definition of the observability (2008). Their result has established a connection between peak- index of .A; C/, which is taken from Antsaklis and Michel(2006). covariance stability, the dynamics of the underlying system and the probability transition matrix of the underlying packet-loss pro- Definition 1. The observability index Io is defined as the smallest 0 0 0 Io−1 0 0 0 cess. In this paper, we consider the same problem as in Xiao et al. integer such that TC ; A C ;:::;.A / C U has rank n. If Io D 1, the (2009) and improve the stability condition thereof. The main con- system .A; C/ is called one-step observable. tributions of this work are summarized as follows: (1) We present a sufficient condition for peak-covariance stability 2.2. Bounded Markovian packet-loss process of the Kalman filtering subjected to bounded Markovian packet losses (Theorem 1). Different from that of Xiao et al.(2009), this In this paper, we consider the estimation scheme, where the f g stability check can be recast as a linear matrix inequality (LMI) raw measurements yk k2N of the sensor are transmitted to the feasibility problem (Proposition 1). estimator over an erasure communication channel: packets may (2) We compare the proposed condition with that of Xiao et al. be randomly dropped or successively received by the estimator. 2 f g (2009). We show both theoretically and numerically that Denote by a random variable γk 0; 1 whether or not yk is D the proposed stability condition is invariant with respect received at time k. If γk 1, it indicates that yk arrives error-free D to similarity state transformations, while the one given in at the estimator; otherwise γk 0. Whether γk equals 0 or 1 is C Xiao et al.(2009) may generate opposite conclusions under assumed to have been known by the estimator before time k 1. different similarity transformations. Moreover, the analysis In order to introduce the packet loss model, we further define a also suggests that our condition is less conservative than the sequence of stopping times, the time instants at which packets are former one. received by the estimator: The remaining part of the paper is organized as follows. Section2 t1 , minfk V k 2 N; γk D 1g; presents the mathematical models of the system and packet losses, t2 , minfk V k > t1; γk D 1g; and introduces the preliminaries of Kalman filtering. Section3 : provides the main results. Comparison with Xiao et al.(2009) and : (2) numerical examples are presented in Section4. Some concluding t minfk V k > t − ; γ D 1g; (3) remarks are drawn in the end. j , j 1 k where we assume t0 D 0 by convention. The packet-loss process, Notations. N is the set of positive integers and C is the set of τj, is defined as n complex numbers. SC is the set of n by n positive semi-definite n×n τ t − t − − 1: matrices over the field C. For a matrix X 2 C , σ .X/ denotes j , j j 1 the spectrum of X, i.e., σ .X/ D fλ V det(λI − X/ D 0g, and ρ.X/ As for the model of packet losses, we assume that the packet- denotes the spectrum radius of X, X ∗, X 0 and X are the Hermitian f g loss process τj j2N is modeled by a time-homogeneous ergodic conjugate, transpose and complex conjugate of X, respectively. k·k Markov chain, where S D f0;:::; sg is the finite-state space of n means the L2-norm on C or the matrix norm induced by L2-norm.