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]Ð[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]Ð[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- encoderÐdecoder 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]Ð[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 GilbertÐElliott 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]Ð[27], the problem of networked control Hangzhou, China, July 2015. Recommended by Associate Editor S. over GilbertÐElliott 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
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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]Ð[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 GilbertÐElliott 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 GilbertÐElliott 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 failure rate, making use of the relationship between ability distribution function. Motivated by this, Shi et al. [33] the two stability criteria we established. This lower bound studied Kalman filtering with random packet losses from a prob- holds without relying on the restriction that the system abilistic perspective where the performance metric was defined matrix has no defective eigenvalues on the unit circle. In using the error covariance matrix distribution function, instead other words, we obtain an MSS condition for general lin- of the mean. Mo and Sinopoli [34] studied the decay rate of the ear time-invariant (LTI) systems under Markovian packet estimation error covariance matrix and derived the critical ar- drops. rival probability for nondegenerate systems based on the decay We believe these results add to the fundamental understanding rate. Weak convergence of Kalman filtering with packet losses, of Kalman filtering under random packet drops. i.e., that error covariance matrix converges to a limit distribu- The remainder of this paper is organized as follows. Section II tion, were investigated in [35]Ð[37] for i.i.d., semi-Markov, and presents the problem setup. Section III focuses on the PCS. Markov drop models, respectively. Section IV studies the relationship between the peak-covariance and mean-square stabilities, the critical p − q curve, and presents a sufficient condition for MSS of general LTI systems. B. Contributions and Paper Organization Section V demonstrates the effectiveness of our approach com- In this paper, we focus on the PCS and MSS of Kalman filter- pared with the literature using two numerical examples. Finally, ing with Markovian packet losses. We first derive relaxed and Section VI concludes this paper. N Sn explicit PCS conditions. Then, we establish a result indicating Notations: is the set of positive integers. + is the set of that PCS implies MSS under quite general settings. We even- n by n positive semidefinite matrices over the complex field. tually make use of these results to obtain MSS criteria. The For a matrix X, σ(X) denotes the spectrum of X and λX contributions of this paper are summarized as follows. denotes the eigenvalue of X that has the largest magnitude. 1) A relaxed condition guaranteeing PCS is obtained de- X∗, X, and X are the Hermitian conjugate, transpose, and scribed by an inequality in terms of the spectral ra- complex conjugate of X, respectively. Moreover, · means dius of the system matrix and transition probabilities of the 2-norm of a vector or the induced 2-norm of a matrix. ⊗ is the the Markov chain, rather than an infinite sum of matrix Kronecker product of two matrices. The indicator function of a norms as in [28]Ð[31]. We show that the condition can subset A⊂Ω is a function 1A :Ω→{0, 1}, where 1A(ω)=1 WU et al.: KALMAN FILTERING OVER GILBERT–ELLIOTT CHANNELS: STABILITY CONDITIONS AND CRITICAL CURVE 1005
The estimator computes xˆk|k , the minimum mean-squared error estimate, and xˆk+1|k , the one-step prediction, accord- ing to xˆk|k = E[xk |Fk ] and xˆk+1|k = E[xk+1|Fk ].LetPk|k and Pk+1|k be the corresponding estimation and prediction er- Fig. 1. Estimation over an erasure channel. ror covariance matrices, i.e., Pk|k = E[(xk − xˆk|k )(·) |Fk ] and Pk+1|k = E[(xk+1 − xˆk+1|k )(·) |Fk ]. They can be computed if ω ∈A, otherwise 1A(ω)=0. For random variables, σ(·) is recursively via a modified Kalman filter [16]. The recursions for the σ-algebra generated by the variables. xˆk|k and xˆk+1|k are omitted here. To study the Kalman filtering system’s stability, we focus on the prediction error covariance matrix P | , which is recursively computed as II. KALMAN FILTERING WITH MARKOVIAN PACKET LOSSES k+1 k Consider an LTI system Pk+1|k = APk|k−1 A + Q −1 xk+1 = Axk + wk (1a) −γk APk|k−1 C (CPk|k−1 C + R) CPk|k−1 A .
yk = Cxk + vk (1b) It can be seen that Pk+1|k inherits the randomness of {γt }1≤t≤k . n×n m ×n where A ∈ R is the system matrix and C ∈ R is We focus on characterizing the impact of {γk }k∈N on Pk+1|k . ∈ Rn the observation matrix, xk is the process state vec- To simplify notations in the sequel, let Pk+1 Pk+1|k , and ∈ Rm ∈ Rn k k Sn → Sn tor and yk is the observation vector, wk and define the functions h, g, h , and g : + + as follows m vk ∈ R are zero-mean Gaussian random vectors with auto- E ≥ E covariance [wk wj ]=δkjQ (Q 0), [vk vj ]=δkjR (R> h(X) AXA + Q (3) 0), E[wk vj ]=0∀j, k. Here, δkj is the Kronecker delta func- g(X) AXA + Q − AXC(CXC + R)−1 CXA tion with δkj =1if k = j and δkj =0otherwise. The initial state x is a zero-mean Gaussian random vector that is uncor- k ◦ ◦···◦ k ◦ ◦···◦ 0 h (X) h h h (X) and g (X) g g g (X) related with wk and vk and has covariance Σ0 ≥ 0. We assume ktimes that (C, A) is detectable and (A, Q1/2 ) is stabilizable. By apply- ktimes ing a similarity transformation, the unstable and stable modes (4) of the considered LTI system can be decoupled. An open-loop prediction of the stable mode always has a bounded estimation where ◦ denotes the function composition. The following sta- error covariance, therefore, this mode does not play any key role bility notion is standard. in the stability issues considered in this paper. Without loss of Definition 1: The Kalman filtering system with packet losses E ∞ generality, we assume that (A1) All the eigenvalues of A have is mean-square stable if supk∈N Pk < . magnitudes no less than 1. Certainly A is nonsingular. Define We consider an estimation scheme illustrated in Fig. 1, where τ min{k : k ∈ N,γ =0} the raw measurements of the sensor {yk }k∈N are transmitted to 1 k the estimator via an erasure communication channel over which β1 min{k : k>τ1 ,γk =1} packets may be dropped randomly. Denote by γ ∈{0, 1} the k . arrival of yk at time k: yk arrives errorfree at the estimator if . γk =1; otherwise γk =0. Whether γk takes value 0 or 1 is τ min{k : k>β− ,γ =0} assumed to be known by the receiver at time k. Define Fk as j j 1 k the filtration generated by all the measurements received by the βj min{k : k>τj ,γk =1}. (5) estimator up to time k, i.e., Fk σ (γt yt ,γt ;1≤ t ≤ k) and ∞ F = σ (∪ Fk ). We will use a triple (Ω, F, P) to denote the k=1 It is straightforward to verify that {τj }j∈N and {βj }j∈N are probability space capturing all the randomness in the model. two sequences of stopping times because both {τj ≤ k} and To describe the temporal correlation of realistic communica- {βj ≤ k} are Fk −measurable; see [38] for details. Due to the tion channels, we assume the GilbertÐElliott channel [23], [24], strong Markov property and the ergodic property of the Markov where the packet loss process is a time-homogeneous two-state chain defined by (2), the sequences {τ } ∈N and {β } ∈N have { } j j j j Markov chain. To be precise, γk k∈N is the state of the Markov finite values P-almost surely. Then, we can define the so- chain with initial condition, without loss of generality, γ1 =1. ∗ journ times at the state 1 and state 0, respectively, by τj and The transition probability matrix for the GilbertÐElliott channel β∗ ∀j ∈ N as is given by j ∗ − ∗ − 1 − qq τj τj βj−1 and βj βj τj P = (2) p 1 − p where β0 =1by convention. A result given by [29, Lemma 2] P | { ∗} { ∗} where p (γk+1 =0γk =1)is called the failure rate, and demonstrates that τk j∈N and βk j∈N are mutually indepen- q P(γk+1 =1|γk =0)the recovery rate. Assume that (A2) dent and have geometric distribution. Let us denote the predic- ∈ The failure and recovery rates satisfy p, q (0, 1). tion error covariance matrix at the stopping time βj by Pβj and 1006 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 4, APRIL 2018
Definition 3: The observability index Io is defined as the − smallest integer such that [C ,AC ,...,(AIo 1 ) C ] has rank n.IfIo =1, the system (C, A) is called one-step observable. L Sn → Sn We also introduce the operator K : + + defined as − I o 1 i−1 i (i) (i) ∗ i (i) (i) LK (X)=p (1−p) (A +K C ) ΦX (A + K C ) i=1 (6) where ΦX is the positive definite solution of the Lyapunov equa- 2 tion (1 − q)A ΦX A + qA XA =ΦX with |λA | (1 − q) < 1, − and K =[K(1),...,K(Io 1)] with each matrix K(i) having compatible dimensions. It can be easily shown that LK (X) is linear and nondecreasing in the positive semidefinite cone. We have the following result. 2 Theorem 1: Suppose |λA | (1 − q) < 1. If any of the follow- Fig. 2. Road map of this work. ing conditions hold. − i) ∃ K [K(1),...,K(Io 1)], where K(i)s are matrices with compatible dimensions, such that |λH (K )| < 1, 1 call it the peak covariance at βj . We introduce the notion of where PCS [29] as follows: − − 1 Definition 2: The Kalman filtering system with packet losses H(K)= qp (A ⊗ A) 1 − (1 − q)I is said to be peak-covariance stable if sup ∈N EP < ∞. j βj − I o 1 Remark 1: In [32], the authors defined stability in stopping · (Ai + K(i)C(i)) times as the stability of Pk at packet reception times. Note that i=1 {βj }j∈N , at which the peak covariance is defined, can also be treated as the stopping times defined on packet reception times. ⊗ (Ai + K(i)C(i))(1 − p)i−1 . (7) Clearly, in scalar systems, the covariance is at maximum when − ii) There exists K [K(1),...,K(Io 1)] with each ma- the channel just recovers from failed transmissions; therefore, trix K(i) having compatible dimensions such that peak covariance sequence gives an upper envelop of covariance Lk ∈ Sn limk→∞ K (X)=0for any X + . matrices at packet reception times. For higher order systems, − iii) There exist K [K(1),...,K(Io 1)] with each matrix the relation between them is still unclear. (i) The results of this paper are organized as follows (cf., Fig. 2). K having compatible dimensions and P>0 such that L First of all, we present a relaxed condition for PCS (Theorem 1). K (P )
⎡ √ ⎤ − I −1 (i) X p(A Y + C F ) ··· p(1 − p)Io 2 (A o ) Y +(C ) F − √ 1 Io 1 ⎢ ∗ ··· ⎥ ⎢ p(YA+ F1 C) Y 0 ⎥ Ψ ⎢ ⎥ ⎣ . . .. . ⎦ . . . . − − ∗ p(1 − p)Io 2 YAIo 1 + F C(i) 0 ··· Y Io −1 (9) WU et al.: KALMAN FILTERING OVER GILBERT–ELLIOTT CHANNELS: STABILITY CONDITIONS AND CRITICAL CURVE 1007
dynamic and channel characteristics on the one hand, and PCS In light of Lemma 1, the spectrum of pn (A ⊗ A) is on the other hand. These results cover the ones in [28], [29], given by σ (pn (A ⊗ A)) = {pn (λi λj ): λi, λj ∈ σ(A)}. Since and [31], as is evident using the subadditivity property of matrix A is a real matrix, its complex eigenvalues, if any, al- 2 norm, and the fact that the spectral radius is the infimum of all ways occur in conjugate pairs. Therefore, |λA | must possible matrix norms. To see this, one should notice that be an eigenvalue of A ⊗ A, and the spectral radius of p ⊗ |λ | n |λ |2i − n (A A) can be computed as pn (A⊗A) = i=1 A (1 |λ | − H (k) i−1 Io 1 |λ |2j − j−1 q) q j=1 A (1 p) p. It is evident that the sequence −1 2 −1 {|λp ⊗ |} ∈N is monotonically increasing. When |λ | (1 − ≤ qp (A ⊗ A) − (1 − q)I n (A A) n A q) < 1,wehave − I o 1 i−1 i (i) (i) lim pn (A ⊗ A)=H(0) (11) (1 − p) (Ai + K(i)C(i)) ⊗ (A + K C ) n→∞ i=1 and ∞ I −1 i−1 i i |λ |2 o ≤ qp (1 − q) A ⊗ A q A 2j j−1 lim |λp ⊗ | = |λ | (1 − p) p. n (A A) 2 A i=1 n→∞ 1 −|λ | (1 − q) A j=1 − I o 1 (12) i−1 i (i) (i) i (i) (i) (1 − p) (A + K C ) ⊗ (A + K C ) As |λX | is continuous with respect to X, (11) and (12) alto- |λ |2 − i=1 |λ | q A Io 1 |λ |2j − j−1 gether lead to = 2 A (1 p) p. H (0) 1−|λA | (1−q) j=1 ∞ − I o 1 K(i) =0∀1 ≤ i ≤ I − 1 − − Letting o , the condition provided in − i 1 i 2 − i 1 i (i) (i)2 2 = q (1 q) A p (1 p) A + K C Theorem 1 becomes: (i) |λA | (1 − q) < 1, and (ii) |λH (0)| < 1. i=1 i=1 Since the left side of (10) is positive, it imposes the positivity of 1 −|λ |2 (1 − q), whereby the conclusion follows. in which the first inequality follows from |λH (K )|≤H(K) A and the submultiplicative property of matrix norms, and Although computationally friendly, Proposition 1 only pro- i ⊗ vides a comparably rough criterion. It can be expected that, the last equality holds because, for a matrix X, X (i) i i 2 given the ability of searching for K s on the positive semidef- X = λ i = λ i ∗ i = X . Compar- (X (X )i )⊗(X i (X ∗)i ) X (X ) inite cone, Theorem 1 is less conservative than Proposition 1; ison with the related results in the literature is also demonstrated this is demonstrated by Example I in Section V. by Example I in Section V. Remark 3: The left side of (10) is strictly positive when In addition, in [28], [29], and [31], the criteria for PCS are Io ≥ 2, while it vanishes when Io =1. In the latter case, plus the 2 difficult to check since some constants related to the operator necessity as shown in [30], |λA | (1 − q) < 1 thereby becomes g are hard to explicitly compute. A thorough numerical search a necessary and sufficient condition for PCS. This observation may be computationally demanding. In contrast, the stability is consistent with the conclusion of [31, Corollary 2]. check of Theorem 1 uses an LMI feasibility problem, which can Remark 4: Proposition 1 covers the result of [30, Th. 3.1]. often be efficiently solved. To see this, notice that In the following proposition, we present another condition for I −1 q|λ |2 o PCS, which is, despite being conservative, easy to check. The A |λ |2j (1 − p)j−1 p new condition is obtained by making all K(i)s in Theorem 1 1 −|λ |2 (1 − q) A A j=1 take the value zero. ∞ − Proposition 1: If the following condition is satisfied I o 1 2i i−1 2j j−1 = |λA | (1 − q) q |ΛA | (1 − p) p I −1 o i=1 j=1 pq|λ |2 |λ |2i (1 − p)i−1 < 1 −|λ |2 (1 − q), (10) A A A − ∞ I o 1 i=1 ≤ Ai 2 (1 − q)i−1 q Aj 2 (1 − p)j−1 p, then the Kalman filtering system is peak-covariance stable. i=1 j=1 Proof: The proof requires the following lemma. which implies the less conservativity of [30, Th. 3.1]. Lemma 1 ([39, Th. 1.1.6]): Let p(·) be a given polynomial. If λ is an eigenvalue of a matrix A, then, p(λ) is an eigenvalue IV. MEAN-SQUARE STABILITY of the matrix p(A). Define a sequence of polynomials of the matrix A ⊗ A as In this section, we will discuss MSS of Kalman filtering with {pn (A ⊗ A)}n∈N , where Markovian packet losses. n i i−1 A. From PCS to MSS pn (A ⊗ A)= (A ⊗ A) (1 − q) q i=1 Note that the PCS characterizes the filtering system at − I o 1 stopping times defined by (5), whereas MSS characterizes the × (A ⊗ A)j (1 − p)j−1 p. property of stability at all sampling times. In the literature, j=1 the relationship between the two stability notations is still an 1008 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 4, APRIL 2018 open problem. In this section, we aim to establish a connection between PCS and MSS. First, we need the following definition for the defective eigenvalues of a matrix. Definition 4: For λ ∈ σ(A) where A is a matrix, if the alge- braic multiplicity and the geometric multiplicity of λ are equal, then λ is called a semisimple eigenvalue of A.Ifλ is not semisim- ple, λ is called a defective eigenvalue of A. We are now able to present the following theorem indicating that as long as A has no defective eigenvalues on the unit cir- cle, i.e., the corresponding Jordan block is 1 × 1, PCS always implies MSS. In fact, we are going to prove this connection for general random packet drop processes {γk }k∈N , instead of limiting to the GilbertÐElliott model. Theorem 2: Let {γk }k∈N be a random process over an un- derlying probability space (S , S,μ) with each γk taking its value in {0, 1}. Suppose {βj }j∈N take finite values μ−almost surely, and that A has no defective eigenvalues on the unit circle. Fig. 3. Relationships between the PCS and MSS over the space of Then, the PCS of the Kalman filter always implies MSS, i.e., the filtering systems under consideration. Theorem 2 indicates that the E ∞ E ∞ supk∈N Pk < whenever supj∈N Pβj < . intersection of the two sets, PCS and NoDEUC, is contained in the set MSS. Note that {βj }j∈N can be defined over any random packet loss processes, therefore, the PCS with packet losses that the filtering system is undergoing remains in accord with Definition 2. Theorem 2 bridges the two stability notions of Kalman fil- tering with random packet losses in the literature. Particu- larly, this connection covers most of the existing models for packet losses, e.g., i.i.d. model [16], bounded Markovian [40], GilbertÐElliott [28], and finite-state channel [25], [26]. Although E E supk∈N Pk and supj∈N Pβj are not equal in general, this connection is built upon a critical understanding that, no matter to which interarrival interval between two successive βj s the time k belongs, Pk is uniformly bounded from above by an affine function of the norm of the peak covariances at the starting and ending points thereof. This point holds regardless of the model of packet loss process. The proof of Theorem 2 was given in Appendix C. We also remark that there is difficulty in relaxing the assump- tion that A has no defective eigenvalues on the unit circle in Theorem 2. This is due to the fact that As defective eigenvalues Fig. 4. p − q plane is divided into MSS and non-MSS regions by the on the unit circle will influence both the PCS and MSS in a critical curve fc (p, q)=0. When p + q =1, the Markovian packet loss process is reduced to an i.i.d. process. As a result, the intersection point nontrivial manner (see Fig. 3). of the curves fc (p, q)=0and p + q =1gives the critical packet drop Remark 5: In [29], for a scalar model with i.i.d. packet probability established in [16]. losses, it has been shown that the PCS is equivalent to MSS, whereas for a vector system even with i.i.d. packet losses, the 2 relationship between the two is unclear. In [31], the equivalence Proposition 2: Let the recovery rate q satisfy |λA | (1 − q) between the two stability notions was established for systems < 1. Then, there exists a critical value pc ∈ (0, 1] for the failure that are one-step observable, again for the i.i.d. case. Theorem 2 rate in the sense that E ∞ ≥ now fills the gap for a large class of vector systems under general i) supk∈N Pk < for all Σ0 0 and 0 Theorem 3: There exists a critical curve defined by fc (p, q)=0, which reads two nondecreasing functions p = · −1 · 2 pc (q) and q = qc (p) with qc ( )=pc ( ), dividing (0, 1) into two disjoint regions such that ∈ E ∞ i) If (p, q) fc (p, q) > 0 , then, supk∈N Pk < for all Σ0 ≥ 0; ii) If (p, q) ∈ fc (p, q) < 0 , then, there exists Σ0 ≥ 0 un- E ∞ der which supk∈N Pk = . Remark 6: If the packet loss process is an i.i.d. process, where p + q =1 in the transition probability matrix defined Fig. 5. Sample path of P with p =0.99, q =0.65 in Example I. in (2), Proposition 2 and Theorem 3 recover the result of k [16, Th. 2]. It is worth pointing out that whether MSS holds or not exactly on the curve fc (p, q)=0is beyond the reach of the current analysis (even for the i.i.d. case with p + q =1): such an understanding relies on the compactness of the stability or nonstability regions. C. MSS Conditions We can now make use of the PCS conditions we obtained in the last section, and the connection between PCS and MSS Fig. 6. Sample path of Pk with p =0.45, q =0.5 in Example II. indicated in Theorem 2, to establish MSS conditions for the con- sidered Kalman filter. It turns out that the assumption requiring p, [29] concludes that p<0.04 guarantees PCS; while Proposi- 2 no defective eigenvalues on the unit circle can be relaxed by an 1−|λA | (1−q) tion 1 requires p< 4 , which generates a less conser- |λA | q approximation method. We present the following result. vative condition p<0.22. Similarly, the example in [30] allows |λ |2 − Theorem 4: Let the recovery rate q satisfy A (1 q) < 1. p =0.1191 at most. We also note that it is rather convenient to ≤ Then, there holds p pc , where check the condition in Proposition 1 even with manual calcula- tion; in contrast, to check the conditions in [29] and [30] involves p sup p : ∃(K, P)s.t. L (P ) connection holds for general random packet drop processes. We also proved that there exists a critical region in the p − q plane such that if and only if the pair of recovery and failure rates falls into that region the expected prediction error covariance matrices are uniformly bounded. By fixing the recovery rate, a lower bound for the critical failure rate was obtained making use of the relationship between two stability criteria for general LTI systems. Numerical examples demonstrated significant im- provement on the effectiveness of our approach compared with Fig. 7. Divergence of EP with p =0.99, q =0.5 in Example II. k the existing literature. APPENDIX A AUXILIARY LEMMAS In this section, we collect some lemmas that are used in the proofs of our main results. Lemma 2 (Lemma A.1 in [33]): For any matrices X ≥ Y ≥ 0, the following inequalities hold h(X) ≥ h(Y ) (15) g(X) ≥ g(Y ) (16) h(X) ≥ g(X) (17) where the operators h and g are defined in (3) and (4), Fig. 8. Blue-colored crosses represent (p, q) points where the Kalman filter is mean-square stable; the gray dots represent (p, q) points at respectively. which the Kalman filter is mean-square unstable. Clearly a critical curve Lemma 3: Consider the operator emerges. (i) i (i) (i) ∗ φi(K ,P) (A + K C )X(·) (i) (i) (i) when (p, q)=(0.99, 0.5) the criterion in Theorem 4 is violated. Q Q (D ) ∗ +[A(i) K(i)] [A(i) K(i)] This verifies the contrapositive of Theorem 4. ∗ D(i)(Q(i))(D(i)) + R(i) 2) Critical Curve: We now illustrate the result of (i) i−1 (i) Theorem 3 by Monte Carlo simulations. We quantize the p − q for all i ∈ N, where C =[C ,AC ,...,(A ) C ] , A = i−1 (i) plane with a step size 0.025 along each axis. For each discretized [A ,...,A, I], D =0for i =1otherwise (p, q) N = 100 000 ⎡ ⎤ value, we run Monte Carlo simulations for 00··· 0 repeated rounds, and each sample goes through 70 time steps. ⎢ C 0 ··· 0 ⎥ (i) ⎢ ⎥ Noticing that without packet loss, the norm of the steady esti- D = ⎢ . . . . ⎥ , mation error covariance is around 3.7, we use ⎣ . . .. . ⎦ CAi−2 CAi−3 ··· 0 1 N P = P (ω ) app N k i Q(i) = diag(Q,...,Q), R(i) = diag(R,...,R), and K(i) are i=1 E i i with k =70as the empirical approximation of [Pk ] and set of compatible dimensions. For any X ≥ 0 and K(i),italways a criterion Papp ≤ 100 for MSS. Clearly the p − q plane con- i (i) ≤ (i) holds that g (X) = minK ( i ) φi(K ,X) φi(K ,X). sists of a stable and another unstable regions, separated by an Proof: The result is readily established when setting B = I emerging critical curve (see Fig. 8). in [40, Lemmas 2 and 3]. For i =1, The result is well known as in [16, Lemma 1]. VI. CONCLUSION ∈ Cn×n ∈ N Lemma 4 ([41]): For any A , >0 and k ,it √ k k n−1 We have investigated the stability of Kalman filtering over holds that A ≤ n(1 + 2/) |λA | + A . GilbertÐElliott channels. Random packet drops follow a time- × Lemma 5 (Lemma 2 in [42]): For any G ∈ Cn n , there ex- homogeneous two-state Markov chain where the two states ist G ∈ Sn ,i=1, 2, 3, 4 such that G =(G − G )+(G − indicate successful or failed packet transmissions. We estab- i + √ 1 2 3 G )i, where i = −1. lished a relaxed condition guaranteeing PCS described by an 4 inequality in terms of the spectral radius of the system ma- trix and transition probabilities of the Markov chain, and then APPENDIX B PROOF OF THEOREM 1 showed that the condition can be reduced to an LMI feasibility problem. It was proved that PCS implies MSS if the system The proof is lengthy and is divided into two parts. In the first matrix has no defective eigenvalues on the unit circle. This part, we will show that condition i) is a sufficient condition for WU et al.: KALMAN FILTERING OVER GILBERT–ELLIOTT CHANNELS: STABILITY CONDITIONS AND CRITICAL CURVE 1011 PCS. In the second part, we will show the equivalence between is bounded. The first inequality is from Lemmas 3 and 6. By conditions i)Ðiv). The proof relies on the following two lemmas. substituting (19) into (18), it yields Lemma 6 ( [29, Lemma 5]): Assume that (C, A) is observ- ∞ 1/2 i able and (A, Q ) is controllable. Define P ≤ W + 1{ ∗ }A βj +1 βj +1=i i=1 n S {P :0≤ P ≤ AP A + Q, for some P ≥ 0}. − 0 0 0 I o 1 l (l) (l) ∗ i × 1{ ∗ }(·)P (A + K C ) (A ) (20) τj +1=l βj Then, there exists a constant L > 0 such that l=1 ∈ Sn k ≤ ≥ i) for any X 0 , g (X) LI for all k Io; and ∞ i i ∞ n k+1 where W 1{β ∗ =i}A U(A ) + 1{β ∗ =i}Vi. ii) for any X ∈ S , g (X) ≤ LI for all k ≥ Io i=1 j +1 i=1 j +1 + To facilitate discussion, we force P in (20) to take the where the operator g is defined in (4). βj +1 n×n maximum. For other cases in (20), the subsequent conclusion Lemma 7: For q ∈ (0, 1) and A ∈ R , the series of matri- ∞ − ∞ i−1 { } ⊗ i − i 1 ⊗ j − still holds as it renders an upper envelop of Pβj j∈N by ces i=1(A A) (1 q) q and i=1 j=0(A A) (1 i−1 |λ |2 − imposing (20) to take equality. q) q converge if and only if A (1 q ) < 1. ∞ i−1 ⊗ j − We introduce the vectorization operator. Let X = Proof: First observe that i=1 j=0(A A) (1 ··· ∈ Cm ×n ∈ Cm − ∞ [x1 x2 xn ] where xi . Then, we define q)i 1 q = (A ⊗ A)i (1 − q)i . i=0 The geometric series vec(X) [x ,x ,...,x ] ∈ Cmn. Notice that vec(AXB)= ⊗ − 1 2 n generated by (A A)(1 q) converges if and only if (B ⊗ A)vec(X). For Kronecker product, we have (A A ) ⊗ |λ | − 1 2 A⊗A (1 q) < 1. Therefore, the conclusion follows from the (B B )=(A ⊗ B )(A ⊗ B ). Take expectations and vec- |λ | {|λ λ | λ λ ∈ } |λ |2 1 2 1 1 2 2 fact that A⊗A = max i j : i, j σ(A) = A . torization operators over both sides of (20). From [29, ≥ ∈ − Now, fix j 1. First note that, for any k [τj+1,βj+1 1], Lemma 2], we obtain γk =0. Hence, we have E E [vec(Pβj +1)] = [vec(W )] ∞ ∞ i Pβ = 1{β ∗ =i}h (Pτ ) i i−1 j +1 j +1 j +1 + (A ⊗ A) (1 − q) q i=1 i=1 ∞ ∞ I −1 ∗ i i ∗ o 1{β =i}A Pτj +1(A ) + 1{β =i}Vi (18) − j +1 j +1 · l (l) (l) ⊗ · − l 1 E i=1 i=1 (A + K C ) ( )p(1 p) [vec(Pβj )]. l=1 i−1 l l (21) where Vi l=0 A Q(A ) . Now, let us consider the inter- − ∗ val [βj ,τj+1 1] over which τj+1 packets are successfully re- In the above-mentioned equation E[vec(W )] can be written as ceived. We will analyze the relationship between Pτ and Pβ j +1 j ∞ in two separate cases, which are τ ∗ ≤ I − 1 and τ ∗ ≥ I . j+1 o j+1 o E ⊗ i − i−1 Computation yields the following result [vec(W )] = (A A) (1 q) q vec(U) i=1 ∞ − I −1 ∞ i 1 o − i l ⊗ l − i 1 P = 1{ ∗ }g (P )+ 1{ ∗ }g (P ) + (A A) (1 q) q vec(Q). (22) τj +1 τj +1=i βj τj +1=l βj i=1 l=0 i=1 l=Io − In Lemma 7, we show that both of the two terms in (22) converge I o 1 ∞ |λ |2 − ≤ ∗ (i) ∗ if A (1 q) < 1. 1{τ =i}φi(K ,Pβj )+LI 1{τ =l} j +1 j +1 For j =1, following the similar argument as abovemen- i=1 l=Io tioned, we have − I o 1 i (i) (i) i (i) (i) ∗ ∞ = 1{ ∗ }(A + K C )P (A + K C ) − τj +1=i βj i 2 i 1 EPβ ≤ EPτ A (1 − q) q i=1 1 1 i=1 ∞ − I o 1 ∞ i (i) i (i) ∗ + LI 1{τ ∗ =l} + 1{τ ∗ =i}[A K ]Ji[A K ] i−1 j +1 j +1 + (1 − q) qVi j=I i=1 o i=1 i (i) (i) i (i) (i) ∗ (A + K C )Pβj (A + K C ) + U (19) where Vis are defined in (18). Moreover, by Lemma 6 and (17), it holds that (i) (i) (i) Q Q (D ) Io where Ji (i) (i) (i) (i) (i) (i) and E ≤ i − i−1 D (Q ) D (Q )(D ) + R Pτ1 LI + g (Σ0 ) (1 p) p i=1 ∞ I −1 o Io i (i) i (i) ∗ − U LI 1{ ∗ } + 1{ ∗ }[A K ]J [A K ] ≤ i 2 − i 1 τj +1=l τj +1=i i LI + Σ0 A (1 p) p + VIo j=Io i=1 i=1 1012 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 4, APRIL 2018 E E (1) (Io−1) showing that Pτ1 is bounded. To sum up, Pβ1 is iii) ⇒ ii). If there exist K =[K ,...,K ] with each 2 (i) bounded if |λA | (1 − q) < 1. By applying the CauchyÐSchwarz matrix K having compatible dimensions and P>0 such inequality to the inner product of random variables, the bound- that LK (P ) k ∈ [τj+1,βj+1 − 1], the relation is given by ·μ(βj ≤ k ≤ βj+1) β −k−1 k−1 j +1 βj +1−j βj +1−k i i ≤ E E Pβj +1 = A Pk (A ) + A Q(A ) d1 Pβj + d1 Pβj +1 + d0 i=0 j=0 β −k β −k from which we obtain Pβ ≥ A j +1 Pk (A ) j +1 . Then, it ·μ(βj ≤ k ≤ βj+1) j +1 yields k−1 βj +1−k βj +1−k ≤ E ≤ ≤ Pβ ≥ A Pk (A ) 2d1 sup Pβj + d0 μ(βj k βj+1) j +1 j≤k j=0 1 βj +1−k βj +1−k ≥ Tr(A Pk (A ) ) ≤ 2d1 α + d0 n 1 − − 1/2 βj +1 k βj +1 k 1/2 which completes the proof. = Tr(Pk (A ) A Pk ) Before proceeding to the proof of the theorem, let us provide n 1 some properties related to the discrete-time algebraic Riccati k−βj +1 −2 ≥ A Tr(Pj ) equation (DARE). The proof, provided in [43], is omitted. n Lemma 9: Consider the following DARE 1 k−βj +1 −2 ≥ A Pk − n P = AP A + Q − AP C (CPC + R) 1 CPA . (29) where the second and the last inequality allows from the fact that If (A, Q1/2 ) is controllable and (C, A) is observable, then, it λ ≥ 1 ≥ ∈ Sn X = X n Tr(X) and Tr(X) X for any X + ;the has a unique positive definite solution P˜ and A + KC˜ is stable, third one holds since ˜ ˜ ˜ −1 where K = −APC (CPC + R) . − − − (A )βj +1 k Aβj +1 k ≥ min σ Aβj +1 k (·) I Fix j ≥ 0. First of all, we shall show that, for k ∈ [βj + 1,τj+1], Pk is uniformly bounded by an affine function of 1 ˜ = I P . By Lemmas 3 and 9, we have g(P − ) ≤ φ (K,P − ) λ k −β βj k 1 1 k 1 A j +1(·) and that A + KC˜ is stable. In light of (16) in Lemma 2, we fur- k−β −2 i ≤ i ˜ ∈ N = A j +1 I. ther obtain g (Pk−1 ) φ1 (K,Pk−1 ) for all i . Therefore, an upper bound of P is given as follows: k If A has no eigenvalues on the unit circle, by√ Lemma 4, there − β −k − − k−β k βj +1≤ j +1 n 1 k βj ≤ j ˜ holds A n1 α1 where n1 n(1 + 2/1 ) Pk = g (Pβj ) φ1 (K,Pβj ) −1 and α1 |λA −1 | + 1 A with a positive number 1 so that − ∗ − k βj k βj α < 1 (such an must exist since |λ −1 | < 1 by assump- ≤ (A + KC˜ ) Pβ (A + C K˜ ) 1 1 A j − β −k k βj +1≤ j +1 tion (A1)). As α1 < 1, A n1 α1 for all i, j = {0, 1} and k ∈ N. It can be seen that the Markov chain in Fig. 9 is ergodic and has a unique stationary distribution p2 p1 p2 π∞ ((0, 0)) = ,π∞ ((0, 1)) = − p2 + q p1 + q p2 + q q π∞ ((1, 1)) = . (32) p1 + q We assume that the Markov chain starts at the stationary distri- bution. Then, the distribution of (zk , z˜k ) for k ≥ 2 is the same as (z1 , z˜1 ), which gives E∞ ◦···◦ P p1 Pk = ψγk (p1 ) ψγ1 (p1 )(Σ0 ) d p1 Ω = ϕ1 ({zj , z˜j }j=1:k ) dπ G ≥ { } Fig. 9. Transitions of the Markov chain {(zk , z˜k )}k ∈N when p2 + q ≤ 1. ϕ2 ( zj , z˜j j=1:k ) dπ G APPENDIX D ◦···◦ P = ψγk (p2 ) ψγ1 (p2 )(Σ0 ) d p2 PROOF OF PROPOSITION 2 Ω = E∞ P If p =0, we have the standard Kalman filter, which evidently p2 k converges to a bounded estimation error covariance. On the other where E∞ means that the expectations is taken conditioned on hand, if p =1, then the Kalman filter reduces to an open-loop the stationarily distributed γ1 . predictor after time step k =2, which suggests that there exists When p2 + q>1, we abuse the definition of probability mea- a transition point for p beyond which the expected prediction sure and allow the existence of negative probabilities in the error covariance matrices are not uniformly bounded. It remains Markov chain described in Fig. 9, generating {(zk , z˜k )}k∈N .It to show that with a given q this transition point is unique. Fix can be easily shown by direct computation that the eigenvalues ≤ E ∞∀ ≥ 3×3 0 p1 < 1 such that supk∈N p1 Pk < Σ0 0.Itsuf- of transition probability matrix, denoted by M ∈ R ,ofthis E ∞ fices to show that, for any p2 APPENDIX F Taking limitation on both sides, we obtain that X˜ − X ≤ (η − ∞ − i i ˜ i PROOF OF THEOREM 4 1) i=0(1 q) A X(A ) , which by Lemma 4 and (34) gives ∞ We shall first show that cd X˜ − X ≤ (η − 1)Q˜ (1 − q)i Ai 2 I d − 1 lim p(η)=p (33) i=0 η→1+ c2 d3 ≤ (η − 1)Q˜ I 2 − − 2 2 (d 1)(d 1) holds where η>1 is properly taken so that η |λA | (1 − q) < 1, 2 3 and c d − ≤ (η − 1)Q˜Q˜ 1 X 2 − − (d 1)(d 1) p(η) sup p : ∃(K, P) s.t. L(ηA,K,P,p) 0 L(A, K, P, p) ˜ 0 are the solutions to the fol- and satisfying ; otherwise it contra- >0 (1 + )L(A, K, P, p) 1 satisfying Φ ≤ 1+Φ where Φ and Φ are the positive X =(1− q)AXA + Q,˜ η−1 X˜ =(1− q)AXA˜ + Q˜ η0 η0 definite solutions to the following equations, respectively: where Q>˜ 0, 0 p− ε.As ˜ i − i i ˜ i ≤˜ i − i i 2 X = η (1 q) A Q(A ) Q η (1 q) A I ε is any positive real number, limη→1+ p(η)=p consequently i=0 i=0 holds. Since ηA has no defective eigenvalues on the unit circle, ∞ combining Theorems 1 and 2, we obtain that pc (ηA,C,q) ≥ 2n−2 i i 2i ≤ nQ˜(1 + 2/ε) d (1 − q) (|λA | + εA) I p(η) holds. i=0 To conclude, we also need to establish the following lemma. 2 Lemma 12: For a given recovery rate q satisfying |λA | (1 − − |λ | (d 1) A ˜ ≤ cd ˜ q) < 1, denote q (A, C, q) as the critical value of q for a system for any ε>0. Taking ε = A ,ityieldsthatX d−1 Q c c with c = n(1 + 2/ε)2n−2 . Note that X˜ − X is bounded in the (C, A). Then, we have following way pc (A, C, q) ≥ lim pc (ηA,C,q). (35) η→1+ X˜ − X =(1− q)A(X˜ − X)A Proof: To emphasize the relevance of h(X) and g(X) to A, we will change the notations h(X) and g(X) into h(A, X) ˜ ˜ +(η − 1) (1 − q)AXA + Q and g(A, X), respectively, in the proof. Note that h(ηA,X) 1016 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 63, NO. 4, APRIL 2018 and g(ηA,X) are both nondecreasing function of η>1, where [14] A. C. Harvey, Forecasting, Structural Time Series Models and the Kalman 2 2 Filter. Cambridge, U.K.: Cambridge Univ. Press, 1990. η is properly chosen so that η |λA | (1 − q) < 1, since, for all ≤ ≤ ≤ ≥ [15] S. Haykin, Kalman Filtering and Neural Networks, vol. 47. New York, 1 η1 η2 1+δ and X 0, one has NY, USA: Wiley, 2004. − 2 − 2 ≥ [16] B. Sinopoli, L. Schenato, M. Franceschetti, K. Poola, M. Jordan, and S. h(η2 A, X) h(η1 A, X)=(η2 η1 )AXA 0 Sastry, “Kalman filtering with intermittent observations,” IEEE Trans. Automat. 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Press, 1995. ney, NSW, Australia, and the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, USA, in 1966. He is an Emeritus Professor in the Australian National University, Canberra, ACT, Australia (having retired as a Distinguished Professor in Junfeng Wu received the B.Eng. degree from 2016), a Distinguished Professor in Hangzhou the Department of Automatic Control, Zhejiang Dianzi University, and a Distinguished Researcher in the National In- University, Hangzhou, China, and the Ph.D. de- formation and Communications Technology Australia. He is a Past gree in electrical and computer engineering from President of the International Federation of Automatic Control and the Hong Kong University of Science and Technol- Australian Academy of Science. His research interests include dis- ogy, Hong Kong, in 2009, and 2013, respectively. tributed control, sensor networks, and econometric modeling. From September to December 2013, he was a Dr. Anderson is a Fellow of the Australian Academy of Science, the Research Associate in the Department of Elec- Australian Academy of Technological Sciences and Engineering, the tronic and Computer Engineering, Hong Kong Royal Society, and a Foreign Member of the U.S. National Academy of University of Science and Technology. From Jan- Engineering. He received the IEEE Control Systems Award of 1997, the uary 2014 to June 2017, he was a Postdoctoral 2001 IEEE James H. Mulligan, Jr., Education Medal, and the Bode Prize Researcher in the ACCESS (Autonomic Complex Communication nEt- of the IEEE Control System Society in 1992, as well as several IEEE and works, Signals and Systems) Linnaeus Center, School of Electrical En- other best paper prizes. He holds honorary doctorates from a number gineering, KTH Royal Institute of Technology, Stockholm, Sweden. He of universities, including Universit Catholique de Louvain, Belgium, and is currently in the College of Control Science and Engineering, Zhe- ETH, Zrich. jiang University, Hangzhou, China. His research interests include net- worked control systems, state estimation, and wireless sensor networks, multiagent systems. Karl Henrik Johansson (F’13) received the Dr. Wu received the Guan Zhao-Zhi Best Paper Award at the 34th M.Sc. and Ph.D. degrees in electrical engineer- Chinese Control Conference in 2015. He was selected to the national ing from Lund University, Lund, Sweden. “1000-Youth Talent Program” of China in 2016. He is the Director of the Stockholm Strate- gic Research Area, Information and Communi- cations Technology, The Next Generation, and a Professor in the School of Electrical En- gineering, KTH Royal Institute of Technology, Stockholm, Sweden. He has held visiting posi- Guodong Shi received the B.Sc. degree in tions at the University of California, Berkeley, the mathematics and applied mathematics from the California Institute of Technology, the NTU, the School of Mathematics, Shandong University, Institute of Advanced Studies, Hong Kong University of Science and Ji’nan, China, and the Ph.D. degree in sys- Technology, and the Norwegian University of Science and Technology. tems theory from the Academy of Mathemat- He is also an IEEE Distinguished Lecturer. His research interests include ics and Systems Science, Chinese Academy networked control systems, cyber-physical systems, and applications in of Sciences, Beijing, China, in 2005 and 2010, transportation, energy, and automation. respectively. Dr. Johansson is a member of the IEEE Control Systems Society From 2010 to 2014, he was a Postdoctoral Board of Governors, the International Federation of Automatic Con- Researcher in the ACCESS Linnaeus Centre, trol (IFAC) executive board, the European Control Association Council, KTH Royal Institute of Technology, Stockholm, and the Royal Swedish Academy of Engineering Sciences. He received Sweden. Since May 2014, he has been with the Research School of En- several best paper awards and other distinctions, including a ten-year gineering, The Australian National University, Canberra, ACT, Australia, Wallenberg Scholar Grant, a Senior Researcher Position with the where he is now a Senior Lecturer and Future Engineering Research Swedish Research Council, the Future Research Leader Award from Leadership Fellow. His research interests include distributed control sys- the Swedish Foundation for Strategic Research, and the triennial Young tems, quantum networking and decisions, and social opinion dynamics. Author Prize from the IFAC.