An Improved Stability Condition for Kalman Filtering with Bounded Markovian Packet Losses✩

Automatica 62 (2015) 32–38

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Automatica

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Brief paper An improved stability condition for Kalman filtering with bounded Markovian packet losses✩

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 integer. 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- ✩ 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 = 1 if k = j and δkj = 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 ′ ′ ′ Io−1 ′ ′ ′ cess. In this paper, we consider the same problem as in Xiao et al. integer such that [C , A C ,...,(A ) C ] has rank n. If Io = 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 { } stability check can be recast as a linear matrix inequality (LMI) raw measurements yk k∈N 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. ∈ { } (2009). We show both theoretically and numerically that Denote by a random variable γk 0, 1 whether or not yk is = the proposed stability condition is invariant with respect received at time k. If γk 1, it indicates that yk arrives error-free = to similarity state transformations, while the one given in at the estimator; otherwise γk 0. Whether γk equals 0 or 1 is + 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 , min{k : k ∈ N, γk = 1}, presents the mathematical models of the system and packet losses, t2 , min{k : k > t1, γk = 1}, 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 min{k : k > t − , γ = 1}, (3) remarks are drawn in the end. j , j 1 k where we assume t0 = 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. S+ 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 ∈ C , σ (X) denotes j , j j 1 the spectrum of X, i.e., σ (X) = {λ : det(λI − X) = 0}, and ρ(X) As for the model of packet losses, we assume that the packet- denotes the spectrum radius of X, X ∗, X ′ and X are the Hermitian { } loss process τj j∈N is modeled by a time-homogeneous ergodic conjugate, transpose and complex conjugate of X, respectively. ∥·∥ Markov chain, where S = {0,..., s} is the finite-state space of n means the L2-norm on C or the matrix norm induced by L2-norm. the Markov chain with s being the maximum length of consecutive The symbol ⊗ represents the Kronecker product operator of two lost packets allowed. Here the Markov chain is characterized by a matrices. For any matrices A, B, C with compatible dimensions, we known transition probability matrix 5 , [πij]i,j∈S in which have vec(ABC) = (C′ ⊗ A)vec(B), where vec(·) is the vectorization of a matrix. Moreover, the indicator function of a subset A ⊂ Ω πij , P(τk+1 = j|τk = i) ≥ 0. (4) is a function 1A : Ω → {0, 1} where 1A(ω) = 1 if ω ∈ A, Denote the initial distribution as p , [p0,..., ps], where pj = otherwise 1A(ω) = 0. The symbol E[·] (resp., E[·|·]) represents the P(τ1 = j). expectation (resp., conditional expectation) of a random variable. 2.3. Kalman filtering with packet losses 2. Problem setup Sinopoli et al.(2004) shows that, when performed with intermittent observations, the optimal linear estimator is a modified 2.1. System model Kalman filter. The modified Kalman filter is slightly different from the standard one in that only time update is performed in the pres- Consider the following discrete-time LTI system: ence of the lost packet. Define the minimum mean-squared error xk+1 = Axk + wk, (1a) estimate and the one-step prediction at the estimator respectively as xˆk|k [xk|γ1y1, . . . , γkyk] and xˆk+1|k [xk+1|γ1y1, . . . , γkyk]. yk = Cxk + vk, (1b) , E , E Let Pk|k and Pk+1|k be the corresponding estimation and prediction n×n m×n n where A ∈ R and C ∈ R , xk ∈ R is the process state vector, error covariance matrices, i.e., m n m yk ∈ R is the observation vector, wk ∈ R and vk ∈ R are zero- ′ Pk|k , E[(xk − xˆk|k)(·) |γ1y1, . . . , γkyk] mean Gaussian random vectors with E[wkwj′] = δkjQ (Q ≥ 0), ′ E[vkvj′] = δkjR (R > 0), E[wkvj′] = 0 ∀j, k. Note that δkj is Pk+1|k , E[(xk+1 − xˆk+1|k)(·) |γ1y1, . . . , γkyk]. 34 J. Wu et al. / Automatica 62 (2015) 32–38

These parameters can be computed recursively by a modified 3. Main result Kalman filter (see Sinopoli et al., 2004 for more details). In partic- ular, In the following theorem, we will present a sufficient condition ′ for peak-covariance stability of Kalman filtering with bounded Pk+1|k = APk|k−1A + Q Markovian packet-loss process. ′ ′ −1 ′ − γkAPk|k−1C (CPk|k−1C + R) CPk|k−1A . (5) Theorem 1. Consider the system described in (1a) and (1b), and the To simplify notations, we denote Pk , Pk|k−1 for shorthand and define the functions h, g, hk and gk: n → n as follows: bounded Markovian packet-loss process described by a probability S+ S+ (1) (I −1) transition matrix 5 in (4). If there exists K , [K ,..., K o ], ′ (i) h(X) , AXA + Q , (6) where K ’s are matrices with compatible dimensions, such that ′ ′ ′ −1 ′ ρ(HK ) < 1, where g(X) , AXA + Q − AXC (CXC + R) CXA , (7) k k  s  ′ ′  h (X) , h ◦ h ◦ · · · ◦ h(X) and g (X) , g ◦ g ◦ · · · ◦ g(X), where ◦ HK , diag A ⊗ A,...,(A ⊗ A) P ⊗ H + Q ⊗ K , (10)       k times k times denotes the function composition. P , [πij]i,j∈S/{0} and Q , [πi0π0j]i,j∈S/{0},

H A + K (1)C ⊗ A + K (1)C 2.4. Problems of interest , ( ) ( ), Io−1  l−2 l (l) (l) l (l) (l) To study the stability of Kalman filtering with packet losses, one K , (π00) (A + K C ) ⊗ (A + K C ) way is to study the asymptotic behavior of the expected prediction l=2 error covariance. In the following we introduce the concept of with peak-covariance stability, which is first studied in Huang and Dey (i) ′ ′ ′ ′ i−1 ′ ′ (2007). To this end, we need the following auxiliary definitions, C , [C , A C ,...,(A ) C ] ; (11) which were introduced in Huang and Dey(2007), then the state estimator is peak-covariance stable, i.e., supj∈N α1 min{k : k ∈ , γk = 0}, ∥ ∥ ∞ , N E Pβj < . β1 , min{k : k > α1, γk = 1}, (8) Before proceeding to the proof, we first give a few supporting . . definitions and lemmas. Consider k compositions of g together. We introduce the { : = } αj , min k k > βj−1, γk 0 , following lemma. βj , min{k : k > αj, γk = 1}, (9) Lemma 2. Consider the operator where β0 = 0 by convention. It is straightforward to verify that {α } ∈ and {β } ∈ are two sequence of stopping times (see Durrett, (i) i (i) (i) ∗ j j N j j N φi(K , P) , (A + K C )X(·) 2010).  (i) (i) (i) ′  (i) (i) Q Q (D ) (i) (i) ∗ + [A K ] ′ [A K ] , Definition 2. The Kalman filtering system with packet losses is D(i)(Q (i)) D(i)(Q (i))(D(i)) + R(i) said to be peak-covariance stable if supj∈ E∥Pβ ∥ < ∞. N j (i) i−1 (i) where i ∈ N,A , [A ,..., A, I],D = 0 for i = 1 The above definition is made regardless of the initial distribution p,  0 0 ··· 0 p due to the following lemma. For any initial distribution p, let E [·] C 0 ··· 0 (i)  . . . (i) denote the expectation conditioned on p. Thanks to the following otherwise D , . . .. . ,Q , diag(Q ,..., Q ),  . . . . lemma, we can omit the superscript from then on. − −    CAi 2 CAi 3 ··· 0 i (i) (i) Lemma 1. For any initial distributions, p and p∗, of the packet-loss R , diag(R,..., R), and K ’s are of compatible dimensions. For { } p∥ ∥ ∞ p∗ ∥ ∥    process τj j∈N, supj∈N E Pβj < if and only if supj∈N E Pβj < i ∞ any X ≥ 0 and K (i), the following statement holds: p∥ ∥ = s [∥ ∥ | = ] p∥ ∥ Proof. Since E Pβj i=1 piE Pβj τ1 i , supj∈ E Pβj < i (i) (i) N g (X) , min φi(K , X) ≤ φi(K , X). ∞ [∥ ∥ | = ] ∞ ∈ (i) if and only if E Pβj τ1 i < for any i S. The result K p∗ ∥ ∥ follows using the same argument for E Pβj . In the literature, stability of Kalman filtering with Markovian Proof. The result is readily established when setting B = I in packet losses (driven by a two-state Gilbert–Elliott packet loss Lemmas 2 and 3 in Xiao et al.(2009). For i = 1, The result is well model) (Huang& Dey, 2007; Xie& Xie, 2008; You et al., 2011) known as Lemma 1 in Sinopoli et al.(2004). and with i.i.d. packet losses (Sinopoli et al., 2004; Shi et al., The following lemma is about the nonlinearity of g operator: for 2010) has been intensively studied. The main problem of this k ≥ I +1, gk(X) is uniformly bounded no matter what the positive work is to study stability of Kalman filtering at packet reception o semidefine matrix X is. times subject to bounded Markovian packet-loss process. As the packet loss is modeled differently, the stability also behaves Lemma 3 (Huang& Dey, 2007, Lemma 5 ). Assume that (A, C) is differently. Due to the nonlinearity of the Kalman filter, it seems observable and A Q 1/2 is controllable. Define challenging to find necessary and sufficient stability conditions for ( , ) a general LTI system. In Section3, we manage to give a sufficient n { : ≤ ≤ ′ + ≥ } S0 , P 0 P AP0A Q , for some P0 0 . peak-covariance stability condition for general LTI systems with bounded Markovian packet-loss process. Our result is mainly built Then there exists a constant L > 0 such that on the prior work (Xiao et al., 2009). Compared with the result (i) for any X ∈ n, gk(X) ≤ LI for all k ≥ I ; thereof, ours prevails from at least two aspects. We will discuss S0 o n k+1 in detail later in Section4. (ii) for any X ∈ S+, g (X) ≤ LI for all k ≥ Io. J. Wu et al. / Automatica 62 (2015) 32–38 35

According to the definitions of αj and βj, we can further define the analysis in the following, we will impose (12) to take equality. the sojourn times at the state 1 and 0 respectively as follows: Without loss of generality, the conclusions in this paper still hold ∗ − ∈ without imposing equality as (12) renders us an upper bound of αj , αj βj−1 N, E[Pβ ]. Next we vectorize both sides of (12). ∗ − ∈ { } 1 βj , βj αj 1,..., s . s  ∗ ∗ [ ] = ⊗ b1 + (1) ⊗ + (1) The distributions of the sojourn times αj and βj are given in the E vec(Pβ1 ) (A A) pb1 (A K C) (A K C) following lemma. b1=1

Io Lemma 4 (Xiao et al., 2009, lemma 4). Denote the joint distribution  a1−2 a (a ) (a ) + p0(π00) π0b (A 1 + K 1 C 1 ) of α∗ and β∗ by 1 j j a1=2  ∗ ∗ ∗ ∗   l = a = b = a = b a1 (a1) (a1) π( ) , P α1 1, β1 1, . . . , αl l, βl l , ⊗ (A + K C ) vec(Σ0) + vec(U) for any αj ∈ N and βj ∈ {1,..., s}. Then it holds that = T Ψ vec(Σ0) + vec(U), (13)  = ; pb1 , if a1 1 π(1) = − where a1 2 ≥ p0(π00) π0b1 , if a1 2, T = [1,..., 1] ⊗ I 2× 2  l if a = 1; n n πblbl+1 π( ), l+1    π(l + 1) = − al+1 2 ≥ s numbers πbl0(π00) π0bl+1 π(l), if al+1 2. ′ ′ ′ sn2×n2 and Ψ = [ψ , . . . , ψ ] ∈ C with Proof of Theorem 1. Compute [P ] as follows: 1 s E β1  i (1) (1) ∞ s ψ = (A ⊗ A) p (A + K C) ⊗ (A + K C)   i 1 [Pβ ] = Pβ π(1) E 1 1 Io = = a1 1 b1 1  a1−2 a (a ) (a ) + p0(π00) π01(A 1 + K 1 C 1 ) s  a1=2 = p hb1 ◦ g(Σ ) b1 0  a1 (a1) (a1) b1=1 ⊗(A + K C )

Io s   − + a1 2 b1 ◦ a1 for i ∈ {1,..., s}. p0(π00) π0b1 h g (Σ0) Similarly, for any l ≥ 1, [Pβ + ] can be calculated as a1=2 b1=1 E l 1 ∞ s ∞ s ∞ s   −     + a1 2 b1 ◦ a1 [ ] = ··· + p0(π00) π0b1 h g (Σ0) E Pβl+1 Pβl+1 π(l 1) a1=Io+1 b1=1 a1=1 b1=1 al+1=1 bl+1=1 s ∞ s ∞ s  ∗ ′     ≤ p Ab1 A + K (1)C A + K (1)C Ab1 = ··· + b1 ( )Σ0( ) ( ) Pβj+1 π(l 1) b1=1 a1=1 b1=1 al+1=Io bl+1=1

s  Io ∞ s Io−1 s  b  a −2     − + A 1 p (π ) 1 π + ··· al+1 2 0 00 0b1 πbl0(π00) π0bl+1 = = b1 1 a1 2 a1=1 b1=1 al+1=2 bl+1=1  · hbl+1 ◦ gal+1 P π(l) a1 (a1) (a1) ∗ b1 ′ βl × (A + K C )Σ0( · ) (A ) ∞ s s s     bl+1   + ··· πb b + h ◦ g Pβ π(l) s l l 1 l a =1 b =1 b =1 b + =1  b1  (1) (1) ∗ b1 ′ 1 1 l l 1 + pb A [AK ]J1[AK ] (A ) 1 Γ + Γ + Γ . b1=1 , 1 2 3 s I o Next we will analyze the boundedness of Γ1, Γ2 and Γ3 one by one.   a1−2 + p0(π00) π0b 1 ∞ s ∞ s b1=1 a1=2     ≤ ··· bl+1 L + ∗ ′ Γ1 h ( I)π(l 1) , W1, (14) × Ab1 [Aa1 K (a1)]J [Aa1 K (a1)]  Ab1 a1 ( ) a1=1 b1=1 al+1=Io bl+1=1

s  Io  b1−1   −  ′ where the inequality is derived from Lemma 3 and W1 is a bounded + p (π )a1 2π + p AiQ (Ai) 0 00 0b1 b1 matrix. b1=1 a1=2 i=0 ∞ s Io−1 s ∞ s     al+1−2 bl+1   a −2 b Γ ≤ ··· π (π ) π h + 1 1 2 bl0 00 0bl+1 p0(π00) π0b1 h (LI) a1=1 b1=1 al+1=2 bl+1=1 a1=Io+1 b1=1  (al+1)  ◦ φa K , Pβ π(l) , Λ1 + Λ2 + Λ3 + Λ4 + Λ5 + Λ6, (12) l+1 l ∞ s Io−1 s  (i) (i) (i) ′  Q Q (D )     al+1−2 where Ji ′ , the partition of Λ1 to Λ6 = ··· πb 0(π00) π0b , D(i)(Q (i)) D(i)(Q (i))(D(i)) + R(i) l l+1 a =1 b =1 a + =2 b + =1 is based on whether or not gk+1(X) ≤ LI has a uniform bound 1 1 l 1 l 1 ∗ ′ × bl+1 al+1 + (al+1) al+1 · bl+1 when k ≥ Io or k ≤ Io − 1 by Lemma 3 and the distribution A (A K C )Pβl ( ) (A ) π(l) of π(1) given in Lemma 4, and the inequality is from Lemmas 2 ∞ s Io−2 s     a −2 and3. One can verify that Λ3, Λ4, Λ5 and Λ6 are all bounded + ··· l+1 πbl0(π00) π0bl+1 matrices. Then U Λ +Λ +Λ +Λ is also bounded. To facilitate , 3 4 5 6 a1=1 b1=1 al+1=2 bl+1=1 36 J. Wu et al. / Automatica 62 (2015) 32–38

bl+1−1  ′ It is worth noting the one-step observable (A, C) system, see × AiQ (Ai) π(l) Definition 1. In this case, we can find K1 such that A + K1C = 0. By i=0 Proposition 1, any X > 0 will automatically satisfy condition (ii), ∞ s Io−1 s     − thereby implying that the Kalman filtering system with bounded + ··· al+1 2 πbl0(π00) π0bl+1 Markovian packet losses is always peak-covariance stable. This = = = = a1 1 b1 1 al+1 2 bl+1 1 observation is consistent with Xiao et al.(2009, Theorem 3.1). ∗ ′ × bl+1 [ al+1 (al+1)] [·]  bl+1 A A K Jal+1 (A ) π(l) ′ + + 4. Comparison with Xiao et al.(2009) , Γ2 W2 W3.

It is straightforward to verify that W2 and W3 is bounded. In this part, we compare our result with that of Xiao et al.(2009) ∞  s s and show the advantages of ours. Recall that the sufficient stability ≤ ··· bl+1 ◦  (1)  Γ3 πblbl+1 h φ1 K , Pβl π(l) condition in Xiao et al.(2009) is ρ(Φ) < 1 where a1=1 b1=1 bl+1=1  Io−1  ∞ s s (1) +  l−1 (1) ∥ ∥2 ∥ s∥2    Φ , d1 P (π00) dl Q diag A ,..., A , = ··· πb b l l+1 l=2 a1=1 b1=1 bl+1=1 (1) (l) (l) (l) 2 b + (1) ∗ b + ′ ∥ + ∥ × l 1 + · l 1 with P, Q being defined in (10) and dl , minK(l) A K C . A (A K C)Pβl ( ) (A ) π(l)

∞ s s bl+1−1 +   ···   i i ′ 4.1. Invariance with respect to similarity transformations πblbl+1 A Q (A ) π(l) a1=1 b1=1 bl+1=1 i=0 ∞ s s Theoretically, a state variable transformation (i.e., a similarity +   ···  transformation from a linear system (A, B, C, D) to (S−1AS, S−1B, πbl0πblbl+1 a1=1 b1=1 bl+1=1 CS, D) through the nonsingular matrix S) does not change the ∗ ′ stability considered in this work. However, different state variable × Abl+1 [AK (1)] J [AK (1)]  (Abl+1 ) π(l) 1 transformations may generate opposite conclusions from the ′ + + , Γ3 W4 W5, stability condition given in Xiao et al.(2009). The invariance of stability behavior with respect to state variable transformations where W and W can be readily shown to be bounded. In 4 5 can be reflected well from the stability conditions presented by this summary, work. ′ ′ [Pl+1] ≤ Γ + Γ + V , for j ≥ 1, (15) E 2 3 n×n Proposition 2. Let S ∈ C be nonsingular. Suppose there exists where V W +W +W +W +W . By a similar argument, we im- (1) (Io−1) (i) , 1 2 3 4 5 K , [K ,..., K ], where K ’s are matrices with compatible pose (15) to take equality and take vectorization. By (13) and (15), dimensions, such that ρ(HK ) < 1, where HK is defined in (10) for we calculate E[vec(Pβ + )] and convert it to the following form: ˜ −1 (1) (Io−1) l 1 (A, C). Then, there always exists K , S [K ,..., K ] such ˜ ˜ ˜ ˜ −1 [ ] = l + l that ρ(H ˜ ) < 1, where H ˜ is defined for (A, C) , (S AS, CS) in E vec(Pβl+1 ) T (HK ) Ψ vec(Σ0) T (HK ) vec(Θl) K K accordance with (10). l−1 + T (HK ) vec(Θl−1) + · · · + vec(Θ0), The proof follows from Proposition 1 and direct calculation. We use (i) where Θ0,..., Θl are the functions of Q , A, K ’s and are bounded. the following example to illustrate this idea. Similar to U and V , Θ0,..., Θl can be readily shown to be bounded in a similar way. Therefore, sup [vec(P )] is bounded as l → k≤l E βk Example 1. Consider the system ∞ if ρ(HK ) < 1. By some basic algebraic manipulations, one ob-   tains that E∥Pβ + ∥ is uniformly bounded if ρ(HK ) < 1, which com- 1.3 0.3 l 1 A = , C = [ 1 1 ]. pletes the proof. 0 1.2 The stability condition in Theorem 1 is difficult to test. Q = I2×2 and R = 1, and the bounded Markovian packet-loss In the following proposition, we provide an equivalent condi- process with transition probability matrix given by tion. The proof has been omitted for the sake of limited space and can be found in the extended version of this work (see 0.6 0.2 0.2 http://arxiv.org/abs/1501.05469). In view of this result, Theorem 1 5 = 0.8 0.1 0.1 . (16) can be recast as an LMI feasibility problem. As for the conversion 0.8 0.1 0.1 to LMIs using Schur complements, we refer readers to Boyd, El Ghaoui, and Feron(1994) for details. (1) = = From Xiao et al.(2009), we have d1 1.2200 and ρ(Φ) 0.7352 < 1. Let Proposition 1. The following statements are equivalent: 1 5 (i) There exists K [K (1),..., K (Io−1)], where K (i)’s are matrices S = . , 0 1 with compatible dimensions, such that ρ(HK ) < 1, where HK is defined in (10); ˜ ˜ −1 ˜(1) = For the system (A, C) , (S AS, CS), we have d1 1.3632 and (ii) There exist X1 > 0,..., Xs > 0 and K1,..., KI −1 such that o ρ(Φ˜ ) = 1.5202 > 1. s s Io−1  j ∗ j ′ +   l−2 j ∗ j ′ πijA ∆1Xi∆1(A ) πi0π0j (π00) A ∆lXi∆l (A ) i=1 i=1 l=2 4.2. Conservativity comparison

< Xj The stability condition given in this work is less conservative l (l) (l) for all j ∈ S/{0}, where ∆l = A + KlC with C defined in (11). compared with that in Xiao et al.(2009), since the latter condition J. Wu et al. / Automatica 62 (2015) 32–38 37 implies the former one. To show this, we need the following 5. Conclusion proposition. We have considered the bounded Markovian packet-loss Proposition 3. Define process model and the notion of the peak-covariance stability for

 Io−1  the Kalman filtering. A sufficient stability condition with bounded  l−1  2 s 2 ΦK , d1P + (π00) dlQ diag ∥A∥ ,..., ∥A ∥ , Markovian packet losses was established. Different from that of l=2 Xiao et al.(2009), this stability check can be recast as an LMI feasibility problem. Then we compared the proposed condition where P and Q are defined in (10) and d ∥A(l) + K (l)C(l)∥2, and l , with that of Xiao et al.(2009), showing that our condition prevails K [K (1),..., K (Io−1)] with K (l)’s of compatible dimensions. If there , from at least two aspects: (1) Our stability condition is invariant exists K such that ρ(Φ ) < 1, then ρ(H ) < 1. K K with respect to similarity state transformations, while the previous Proof. If treating a scalar as the Kronecker product of two other result is not; (2) Our condition is proved to be less conservative ′ scalars, similar to Proposition 1, the condition ρ(ΦK ) < 1 is than the previous one. Numerical examples were provided to equivalent to that there exists a vector demonstrate the effectiveness of our result compared with the literature. x , [x1,..., xs], where xj > 0 for all j ∈ S/{0}, such that Acknowledgments

s Io−1 s   l−2 j 2  j 2 The work by J. Wu and K. H. Johansson was partially supported πi0π0j (π00) dl∥A ∥ xi + πijd1∥A ∥ xi < xj. i=1 l=2 i=1 by the Swedish Research Council and the Knut and Alice Wallen- berg Foundation. The work by L. Xie was partially supported by the The submultiplicativity and subadditivity of a matrix norm result National Research Foundation of Singapore under NRF2011NRF- in the following inequality: CRP001-090 and NRF2013EWT-EIRP004-012, and the Natural Sci-  s Io−1 ence Foundation of China under NSFC 61120106011. The work by L.   l−1 j l + (l) (l) · ∗ j ′  πi0π0j (π00) xiA (A K C )( ) (A ) Shi was supported by an HK RGC General Research Fund 16209114. i=1 l=2 s  References +  j + (1) + (1) ∗ j ′ πijxiA (A K C)(A K C) (A )  < xj (17) i=1 Antsaklis, Panos J., & Michel, Anthony N. (2006). Linear systems. Springer. Boyd, Stephen P., El Ghaoui, Laurent, Feron, Eric, & Balakrishnan, Venkataramanan for all j ∈ S/{0}. Let X = x I × . Then we obtain from (17) that j j n n (1994). Linear matrix inequalities in system and control theory. Vol. 15. SIAM. Durrett, Rick (2010). Probability: theory and examples. Vol. 3. Cambridge University s Io−1   l−2 j l (l) (l) ∗ j ′ Press. πi0π0j (π00) A (A + K C )Xj(·) (A ) Elliott, E. O. (1963). Estimates of error rates for codes on burst-noise channels. Bell i=1 l=2 System Technical Journal, 42(5), 1977–1997. s Gilbert, Edgar N. (1960). Capacity of a burst-noise channel. Bell System Technical +  j + (1) + (1) ∗ j ′ Journal, 39(5), 1253–1265. πijA (A K C)Xi(A K C) (A ) < Xj. Huang, M., & Dey, S. (2007). Stability of Kalman filtering with Markovian packet i=1 losses. Automatica, 43, 598–607. Kar, Soummya, Sinopoli, Bruno, & Moura, José M. F. (2012). Kalman filtering with Therefore ρ(HK ) < 1, which completes the proof. intermittent observations: Weak convergence to a stationary distribution. IEEE Transactions on Automatic Control, 57(2), 405–420. In virtue of Proposition 3, it is evident that ρ(Φ) < 1 implies Mo, Yilin, & Sinopoli, Bruno (2010). Towards finding the critical value for Kalman ⋆ ⋆ ⋆ ⋆ ρ(H ⋆ ) < 1, where K [K ,..., K ] with K filtering with intermittent observations. ArXiv Preprint arXiv:1005.2442. K , 1 Io−1 i , ∥ (i) + (i) (i)∥2 Mo, Yilin, & Sinopoli, B. (2012). Kalman filtering with intermittent observations: arg minK(i) A K C . Tail distribution and critical value. IEEE Transactions on Automatic Control, 57(3), 677–689. Example 1 (Cont’d). We continue to consider Example 1 with an Plarre, K., & Bullo, F. (2009). On Kalman filtering for detectable systems with alternative transition probability matrix intermittent observations. IEEE Transactions on Automatic Control, 54(2), 386–390. 0.6 0.2 0.2 Shi, L., Epstein, M., & Murray, R. M. (2010). Kalman filtering over a packet-dropping network: A probabilistic perspective. IEEE Transactions on Automatic Control, 51 = 0.6 0.2 0.2 . 55(9), 594–604. 0.6 0.2 0.2 Sinopoli, B., Schenato, L., Franceschetti, M., Poola, K., Jordan, M., & Sastry, S. (2004). Kalman filtering with intermittent observations. IEEE Transactions on Automatic From Xiao et al.(2009, Theorem 2), we obtain ρ(Φ) = 1.4704 > 1. Control, 49(9), 1453–1464. By solving an LMI feasibility problem using the cvx in Matlab, we Wu, Jing, & Chen, Tongwen (2007). Design of networked control systems with packet dropouts. IEEE Transactions on Automatic Control, 52(7), 1314–1319. see that the condition in Theorem 1 still holds with a group of Wu, Junfeng, Shi, Guodong, Anderson, Brian D. O., & Johansson, Karl Henrik (0000). feasible matrices Kalman filtering over Gilbert–Elliott channels: Stability conditions and the     critical curve. IEEE Transactions on Automatic Control, submitted for publication. 0.1081 0.0243 −0.8079 ArXiv Preprint arXiv:1411.1217. X = X = and K = . 1 2 0.0243 0.1042 −0.5914 Xiao, Nan, Xie, Lihua, & Fu, Minyue (2009). Kalman filtering over unreliable communication networks with bounded Markovian packet dropouts. International If we consider the transition probability matrix, which only Journal of Robust and Nonlinear Control, 19(16), 1770–1786. allows the maximum length of consecutive packet losses to be 1, Xie, Li (2012). Stochastic comparison, boundedness, weak convergence, and ergodicity of a random Riccati equation with Markovian binary switching. SIAM i.e., Journal on Control and Optimization, 50(1), 532–558. Xie, Li, & Xie, Lihua (2007). Peak covariance stability of a random Riccati equation 0.6 0.4 5 = , arising from Kalman filtering with observation losses. Journal of Systems Science 2 0.8 0.2 and Complexity, 20(2), 262–272. Xie, Li, & Xie, Lihua (2008). Stability of a random Riccati equation with Markovian then ρ(Φ) = 0.49 < 1 and the condition in our Theorem 1 binary switching. IEEE Transactions on Automatic Control, 53(7), 1759–1764. holds. When we increase π in 5 from 0.2 to 0.5, one can verify Xiong, Junlin, & Lam, James (2007). Stabilization of linear systems over networks 11 2 with bounded packet loss. Automatica, 43(1), 80–87. numerically that ρ(Φ) > 1 while the condition in Theorem 1 still You, Keyou, Fu, Minyue, & Xie, Lihua (2011). Mean square stability for Kalman holds. filtering with Markovian packet losses. Automatica, 47(12), 2647–2657. 38 J. Wu et al. / Automatica 62 (2015) 32–38

Junfeng Wu received the B.Eng. degree from the De- Technology from 1986 to 1989. His research interests include robust control and partment of Automatic Control, Zhejiang University, estimation, networked control systems, multi-agent networks, and unmanned Hangzhou, China, in 2009 and the Ph.D. degree in Electri- systems. He has served as an Editor-in-Chief of Unmanned Systems, an Editor of cal and Computer Engineering from the Hong Kong Uni- IET Book Series in Control and an Associate Editor of a number of journals including versity of Science and Technology, Hong Kong, in 2013. IEEE Transactions on Automatic Control, Automatica, IEEE Transactions on Control From September to December 2013, he was a Research Systems Technology, and IEEE Transactions on Circuits and Systems-II. He is a Associate in the Department of Electronic and Computer Fellow of IEEE and Fellow of IFAC. Engineering at the Hong Kong University of Science and Technology, Hong Kong. He is currently a Postdoctoral Re- searcher at ACCESS (Autonomic Complex Communication nEtworks, Signals and Systems) Linnaeus Center, School Karl Henrik Johansson is Director of the ACCESS Linnaeus of Electrical Engineering, KTH Royal Institute of Technology, Sweden. His research Centre and Professor at the School of Electrical Engineer- interests include networked control systems, state estimation, and wirelesssensor ing, KTH Royal Institute of Technology, Sweden. He is a networks, multi-agent systems. He received the Guan Zhao-Zhi Best Paper Award Wallenberg Scholar and has held a Senior Researcher Po- at the 34th Chinese Control Conference in 2015. sition with the Swedish Research Council. He also heads the Stockholm Strategic Research Area ICT The Next Gen- eration. He received M.Sc. and Ph.D. degrees in Electrical Ling Shi received the B.S. degree in electrical and elec- Engineering from Lund University. He has held visiting po- tronic engineering from the Hong Kong University of Sci- sitions at UC Berkeley, California Institute of Technology, ence and Technology, Kowloon, Hong Kong, in 2002 and Nanyang Technological University, and Institute of Ad- the Ph.D. degree in control and dynamical systems from vanced Studies Hong Kong University of Science and Tech- the California Institute of Technology, Pasadena, CA, USA, nology. His research interests are in networked control systems, cyber–physical in 2008. He is currently an Associate Professor at the De- systems, and applications in transportation, energy, and automation systems. He partment of Electronic and Computer Engineering, Hong has been a member of the IEEE Control Systems Society Board of Governors and Kong University of Science and Technology. His research the Chair of the IFAC Technical Committee on Networked Systems. He has been on interests include networked control systems, wireless sen- the editorial boards of several journals, including Automatica, IEEE Transactions on sor networks, event-based state estimation and sensor Automatic Control, and IET Control Theory and Applications. He is currently a Se- scheduling, and smart energy systems. He has been serv- nior Editor of IEEE Transactions on Control of Network Systems and Associate Ed- ing as a Subject Editor for International Journal of Robust and Nonlinear Control itor of European Journal of Control. He has been Guest Editor for a special issue of from 2015. He also served as an Associate Editor for a special issue on Secure Control IEEE Transactions on Automatic Control on cyber–physical systems and one of IEEE of Cyber–Physical Systems in the IEEE Transactions on Control of Network Systems Control Systems Magazine on cyber–physical security. He was the General Chair in 2015. of the ACM/IEEE Cyber–Physical Systems Week 2010 in Stockholm and IPC Chair of many conferences. He has served on the Executive Committees of several Euro- Lihua Xie received the B.E. and M.E. degrees in electrical pean research projects in the area of networked embedded systems. He received engineering from Nanjing University of Science and the Best Paper Award of the IEEE International Conference on Mobile Ad hoc and Technology in 1983 and 1986, respectively, and the Ph.D. Sensor Systems in 2009 and the Best Theory Paper Award of the World Congress degree in electrical engineering from the University of on Intelligent Control and Automation in 2014. In 2009 he was awarded Wallen- Newcastle, Australia, in 1992. Since 1992, he has been berg Scholar, as one of the first ten scholars from all sciences, by the Knut and with the School of Electrical and Electronic Engineering, Alice Wallenberg Foundation. He was awarded Future Research Leader from the Nanyang Technological University, Singapore, where he is Swedish Foundation for Strategic Research in 2005. He received the triennial Young currently a Professor. He served as the Head of Division Author Prize from IFAC in 1996 and the Peccei Award from the International Insti- of Control and Instrumentation from July 2011 to June tute of System Analysis, Austria, in 1993. He received Young Researcher Awards 2014. He held teaching appointments in the Department from Scania in 1996 and from Ericsson in 1998 and 1999. He is a Fellow of the of Automatic Control, Nanjing University of Science and IEEE.