Distributed filtering for uncertain systems under switching sensor networks and quantized communications

Automatica 114 (2020) 108842

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Distributed filtering for uncertain systems under switching sensor networks and quantized communications✩ ∗

Xingkang He a,c , Wenchao Xue a,b , , Xiaocheng Zhang a,b , Haitao Fang a,b a

LSC, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China c Division of Decision and Control Systems, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden b

article

info

Article history: Received 7 November 2018 Received in revised form 29 September 2019 Accepted 6 January 2020 Available online xxxx Keywords: Sensor network Uncertain system Distributed Kalman filtering Biased observation Quantized communications

a b s t r a c t This paper considers the distributed filtering problem for a class of stochastic uncertain systems under quantized data flowing over switching sensor networks. Employing the biased noisy observations of the local sensor and interval-quantized messages from neighboring sensors successively, an extended state based distributed Kalman filter (DKF) is proposed for simultaneously estimating both system state and uncertain dynamics. To alleviate the effect of observation biases, an event-triggered update based DKF is presented with a tighter mean square error (MSE) bound than that of the time-driven one by designing a proper threshold. Both the two DKFs are shown to provide the upper bounds of MSE online for each sensor. Under mild conditions on systems and networks, the MSE boundedness and asymptotic unbiasedness for the proposed two DKFs are proved. Finally, numerical simulations demonstrate the effectiveness of the developed filters. © 2020 Elsevier Ltd. All rights reserved.

1. Introduction In recent years, sensor networks are broadly studied and applied to environment sensing, target tracking, smart grid, etc. As is well known, state estimation problems over sensor networks are usually modeled as distributed filtering studies. Thus more and more researchers and engineers around the world are paying their attention to the methods and theories of distributed filtering. In the existing literature on distributed filtering over sensor networks, many effective approaches and analysis tools have been provided. For linear time-invariant systems, distributed filters with constant parameter gains were investigated in Khan and Jadbabaie (2014) and Khan, Kar, Jadbabaie, and Moura (2010), which yet confined the scope of the system dynamics to be considered. As is known, the optimal centralized Kalman filter for linear stochastic systems can provide the estimation error covariances online. However, in distributed Kalman filters ✩ This work was supported in part by the National Natural Science Foundation of China under Grants 61633003-3, 11931018 and 61973299. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Erik Weyer under the direction of Editor Torsten Söderström. ∗ Corresponding author at: LSC, NCMIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China. E-mail addresses: [email protected] (X. He), [email protected] (W. Xue), [email protected] (X. Zhang), [email protected] (H. Fang). https://doi.org/10.1016/j.automatica.2020.108842 0005-1098/© 2020 Elsevier Ltd. All rights reserved.

(DKFs) (Battistelli & Chisci, 2014, 2016; Cattivelli & Sayed, 2010; He, Xue, & Fang, 2018; Liu, Tian, & Zhang, 2018; Olfati-Saber, 2007; Wang & Ren, 2018), the covariances cannot be obtained by each sensor due to the unknown correlation between state estimates of sensors, which essentially hinders the optimality of distributed filters. Then the distributed Kalman filters (DKFs) based on consensus or diffusion strategies were studied in Cattivelli and Sayed (2010) and Olfati-Saber (2007), where the state estimates were fused by scalar weights. As a result, the neighboring information cannot be well utilized. Owing to the important role of filtering consistency1 in real-time precision evaluation, the consistent algorithms in Battistelli and Chisci (2014), He et al. (2018), Liu, Wang, He, and Zhou (2017) and Wang and Ren (2018), enabled their applications to design covariance weight based fusion schemes, though the system models were limited to be linear and the communications were required to be perfect. In this paper, we will propose consistent filters for general system models, communication schemes and network topologies. Stability is one of fundamental properties for filtering algorithms. In the existing results on stability analysis, local observability conditions of linear systems were assumed in Cattivelli and Sayed (2010) and Yang, Chen, Wang, and Shi (2014), which confined the application scope of distributed filters. On the other hand, the sensor observation bias, which prevalently exists owing 1 The filtering consistency means that an upper bound of mean square estimation error can be calculated online.

2

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to factors like calibration error, sensor drift, and registration error, can directly influence the consistency as well as the stability of filters. This is attributed to the difficulty in dealing with biased observations of the local sensor and fusing the biased estimates from neighboring sensors. The state estimation problems in the presence of observation bias were investigated in Caglayan and Lancraft (1983) and Keller and Darouach (1997) by assuming the independence between the system state and the random bias, which yet is hard to be satisfied for feedback control systems with colored random bias processes. More importantly, owing to the existence of outer disturbances or unmodeled dynamics, many practical systems contain uncertain dynamics, which may be nonlinear. To deal with the unknown dynamics, some robust estimation methods, such as H∞ filters and set valued filters, were studied by researchers (Ding, Wang, Dong, & Shu, 2012; Yang & Che, 2008). An extended state based Kalman filter was proposed in Bai, Xue, Huang, and Fang (2018) for a class of nonlinear uncertain systems. However, the relation between the original system and the formulated system still needs further investigation. Compared with the centralized filter (Bai et al., 2018), more general system models and noise conditions will be studied in this work under a distributed framework. Communication scheme between sensors is one of the essential features for decentralized algorithms. In the past years, a considerable number of results have analyzed topology conditions in terms of network connectivity and graph types (Battistelli & Chisci, 2014, 2016; Cattivelli & Sayed, 2010; He et al., 2018; Liu et al., 2018; Olfati-Saber, 2007; Wang & Ren, 2018; Yang et al., 2014). Most of these results assumed that the network is fixed over time. However, due to the network vulnerability (e.g., link failure Liu et al., 2017), the topologies of sensor networks may be changing with time. Another significant aspect is on the message transmission between neighboring sensors. A majority of the existing literature on distributed filters required the accurate transmission. Nevertheless, due to limitations of energy and channels in practical networks, such as wireless sensor network, it is difficult to ensure perfect sensor communications. Thus, the filter design under quantized sensor communications seems to be an important issue of practice. The main contributions of this paper are summarized in the following. (1) By utilizing the techniques of interval quantization and state extension, we propose a quantized communication based DKF for a class of stochastic systems suffering uncertain dynamics and observation biases. The filter enables the upper bounds of mean square estimation errors to be available online. (2) Under some mild conditions including uniformly collective observability of the system and jointly strong connectivity of the switching networks, we prove that the mean square estimation errors are uniformly upper bounded. Furthermore, it is shown that the estimation biases tend to zero under certain decaying conditions of uncertain dynamics and observation biases. (3) An event-triggered observation update based DKF is presented with a tighter mean square error (MSE) bound than that of the time-driven one. Also, the mean square boundedness of the estimation error for the event-triggered filter is proved. More importantly, we reveal that the estimation biases of the event-triggered filter can tend to zero even if the observation biases of some sensors are not decaying over time. Compared with the existing literature (Battistelli & Chisci, 2014, 2016; Cattivelli & Sayed, 2010; He et al., 2018; Khan & Jadbabaie, 2014; Khan et al., 2010; Liu et al., 2018; Olfati-Saber,

2007; Wang & Ren, 2018), the studied systems are more general by considering uncertain dynamics and observation biases. Furthermore, although there are some results on quantized distributed consensus (Kar & Moura, 2010; Zhu & Martínez, 2011), the distributed filtering problems with quantized sensor communications have not been well investigated in the existing literature, especially for the scenario that the system is unstable and collectively observable. Moreover, the conditions of noise in Wang and Ren (2018), the initial estimation error in Battistelli and Chisci (2016) and Reif, Gunther, Yaz, and Unbehauen (1999), and the non-singularity of time-varying system matrices in Battistelli and Chisci (2016) and Wang and Ren (2018) are all relaxed in this paper. The remainder of the paper is organized as follows. Section 2 is on the model description and preliminaries. Section 3 studies the system reconstruction and filtering structure. Section 4 analyzes the distributed filter with time-driven update scheme. Section 5 studies the distributed filter with event-triggered update scheme. Section 6 shows the numerical simulations. The conclusion of this paper is given in Section 7. Some proofs are given in Appendix. Notations. The superscript ‘‘T’’ represents the transpose. In stands for the identity matrix with n rows and n columns. E {x} denotes the mathematical expectation of the stochastic variable x, and blockdiag{·} means that the block elements are arranged in diagonals. diag{·} represents the diagonalization of scalar elements. tr(P) is the trace of the matrix P. N+ denotes the set of positive natural numbers. Rn stands for the set of n-dimensional real vectors. [a : b) stands for the set of integers a, a + 1, . . . , b − 1. We denote [a : b] = [a : b) ∪ b. And, sat(f , b) means max{min{f , b}, −b}. We assume that λmin (A) and λmax (A) are the minimal eigenvalue and maximal eigenvalue of a real-valued square A, respec( matrix ) tively. supk∈N Ak ATk < ∞ means supk∈N λmax Ak ATk < ∞. Let ∥·∥2 be the standard Euclidean norm. 2. Model description and preliminaries 2.1. Network topology and definitions We model the communication topologies of sensor networks by switching weighted digraphs {Gs = (V , Es , As )}, where V , Es and As stand for the node set, the edge set and the weighted G adjacency matrix, respectively. We assume that As = [ai,js ]i,j∈V is row stochastic with nonnegative off-diagonal ∑ elements and G G G positive diagonal elements, i.e., ai,is > 0, ai,js ≥ 0, j∈V ai,js = 1. As node i can receive information from its neighboring sensors, G G the neighbor set of node i is denoted by Ni s ≜ {j ∈ V |ai,js ⟩0}, G

G

which includes node i. We denote ai,js = ai,j (k) and Ni s = Ni (k), if the graph is Gs at time k. Gs is called strongly connected if for any pair nodes (i1 , il ), there exists a direct path from i1 to il consisting of edges (i1 , i2 ), (i2 , i3 ), . . . , (il−1 , il ). We⋃say that {Gs }Ks=1 is jointly strongly connected if the union graph Ks=1 Gs is strongly connected, where K ∈ N+ . The following definitions are needed in this paper. Definition 2.1 (Julier & Uhlmann, 1997). Suppose that xk is a random vector and xˆ k is the estimate of xk . Then the pair (xˆ k , Πk ) is said to be consistent if E {(xˆ k − xk )(xˆ k − xk )T } ≤ Πk . An algorithm is of consistency if it provides consistent pairs (xˆ k , Πk ) for all k ∈ N. Definition 2.2. Let ek,i be the state estimation error of sensor i at time k, then the sequence E {ek,i eTk,i }, k ∈ N, is said to be stable if supk∈N E {ek,i eTk,i } < ∞. And, the sequence of estimates is said to be asymptotically unbiased if limk→∞ E {ek,i } = 0.

X. He, W. Xue, X. Zhang et al. / Automatica 114 (2020) 108842

Let (Ω , F , P) be the basic probability space. Fk stands for a filtration of σ -algebra F , i.e., for Fk ⊂ F , Fi ⊂ Fj if i < j. Here, the σ -algebra is a collection of subsets of Ω and satisfies certain algebraic structure. A discrete-time sequence {ξk , Fk } is said to be adapted if ξk is measurable to Fk . In principle, ξk is a function of past events within Fk . We refer the readers to the formal definitions of ‘filtration’, ‘σ -algebra’ and ‘measurable’ in Chow and Teicher (2012). Definition 2.3. A discrete-time adapted sequence {ξk , Fk } is called a martingale difference sequence (MDS), if E {∥ξk ∥2 } < ∞ and E {ξk |Fk−1 } = 0, almost surely. Definition 2.4. A time sequence {Tl , l ∈ N} is called an L-step supporting sequence (L-SS) of a matrix sequence {Mk , k ∈ N}, if there exists a scalar β > 0, such that the sequence {Tl , l ∈ N} is well defined in the following manner

⏐ ⎧ { } ⏐ ⎪ ⏐λmin (Mk+s M T ) ≥ β, ∀s ∈ [0 : L) ⎪ T = inf k ≥ 0 ⎪ 0 k+s ⏐ ⎪ ⎪ ⏐ ⎪ { ⎪ ⏐ ⎪ ⎪ ⎨Tl+1 = inf k ≥ Tl + L⏐ ⏐ } ⎪ ⎪ T ⎪ ) ≥ β, ∀ s ∈ [ 0 : L) λ (M M ⎪ min k+s k+s ⎪ ⎪ ⎪ ⎪ ⎪ ⎩sup{Tl+1 − Tl } < ∞.

Remark 2.1. The definition of L-SS is introduced to study the nonsingularity of the time-varying transition matrices {A¯ k,i } given in the sequel. In many existing results (Battistelli & Chisci, 2016; Wang & Ren, 2018), A¯ k,i is assumed to be nonsingular for any k ∈ N, which is removed in this paper. 2.2. Model description and assumptions Consider the following model for a class of stochastic systems with uncertain dynamics and biased observations

{

¯ k,i xk + bk,i + vk,i , yk,i = H

i ∈ V,

(2)

where xk ∈ Rn is the unknown system state, A¯ k ∈ Rn×n is the known state transition matrix and ω ¯ k ∈ Rn is the unknown zero-mean white process noise. f (xk , k) ∈ Rp is the uncertain ¯ k ∈ Rn×p is the dynamics (e.g., some unknown disturbance). G T ¯ ¯ known matrix subject to supk∈N {Gk Gk } < ∞. Here, yk,i ∈ Rmi ¯ k,i ∈ Rmi ×n is the is the observation vector obtained via sensor i, H ¯ T H¯ k,i } < ∞, bk,i ∈ known observation matrix, subject to supk∈N {H k,i Rmi is the unknown state-correlated stochastic observation bias of sensor i, and vk,i ∈ Rmi is the stochastic zero-mean observation noise. N is the number of sensors over the system, thus V = {1, 2, . . . , N }. Note that A¯ k , G¯ k are known to all the local sensors, ¯ k,i and yk,i are only known to the local sensor i. Suppose while H that σ {x} stands for the minimal sigma algebra generated by the random vector x. Let Fk ≜ σ {x0 , b0,i , ω ¯ j , vj,i , i ∈ V , 0 ≤ j ≤ k} and fk ≜ f (xk , k) for simplicity. In the following, we will provide several assumptions on the system structure and communication scheme. Assumption 2.1.

T • (3) The sequence {vk,i , Fk }∞ k=0 is MDS and E {vk,i vk,i } ≤ Rk,i as well as supk∈N Rk,i > 0 holds ∀i ∈ V . • (4) It holds that E {(X0 − Xˆ 0,i )(X0 − Xˆ 0,i )T } ≤ P0,i , where Xˆ 0,i is the estimate of X0 ≜ [xT0 , f0T ]T , i ∈ V .

Compared with (Wang & Ren, 2018) where the observation noises of sensors are independent, the MDS assumption on vk,i is milder. The condition (4) of Assumption 2.1 is more general than that in Battistelli and Chisci (2016) and Reif et al. (1999) which the initial estimation error is required to be sufficiently small. Assumption 2.2. There is an L ∈ N+ , such that the sequence {A¯ k , k ∈ N} has an L-SS and supk∈N {A¯ k A¯ Tk } < ∞. Assumption 2.2 poses no requirement on the stability of the original system (2). Besides, within the scope of distributed filtering for time-varying systems, Assumption 2.2 is milder than that in Battistelli and Chisci (2016) and Wang and Ren (2018), where the non-singularity of the system state transition matrix is needed for each time. Assumption 2.3. The following conditions hold.

(1)

l∈N

¯ k f (xk , k) + ω¯ k , xk+1 = A¯ k xk + G

3

The following conditions hold:

∞ • (1) The sequence {ω¯ k }∞ k=0 is independent of x0 and {vk,i }k=0 , i ∈ V , with E {ω ¯ k ω¯ kT } ≤ Qk , where infk Qk > 0 and sup Qk < ∞ , ∀k ∈ N . • (2) The sequences {bk,i }∞ k=0 , i ∈ V , are measurable to Fk−1 , k ≥ 1, and E {bk,i bTk,i } ≤ Bk,i .

• (1) fk is measurable to F¯k , where F¯k = σ {x0 , b0,i , ω¯ j−1 , vj,i , i ∈ V , 1 ≤ j ≤ k} for k ≥ 1, and F¯0 = σ {x0 , b0,i , v0,i , i ∈ V }. • (2) E {uk uTk } ≤ Qˆ k , where uk ≜ fk+1 − fk , subject to infk∈N λmin (Qˆ k ) > 0 and supk∈N λmax (Qˆ k ) < ∞. The first condition of Assumption 2.3 permits fk to be implicitly related with {xj , yj,i }kj=0 , i ∈ V . Under this setting, the model built in (2) also considers the distributed output feedback control systems, such as the system of coupled tanks (Orihuela, Millán, Vivas, & Rubio, 2016). The condition in (2) of Assumption 2.3 relies on the boundedness of the dynamic increment, which is milder than the boundedness of uncertain dynamics required by Cai, Queiroz, and Dawson (2006). Definition 2.5. The system (2) is said to be uniformly collective ¯ steps, M ¯ ∈ N, if there exists a positive observable after M ¯ integer N¯ and[a constant α > 0 such that for ] any k ≥ M, there is

∑N

i=1

∑k+N¯

¯ k,k = In , Φ ¯ k+1,k Φ

¯ j,k ≥ α In , where ¯ jT,k H¯ jT,i (Rj,i + Bj,i )−1 H¯ j,i Φ Φ ¯ ¯ j,k = Φ ¯ j,j−1 × · · · × Φ ¯ k+1,k , if j > k. = Ak , Φ j=k

¯ ∈ N such that the system Assumption 2.4. There is an integer M ¯ steps. (2) is uniformly collectively observable after M ¯ = 0, Assumption 2.4 degenerates to the uniformly If M collectively observability (Jazwinski, 1970). Furthermore, if the system is time-invariant, then Assumption 2.4 degenerates to ¯ being observable (Battistelli & Chisci, 2014), where H¯ = (A¯ , H) T ¯ [H1 , . . . , H¯ NT ]T . Besides, if the local observability conditions are satisfied (Cattivelli & Sayed, 2010; Yang et al., 2014), then Assumption 2.4 holds, but not vice versa. The topologies of the networks are assumed to be switching digraphs {Gσk , k ∈ N}. σk is the switching signal defined by σk : N → Ω , where Ω is the set of the underlying network topology numbers. For convenience, the weighted adjacency matrix of the digraph Gσk is denoted by Aσk = [ai,j (k)] ∈ RN ×N . To analyze the switching topologies, we consider the infinite interval sequence consisting of non-overlapping and contiguous time intervals [kl , kl+1 ), l = 0, 1, . . . , with k0 = 0. It is assumed that there exists an integer k0 , such that kl+1 − kl ≤ k0 . On the switching topologies, the following assumption is needed.

⋃k

l+1 Assumption 2.5. The union graph k=kl Gσk is strongly connected, and the elements of Gσk , i.e., ai,j (k), belong to a set consisting of finite nonnegative real numbers.

4

X. He, W. Xue, X. Zhang et al. / Automatica 114 (2020) 108842

2.3. Quantized communications In sensor networks, such as wireless sensor networks, the signal transmission between two sensors may confront the problems of channel limitation and energy restriction. Thus, without losing too much accuracy, some quantization operators can be considered to reduce the package size in the encoding process with respect to the transmitted messages. In this paper, we study the case that the messages to be transmitted are quantized element by element through a given quantizer before transmission. Let zk,i be a scalar element remaining to be sent out by sensor i, then we consider the following interval quantizer with quantizing step ∆i > 0 and quantizing function g(·) : R −→ Qi g(zk,i ) = m∆i , if (m −

1

)∆i ≤ zk,i < (m +

1

) ∆i (3) 2 where Qi = {m∆i |m ∈ Z} is the quantization alphabet with countably infinite elements. Then we write g(zk,i ) = zk,i + ϑ (zk,i ), where ϑ (zk,i ) is the quantization error, which is deterministic conditioned on the input zk,i . In addition, we have ϑ (zk,i ) ∈ ∆ ∆ (− 2i , 2i ]. A technique to deal with the correlation between ϑ (zk,i ) and zk,i is to consider a dither operator by adding a random sequence {ξk,i }, which can randomize the quantization error and make it independent of the input data. Write 2

ϑ (k, i) = g(zk,i + ξk,i ) − (zk,i + ξk,i ),

(4)

then the quantization error sequence {ϑ (k, i)} is independent and ∆ ∆ identically distributed (i.i.d.), uniformly distributed on (− 2i , 2i ] and independent of zk,i , if the following assumption holds. Assumption 2.6. The sequence {ξk }k≥0 satisfies Schuchman conditions (Kar & Moura, 2010) and it is independent of F¯k defined in Assumption 2.3. A sufficient condition such that {ξk }k≥0 satisfies Schuchman conditions is that {ξk,i }k≥0 is an i.i.d. sequence uniformly dis∆ ∆ tributed on (− 2i , 2i ] and independent of input sequence {zk,i }. 3. System reconstruction and filtering structure

By constructing a new state vector, consisting of the original state xk and the uncertain dynamics fk , a modified system model is given as follows. Xk+1 = Ak Xk + Duk + ωk

¯ 1,1¯ > α In Θ k,N

¯ 2,2¯ − (Θ ¯ 1,2¯ )T (Θ ¯ 1,1¯ − α In )−1 Θ ¯ 1,2¯ > α Ip Θ k,N k,N k,N k,N 1,1

1 ,2

(6)

2,2

¯ ¯,Θ ¯ ¯ , and Θ ¯ ¯ are given in (B.1). where Θ k,N k,N k,N According to the proof of Proposition 3.2 and Definition 2.5, we can obtain the following lemma. Lemma 3.1. Let Assumption 2.4 hold, then there exist M , N¯ ∈ N+ ¯ 1,1¯ > α In , where Θ ¯ 1,1¯ is and α > 0, such that for any k ≥ M, Θ k,N k,N given in (B.1). Remark 3.1. By Proposition 3.2 and Lemma 3.1, the observability gap between the reformulated system (5) and the system ((2) is the ) second equation in (6), which will reduce to rank

¯ In −A¯ −G ¯ H 0

= n + p if the system is time-invariant and

¯ = [H¯ T , . . . , H¯ T ]T , H¯ i ∈ Rmi ×n , A¯ ∈ Rn×n and observable, where H N 1 n × p ¯ ∈R . G Example. To show the feasibility of (6), we consider a(connected ( network) with three sensors. Suppose that bk,i( = )sat sin x2k (1) ) +x2k (2) , 1 . For the system (2), we let A¯ = 10 12 , G¯ = [1, 1]T , ¯ ¯ 2 = [0, 1], H¯ 3 = [1, 1], Ri = 1, Bi = 1, i = 1, 2, 3. H1 = [1, 0], H On one hand, according to (6) and the notations in the proof of Proposition 3.2, by choosing α( = 2 and N¯ = 10, we have

¯ 2,2¯ − (Θ ¯ 1,2¯ )T (Θ ¯ 1,1¯ − α In )−1 ¯ 1,1¯ − α In ) > 5.77, and λmin Θ λmin (Θ k,N k,N k,N k,N ) ¯ 1,2¯ − α Ip > 1.2 × 106 . On the other hand, by Remark 3.1, Θ k,N ( ) ( I −A −G ) we have rank nH¯ 0 = 3. Then due to rank H¯H¯A¯ = 2, the condition (6) holds. In the sequel, we require the following assumption. Assumption 3.1. There exist M , N¯ ∈ N+ and α > 0, such that for any k ≥ M, the second inequality in (6) holds.

3.1. System reconstruction

{

yk,i = Hk,i Xk + bk,i + vk,i

Proposition 3.2. The reformulated system (5) is uniformly collec¯ steps if and only if there exists N¯ ∈ N+ tively observable after M ¯ and α > 0, such that for any k ≥ M,

It is straightforward to prove that if Assumptions 2.4 and 3.1 are both satisfied, there is a common α > 0, such that the two inequalities in (6) hold simultaneously. By Proposition 3.1, Assumptions 2.4 and 3.1 can lead to the uniformly collective observability of the reformulated system (5).

(5) 3.2. Filtering structure

where

( )

( ) ¯k A¯ k G ∈ Rn+p , Ak ≜ ∈ R(n+p)×(n+p) , 0 Ip ( ) ( ) ω¯ k 0 n+p ωk ≜ ∈R , D≜ ∈ R(n+p)×p , 0 Ip ( ) ¯ k,i 0 ∈ Rmi ×(n+p) . uk ≜ fk+1 − fk ∈ Rp , Hk,i ≜ H Xk ≜

xk fk

Considering the system (2) and the reformulated system (5), Propositions 3.1 and 3.2, which are proved in Appendices A– B, show some equivalence between the properties of the two systems. Proposition 3.1. On the relation between A¯ k in (2) and Ak in (5), the following conclusions hold.

• (1) supk∈N {Ak ATk } < ∞ if and only if supk∈N {A¯ k A¯ Tk } < ∞. • (2) {Ak |Ak ∈ R(n+p)×(n+p) , k ∈ N} has an L-SS if and only if {A¯ k |A¯ k ∈ Rn×n , k ∈ N} has an L-SS.

In this paper, we consider two observation update schemes, namely, time-driven update and event-triggered update, whose difference lies in whether the biased noisy observation yk,i is utilized at the update stage or not. We propose the following distributed filter structure of the system (5) for sensor i, ∀i ∈ V ,

⎧ X¯ = Ak−1 Xˆ k−1,i ⎪ ⎪ ⎪ k,i ⎪ ⎪ if yk,i is utilized: ⎪ ⎪ ⎪ ⎪ ⎨ X˜ k,i = X¯ k,i + Kk,i (yk,i − Hk,i X¯ k,i ) if y

is discarded:

k,i ⎪ ⎪ ⎪ X˜ k,i = X¯ k,i ⎪ ⎪ ∑ ⎪ ⎪ ⎪ Xˆ = Wk,i,j X˜k,j ⎪ ⎩ k,i

(7)

j∈Ni (k)

where X˜k,j = X˜ k,i , if i = j, otherwise X˜k,j = G1 (X˜ k,j ), j ∈ Ni , and G1 (X˜ k,j ) is the vector sent out by sensor j through elementwise dither quantization. Also, X¯ k,i , X˜ k,i , and Xˆ k,i are the extended

X. He, W. Xue, X. Zhang et al. / Automatica 114 (2020) 108842 Table 1 Connection between results and assumptions. Assumptions Assumptions Assumptions Assumptions Assumptions

Algorithm 1 Extended State Based Distributed Kalman Filter (ESDKF) with Quantized Communications:

Results 2.1, 2.3, 2.6 2.1–2.3 2.1–2.6, 3.1 2.1–2.6, 3.1

Lemmas 4.1/5.2 (Consistency) Lemma 4.3 Theorems 4.1/5.1 (Stability) Theorems 4.2/5.2 (Unbiasedness)

state’s prediction, update, and estimate by sensor i at the kth moment, respectively. Kk,i and Wk,i,j , j ∈ Ni (k), are the filtering gain matrix and the local fusion matrices, respectively. They remain to be designed. In the sequel, we will study the consistency, stability and asymptotic unbiasedness of the proposed distributed filters under Assumptions 2.1–3.1. Table 1 is to provide the connection between the main results and assumptions. 4. Distributed filter: time-driven update In this section, for the filtering structure (7) with yk,i employed at each time, we will study the design methods of Kk,i and Wk,i,j . Then we will find the conditions to guarantee the stability of the mean square estimation error for the proposed filter with the designed Kk,i and Wk,i,j . 4.1. Filter design

Prediction: Each sensor carries out a prediction operation X¯ k,i = Ak−1 Xˆ k−1,i P¯ k,i = (1 + θk,i )Ak−1 Pk−1,i ATk−1 +

Lemma 4.1. Consider the filtering structure (7) with a nonstochastic filtering gain Kk,i . Under Assumptions 2.1, 2.3 and 2.6, for any i ∈ V , positive scalars θk,i and µk,i , the pairs (X¯ k,i , P¯ k,i ),(X˜ k,i , P˜ k,i ), (X˜k,i , P˜ k,i ) and (Xˆ k,i , Pk,i ) are all consistent, provided by Wk,i,j = ai,j (k)Pk,i P˜ k−,j1 , where P˜ k,j =

P˜ k,j , Pˇ k,j +

n¯ ∆j (2∆j +1) 2

if i = j, j ∈ Ni (k) In¯ ,

if i ̸ = j, j ∈ Ni (k),

and P¯ k,i , P˜ k,i , Pˇ k,i and Pk,i are recursively calculated through ⎧ 1+θ ⎪ P¯ = (1 + θk,i )Ak−1 Pk−1,i ATk−1 + θ k,i Q¯ k−1 + Q˜ k−1 , ⎪ k,i ⎪ k,i ⎪ ⎪ ⎪ P˜ k,i = (1 + µk,i )(I − Kk,i Hk,i )P¯ k,i (I − Kk,i Hk,i )T ⎪ ( ) ⎪ ⎪ ⎪ 1+µ ⎨ +Kk,i Rk,i + µ k,i Bk,i KkT,i , k,i ⎪ ˇ k,i = G2 (P˜ k,i ), ⎪ P ⎪ ⎪ (∑ )−1 ⎪ ⎪ ⎪ − 1 ⎪ ⎪Pk,i = ai,j (k)P˜ k,j , ⎪ ⎩ j∈Ni (k)

with n¯ = n + p, Q˜ k−1 = blockdiag{Qk−1 , 0p×p }, Q¯ k−1 = DQˆ k−1 DT . Here, G2 (P˜ k,i ) is the element-wise quantization operator by (3) without dither. The next lemma, proved in Appendix D, considers the design of the filtering gain matrix Kk,i . Lemma 4.2. Solving the optimization problem Kk∗,i = arg minKk,i tr(P˜ k,i ) yields Kk∗,i

( T ¯ = Pk,i Hk,i Hk,i P¯ k,i HkT,i +

Rk,i 1 + µk,i

+

Bk,i

µk,i

)−1

.

Summing up the results of Lemmas 4.1 and 4.2, the extended state based DKF (ESDKF) with quantized communications is provided in Algorithm 1.

θk,i

Q¯ k−1 + Q˜ k−1 , ∀θk,i > 0,

Update: Each sensor uses its own observations to update the estimation X˜ k,i = X¯ k,i + Kk,i (yk,i − Hk,i X¯ k,i )

(

Kk,i = P¯ k,i HkT,i Hk,i P¯ k,i HkT,i +

Rk,i 1+µk,i

+

Bk,i

)−1

µk,i

P˜ k,i = (1 + µk,i )(I − Kk,i Hk,i )P¯ k,i . Quantization: Each sensor uses element-wise dither quantization to X˜ k,i and interval quantization to P˜ k,i , i.e., Xˇ k,i = G1 (X˜ k,i ), Pˇ k,i = G2 (P˜ k,i ), where G1 (·) and G2 (·) are the element-wise quantization operators with and without dither, respectively. Local Fusion: Each sensor receives (Xˇ k,j , Pˇ k,j , ∆j ) from its neighbors, denoting n¯ = n + p, for j ∈ Ni (k),

{ (X˜k,j , P˜ k,j ) =

(X˜ k,j , P˜ k,j ), (Xˇ k,j , Pˇ k,j +

∑ j∈Ni (k)

The next lemma, proved in Appendix C, provides a design method of fusion matrices {Wk,i,j }j∈Ni , which can lead to the consistent estimate pairs of each sensor.

1+θk,i

where Q¯ k−1 and Q˜ k−1 are given in Lemma 4.1.

Xˆ k,i = Pk,i

{

5

( Pk,i =

∑ j∈Ni (k)

if i = j,

n¯ ∆j (2∆j +1)

In¯ ), otherwise

2

ai,j (k)P˜ k−,j1 X˜k,j ,

ai,j (k)P˜ k−,j1

)−1 .

4.2. Stability The next lemma, proved in Appendix E, is useful for further analysis. Lemma 4.3. Under Assumptions 2.1–2.3, if there are positive constants {θ1 , θ2 , µ1 , µ2 } such that θk,i ∈ (θ1 , θ2 ) and µk,i ∈ (µ1 , µ2 ), the following two conclusions hold. P¯ k−,i1 1+µk,i 1+µk,i Bk,i . µk,i

• (1) It holds that P˜ k−,i1 = where ∆Rk,i = Rk,i +

1 + HkT,i ∆R− k,i Hk,i , k ∈ N, i ∈ V ,

• (2) There exists a positive scalar η such that P¯ T−l +1 s+1,i ≥ T −1 −1 ηA− Tl +s PTl +s,i ATl +s , s ∈ [0 : L), where {Tl , l ∈ N} is an L-SS of {Ak , k ∈ N}. Theorem 4.1. Consider the system (5) with Algorithm 1. Under Assumptions 2.1–2.6 and 3.1, if the parameter L given in (1) satisfies L > max{k0 , N }+ N¯ and there are positive constants {θ1 , θ2 , µ1 , µ2 } such that θk,i ∈ (θ1 , θ2 ) and µk,i ∈ (µ1 , µ2 ), then the { sequence of mean square estimation errors is stable, i.e., supk∈N E {(Xˆ k,i − Xk )

}

(Xˆ k,i − Xk )T } < +∞, ∀i ∈ V . Proof. Due to the consistency in Lemma 4.1, we turn to prove supk∈N Pk,i < ∞. Note that the key steps are provided in the following due to page limitation. For the detailed proof, the readers can refer to the arXiv version of this work (He, Xue, Zhang, & Fang, 2019). Under Assumption 2.2, {Ak |Ak ∈ R(n+p)×(n+p) , k ∈ N} has an L-SS, which is supposed to be {Tl , l ∈ N} subject to L ≤ ¯ Tl+1 − Tl ≤ T < ∞, ∀l ≥ 0, where L > max{k0 , N } + N. ¯ where M ¯ is Without loss of generality, we assume T0 ≥ M,

6

X. He, W. Xue, X. Zhang et al. / Automatica 114 (2020) 108842

given in Assumption 2.4. Otherwise, a subsequence of {Tl , l ∈ N} can always be obtained to satisfy the requirement. We divide the sequence set {⋃ Tl , l ≥ 0} into two non-overlapping time set: {Tl + L, l ≥ 0} and l≥0 [Tl + L + 1 : Tl+1 + L − 1]. In the following, we simply consider the time instants k = Tl + L, l ≥ 1. For other time instants, we refer to the arXiv version of this work (He et al., 2019). For convenience, let k¯ ≜ Tl + L. It can be easily shown that ) ( n¯ ∆i (2∆i +1) In¯ 2

−1

≥ ϱP˜ k¯−,i1 , where ϱ ∈ (0, 1] and n¯ = n + p. ϱη According to Lemma 4.3, denoting δ = (1+µ ) , we obtain 2 [ L−1 L−1 amin δ µ1 ∑ Pk¯−,i1 ≥ Φk¯−,k¯T−s (8) µ1 + 1 s=k0 ] ∑ T −1 · Hk¯ −s,j (Rk¯ −s,j + Bk¯ −s,j ) Hk¯ −s,j Φk¯−,k¯1−s , Pˇ k,i +

j∈V k¯ ,k¯ −s

where amin = mini,j∈V ai,j

¯

> 0, s ∈ [max{k0 , N } : L), and aki,,jk−s

4.4. Asymptotically unbiased We write fk = O(gk ) (or fk = o(gk )) if and only if fk ≤ Mgk f for all k ≥ 0 (or limk→∞ gk = 0). The following two lemmas, k proved in He, Hu, Hong, Shi, and Fang (2020) and Appendix G respectively, are to find conditions ensuring the asymptotic unbiasedness of Algorithm 1. Lemma 4.4. Suppose that Π0 and Π1 satisfy 0 ≤ Π0 ≤ Π1 and Π1 > 0, then Π0 Π1−1 Π0 ≤ Π0 . Lemma 4.5. Let {dk } ∈ R be generated by dk+1 = ρ dk + mk with ρ ∈ (0, 1) and d0 < ∞, then (1) if mk = o(1), then dk = o(1), i.e., dk → 0 as k → ∞; (2) if mk = o(δ k ) and ρ < δ < 1, then dk = o(δ k ); (3) if mk = o( k1M ) with M ∈ N+ , then dk = o( kM1−1 ).

¯ ¯

is the (i, j)th element of Πlk=k¯ −s Aσl . Note that amin can be obtained under Assumption 2.5, since the elements of Aσk belong to a finite set and the jointly strongly connected network can lead to k¯ ,k¯ −s ai,j > 0, s ≥ max{k0 , N }. Under Assumptions 2.2 and 2.4, there

˘ such that Pk,i ≤ P˘ , k = exists a constant positive definite matrix P, Tl + L.

An upper bound of MSE can be derived by the proof of Theorem 4.1, from which the influence of quantization interval size ∆i can be reflected through the scalar ϱ. With the increase of ∆i , the upper bound will become larger. 4.3. Design of parameters In this subsection, the design methods for the parameters θk,i and µk,i are considered by solving two optimization problems. (a) Design of θk,i At the prediction stage, the design of the parameter θk,i is aimed to minimize the trace of P¯ k,i , which is an upper bound of MSE by Lemma 4.1. Mathematically, the optimization problem on θk,i is given as

θk∗,i = arg min tr(P¯ k,i ),

(9)

θk,i

Theorem 4.2. Consider the system (5) satisfying the same conditions as Theorem 4.1 and λmin {Ak ATk } ≥ β0 > 0. Denoting Mk := maxj∈V ∥E {bk,j }∥22 + ∥E {uk }∥22 ,

• if Mk = o(1), then ∥E {ek,i }∥2 = o(1); 1−M • if Mk = o(k−M ) with M ∈ N+ , then ∥E {ek,i }∥2 = o(k 2 ); k • if Mk = o(δ k ) and ϱ˜ < δ < 1, then ∥E {ek,i }∥2 = o(δ 2 ), where ϱ˜ ∈ (0, 1) is defined in (13). Proof. Since P¯ k,i is positive definite, we define the following function Vk,i (E {¯ek,i }) = E {¯ek,i }T P¯ k−,i1 E {¯ek,i }. From the fact (iii) of Lemma 1 in Battistelli and Chisci (2014) and the non-singularity T −1 −1 ek+1,i }, of Ak , we have Vk+1,i (E {¯ek+1,i }) ≤ ϱE {¯ek+1,i }T A− k Pk,i Ak E {¯ where 0 < ϱ < 1. Since wk is zero-mean, E {¯ek+1,i } = Ak E {ek,i } − DE {uk }. Then, there exists a scalar α0 > 0, such that ϱ¯ 1 = ϱ(1 + α0 ) ∈ (0, 1), and Vk+1,i (E {¯ek+1,i })

≤ϱ¯ 1 E {ek,i }T Pk−,i1 E {ek,i } + O(∥E {uk }∥22 ).

(11) ∆i

∑

where P¯ k,i = (1 + θk,i )Ak−1 Pk−1,i ATk−1 + θ k,i Q¯ k−1 + Q˜ k−1 . k,i Since (9) is convex, which can be numerically solved by many existing convex optimization methods. The next Proposition 4.1, proved in Appendix F, provides the closed-form solution of (9).

Vk+1,i (E {¯ek+1,i }) ≤ϱ¯ 2

Proposition 4.1.

Kk,j Hk,j )E {¯ek,j } + Kk,j E {bk,j }, then we have

1+θ

√ θk∗,i =

Solving (9) yields the closed-form solution

tr(Q¯ k−1 ) tr(Ak−1 Pk−1,i ATk−1 )

j∈Ni (k)

(12)

Notice that P˜ k,j = (1 + µk,j )(I − Kk,j Hk,j )P¯ k,j and E {ek,j } = (I − P¯ k−,j1

1+µk,j

= P˜ k−,j1 (I − Kk,j Hk,j )

and ≤ There exists a sufficiently small scalar α1 > 0 such that ϱ¯ 2 (1 + α1 ) < 1. Denote P˜ k−,j1 .

ϱ˜ = ϱ¯ 2 (1 + α1 ),

Furthermore, it holds that θk∗,i > 0. (b) Design of µk,i Considering the consistency, we cast the design of µk,i into the following optimization problem: µk,i

ai,j (k)E {˜ek,j }T P˜ k−,j1 E {˜ek,j }

+ O(∥E {uk }∥22 ).

−1 P˜ k,j

, i ∈ V.

µ∗k,i = arg min tr(P˜ k,i ), s.t. µk,i > 0.

∆

As the quantization error is uniformly distributed in (− 2 , 2i ], the quantization error is zero-mean. By (11) and Lemma 2 in Battistelli and Chisci (2014), there exists a scalar ϱ¯ 2 ∈ (0, 1) such that

(10)

Remark 4.1. Although the optimization in (10) is not convex, it can be solved by some existing methods, such as the quasiNewton methods or the nonlinear least square (BartholomewBiggs, 2008).

(13)

For this α1 , we have E {˜ek,j }T P˜ k−,j1 E {˜ek,j }

≤(1 + α1 )E {¯ek,j }T (I − Kk,j Hk,j )T × P˜ −1 (I − Kk,j Hk,j )E {¯ek,j } + O(∥E {bk,j }∥22 ) k,j

≤(1 + α1 )E {¯ek,j }T

P¯ k−,j1 1 + µk,j

P˜ k,j

P¯ k−,j1 1 + µk,j

E {¯ek,j }

+ O(∥E {bk,j }∥22 ) ≤(1 + α1 )E {¯ek,j }T P¯ −1 E {¯ek,j } + O(∥E {bk,j }∥22 ) k,j

X. He, W. Xue, X. Zhang et al. / Automatica 114 (2020) 108842

where the last inequality is obtained by Lemma 4.4 and (1) of Lemma 4.3. By (12) and (13), we have Vk+1,i (E {¯ek+1,i })

≤ϱ˜

∑

ai,j (k)Vk,j (E {¯ek,j })

j∈Ni (k)

7

Remark 5.1. The triggering scheme in (16) shows that if the current information is more sufficient in at least one channel than the prediction information, then it is worth using the available observation in the update stage. The triggering threshold τ is used as a measure of information increment.

(14)

The following lemma, proved in Appendix H, provides an equivalent form of the triggering scheme.

Due to Ak = [ai,j (k)], i, j = 1, 2, . . . , N, summing up (14) for i = 1, 2, . . . , N, then

Lemma 5.1. The event-triggered scheme (16) is satisfied if and only if

+ O(max ∥bk,j ∥22 ) + O(∥uk ∥22 ) j∈ V

Vk+1 (E {¯ek+1 })

(15)

≤ϱ˜ Ak Vk (E {¯ek }) + 1N ⊗ (O(max ∥ j∈V

bk,j 22 )

uk 22 ))

∥ + O(∥ ∥

,

λmax (P˜ k−,i1 − P¯ k−,i1 ) > τ ,

(17)

where P˜ k,i = (1 + µk,i )(I − Kk,i Hk,i )P¯ k,i and τ ≥ 0.

where Vk (E {¯ek }) = [VkT,1 (E {¯ek,1 }), . . . , VkT,N (E {¯ek,N })]T . Taking 2-norm operator on both sides of (15) and considering ∥Ak ∥2 = 1 ( yields)∥Vk+1 (E {¯ek+1 })∥2 ≤ ϱ∥ ˜ Vk (E {¯ek })∥2 + O maxj∈V ∥bk,j ∥22 + ∥uk ∥22 . Due to Mk = maxj∈V ∥bk,j ∥22 + ∥uk ∥22 = o(1) and (1) of Lemma 4.5, the estimate sequence of each sensor by Algorithm 1 is asymptotically unbiased. Furthermore, if Mk satisfies certain convergence rates (i.e., Mk = o(k−M ) or Mk = o(δ k )), by (2) and (3) of Lemma 4.5, the estimation bias is convergent to zero with k 1−M certain rates (i.e., o(k 2 ) or o(δ 2 )), respectively. Remark 4.2. If the observation biases of sensors do not converge to zero or the uncertainty does not converge to a constant vector, one can analyze an upper bound (i.e., b1 ) of estimation bias with the similar procedure as the proof of Theorem 4.2. Additionally, due to E {Xˆ k,i − Xk }E {Xˆ k,i − Xk }T + Cov{Xˆ k,i − Xk } = E {(√ Xˆ k,i − Xk )(Xˆ k,i − T ˆ Xk ) } ≤ Pk,i , it holds that b2 := ∥E {(Xk,i − Xk )}∥2 ≤ ∥Pk,i ∥2 . Then one can utilize the value min{b1 , b2 } to evaluate the estimation bias in real time.

Since Algorithm 1 is consistent, P˜ k−,i1 and P¯ k−,i1 stand for the lower bounds of information matrices at the update stage and the prediction stage, respectively. Then P˜ k−,i1 − P¯ k−,i1 reflects the variation of statistical information resulted from a new observation. In light of Lemma 5.1, the event E is triggered if sufficiently new information is accumulated at the update stage. 5.2. ESDKF with event-triggered update Based on the event-triggered scheme (16) and Algorithm 1, we can obtain the extended state based DKF with event-triggered update scheme in Algorithm 2. Algorithm 2 ESDKF based on event-triggered update: Prediction: the same as the one of Algorithm 1 Event-triggered update: If (16) is satisfied, then

(

Kk,i = P¯ k,i HkT,i Hk,i P¯ k,i HkT,i +

Rk,i 1+µk,i

+

Bk,i

)−1

µk,i

Remark 4.3. If the system is noise-free with unknown but deterministic uncertain dynamics and observation biases, the conclusions of Theorem 4.2 degenerates to the convergence of estimation error with specified rates.

P˜ k,i = (1 + µk,i )(I − Kk,i Hk,i )P¯ k,i . X˜ k,i = X¯ k,i + Kk,i (yk,i − Hk,i X¯ k,i ) Otherwise, X˜ k,i = X¯ k,i , P˜ k,i = P¯ k,i .

5. Distributed filter: event-triggered update

Quantization: the same as the one of Algorithm 1 Local Fusion: the same as the one of Algorithm 1

In this section, we will study an even-triggered update based DKF and analyze the conditions to ensure the mean square boundedness and asymptotic unbiasedness. 5.1. Event-triggered update scheme Due to the influence of random noise and observation bias over the system (2), some corrupted observations may lead to the performance degradation of filters. Thus, we aim to provide a scheme to decide when the observation is utilized or discarded. We introduce the information metric Sk,i , defined as

(

Sk,i ≜ HkT,i Rk,i +

(1+µk,i )

µk,i

Bk,i

)−1

Hk,i . In the following, we define

the update event and the event-triggered scheme. Definition 5.1. We say that an update event E of sensor i is triggered at time k, if sensor i utilizes the observation yk,i to update the estimate as Algorithm 1. The event E is triggered at time k (i.e., yk,i is utilized) if

( λmax Sk,i −

µk,i ¯ −1 P 1 + µk,i k,i

)

> τ,

(16)

where τ ≥ 0 is the preset triggering threshold of the observation update. Otherwise, yk,i will be discarded.

According to Algorithms 1 and 2, we can easily obtain Lemma 5.2. Lemma 5.2. Consider the system (5) with Algorithm 2. Under Assumptions 2.1, 2.3 and 2.6, the pairs (X¯ k,i , P¯ k,i ), (X˜ k,i , P˜ k,i ), (X˜k,i , P˜ k,i ) and (Xˆ k,i , Pk,i ) are all consistent. The next lemma, proved in Appendix H, is on Algorithm 2. Lemma 5.3.

Under the event-triggered update scheme, for

Algorithm 2, it holds that P˜ k−,i1 ≥ where ∆Rk,i = Rk,i +

1+µk,i

µk,i

P¯ k−,i1 1+µk,i

1 + HkT,i ∆R− k,i Hk,i − τ In+p ,

Bk,i .

The next proposition, proved in Appendix H, studies the relation between Algorithms 1 and 2, which shows that the eventtriggered observation update scheme can lead to a tighter bound of MSE than the typical time-driven observation update. Proposition 5.1. Let PkAs,i , s = 1, 2, be the Pk,i matrix of Algorithms 1 and 2, respectively. If the two algorithms share the same initial A1 setting and τ = 0, then PkA2 ,i ≤ Pk,i . Remark 5.2. In the scenarios as (Han, You, Xie, Wu, & Shi, 2015; Weerakkody, Mo, Sinopoli, Han, & Shi, 2016) where the estimator and sensor are distributed at different geographical locations

8

X. He, W. Xue, X. Zhang et al. / Automatica 114 (2020) 108842

with energy-constrained communication channels, it is suggested to judge which observations contain high-quality information and to decide when the observations are transmitted from the sensor to the remote estimator. These tasks can be achieved by the proposed event-triggered update scheme, which serves for transmission reduction and performance improvement. 5.3. Stability and asymptotic unbiasedness

sup E {ek,i eTk,i } < +∞, ∀i ∈ V .

}

Furthermore, if infk λmin Ak ATk > 0, then

)

1

max sup λmax E {ek,i eTk,i } ≤

)

(

i∈V

a0 − τ a1

k≥L

( Rk,i +

1 + µ2

)−1 Bk,i

Hk,i

µ2 T a¯ = sup{λmax (Ak Ak )}, bw = sup{λmax (Q¯ k )} k∈N

r0 = (1 + θ2 )(a0 − τ a1 )a¯ + r1 =

1 + θ1

(20) bw + bq

θ1 µ1 + τ , bq = sup{λmax (Q˜ k )}, (µ1 + 1)r0

Theorem 5.2. Consider the system (5) with Algorithm 2 under the ˜ same conditions as Theorem 5.1. If there ( is an ) integer M > L such that the set S = {i ∈ V |supk≥M˜ λmax Ik+1,i ≤ r1 } is non-empty, and

,

where ek,i = Xˆ k,i − Xk , a0 and a1 are given in (19). Proof. According to Lemma 5.2, we turn to prove supk∈N Pk,i < ∞. Similar to the proof of⋃ Theorem 4.1, with the same notations, we consider the time set l≥0 [Tl + L + 1 : Tl+1 + L − 1]. The rest part can be similarly proved by taking[the method as the ] proof of ∑ k¯ ,k¯ −L −1 Theorem 4.1. We have Pk¯−,i1 ≥ δ L Φk¯−,k¯T−L a P Φk¯−,k¯1−L + j∈V i,j k¯ −L,j ϱη and P˘ −1 , where δ = k¯ ,i

HkT,i

where a0 and a1 are given in (19), {θ1 , θ2 , µ1 , µ2 } are given in Theorem 5.1.

k∈N

(

To analyze the asymptotic unbiasedness of Algorithm 2, for convenience, we denote Ik,i =

Theorem 5.1. Consider the system (5) and Algorithm 2 under Assumption 2.1–2.6 and 3.1. If the parameter L given in (1) satisfies L > max{k0 , N } + N¯ and θk,i ∈ (θ1 , θ2 ), µk,i ∈ (µ1 , µ2 ) for positive constants {θ1 , θ2 , µ1 , µ2 }, then there exists a scalar ϑ > 0, such that for 0 ≤ τ < ϑ ,

{

discarded. As a result, the collective observability condition in Assumption 2.4 may not hold, which means that the boundedness of MSE is not guaranteed.

¯ k = o(1), then ∥E {ek,i }∥2 = o(1); • if M 1−M ¯ k = o(k−M ) with M ∈ N+ , then ∥E {ek,i }∥2 = o(k 2 ); • if M k k ¯ k = o(δ ) and ϱ˜ < δ < 1, then ∥E {ek,i }∥2 = o(δ 2 ), where • if M ϱ˜ ∈ (0, 1) is defined in (13), ¯ k := maxj∈V −S ∥E {bk,j }∥22 + ∥E {uk }∥22 . where M

(1+µ2 )

L−1

P˘ k¯−,i1 =ϱ

∑

[ δ s Φk¯−,k¯T−s

s=0

·

∑

k¯ ,k¯ −s

(18)

ai,j

j∈V

(

1 Hk¯T−s,j ∆R− H¯ k¯ −s,j k−s,j

− τ In+p

)

] Φk¯−,k¯1−s

.

We have Pk¯−,i1 ≥ P˘ k¯−,i1 . Then we turn to prove there is a constant

matrix S > 0, such that P˘ k¯−,i1 ≥ S. Under the conditions of this theorem, it follows from the proof of Theorem 4.1 that there is ] a [

∑ k¯ ,k¯ −s T 1 Hk¯ ,j ∆R− H¯ δ s Φk¯−,k¯T−s j∈V ai,j k¯ ,j k,j ∑ L−1 s −T −1 −1 Φk¯ ,k¯ −s ≥ π In+p . Denote Ξk¯ = s=0 δ Φk¯ ,k¯ −s Φk¯ ,k¯ −s . To guarantee infk¯ P˘ k¯−,i1 > 0, it is sufficient to prove that there exists a constant matrix S > 0, such that π In+p − τ Ξk¯ ≥ S. Under Assumption 2.2, there exists a constant scalar µ ¯ > 0, such that supk¯ Ξk¯ ≤ µ ¯ In+p . Let S = (π − τ µ ¯ )In+p , then a sufficient condition is 0 ≤ τ < πµ¯ . Choosing ϑ = πµ¯ > 0, the boundedness is obtained. Furthermore, if infk λmin {Ak ATk } > 0, the above analysis holds for any k ≥ L with scalar π > 0, such that

∑L−1 s=0

no prediction step in the proof of Theorem 4.1 of the arXiv version of this work (He et al., 2019). Thus, for any k ≥ 0, it holds that Pk−,i1 ≥ P˘ k¯−,i1 ≥ ϱ(π − τ µ ¯ )In+p . Let a0 = ϱπ , a1 = ϱµ, ¯

(19)

1 then we have maxi∈V supk≥L λmax (Pk,i ) ≤ a −τ . In light of the a1 0 consistency in Lemma 4.1, the conclusion holds.

Remark 5.3. If better performance of Algorithm 2 is pursued, one could let τ be zero or sufficiently small, which can ensure a smaller bound of MSE by Proposition 5.1. If one aims to reduce the update frequency with stable estimation error, a relatively large τ can be set by satisfying the requirement in Theorem 5.1. Note that if a too large τ is given such that the triggering condition (16) is hardly satisfied, then most of observation information will be

Remark 5.4. Theorem 5.2 shows that the estimation biases of Algorithm 2 tend to zero even if the observation biases of a sensor set (i.e., S ) do not decay. The proof of Theorem 5.2 is similar to that of Theorem 4.2, by noting that P˜ k,i = P¯ k,i , for i ∈ S , which indicates that for those sensors with non-decaying observation biases, their observations will no longer be used after a finite time. In the following, we show the feasibility of the condition that ˜ > L such that the set S = {i ∈ there is a positive integer M ( ) V |supk≥M˜ λmax Ik+1,i ≤ r1 } is non-empty. Feasibility for a non-empty S To find the condition under which the( set S) is non-empty, we prove that the condition supk≥M˜ λmax Ik+1,i ≤ r1 can be satisfied for some sensors. Note that under the conditions of Theorem 5.2, then the conclusions of Theorem 5.1 holds as well. Thus, recalling the notations in (20), we have a0 −τ a1 > 0, which leads to r0 > 0 and then r1 > τ > 0. Meanwhile, whatever how τ is small, r1 has a uniformly lower bound by noting 0 < 1+θ r0 ≤ (1 + θ2 )a0 a¯ + θ 1 bw + bq . If there is M ≥ L, such that 1

supk≥M˜ λmax Ik+1,i ≤ r1 , then S is non-empty. Simple examples ˜ ≥ L, ∥Hk,i ∥2 being ensuring this condition include ) that for k ≥ M (

(

)

very small, λmax Rk,i +

1+µ2

µ2

Bk,i

being very large, and so on.

The event-triggered scheme and the time-driven scheme have some similar properties in estimation consistency, boundedness of MSE, and asymptotic unbiasedness of estimates. The differences between the two schemes are explicitly shown in Table 2. 6. Numerical simulations In this section, numerical simulations are carried out to demonstrate the aforementioned theoretical results and show the effectiveness of the proposed algorithms.

X. He, W. Xue, X. Zhang et al. / Automatica 114 (2020) 108842

9

Table 2 Differences of time-driven and event-triggered update schemes. Update scheme

Observation

Threshold

Stability condition

Asymptotic unbiasedness

Time-driven Event-triggered

Always Selective

No Yes

System+topology System+topology+threshold

Decaying observation biases of all sensors Non-decaying observation biases of some sensors

6.1. Performance evaluation In this and next subsections, let us consider an object whose motion is described by the following kinematic model (Simon, 2010) with uncertain dynamics:

⎛

xk+1

1 ⎜0 =⎝ 0 0

0 1 0 0

⎞

T 0 1 0

⎛

⎞

0 0 T⎟ ⎜0 x + 0 ⎠ k ⎝T 1 0

0 0⎟ f (xk , k) + ωk , 0⎠ T

(21)

where T = 0.1 is the sampling step, xk is the unknown state vector consisting of four-dimensional components along the coordinate axes and f (xk , k) is the uncertain dynamics. The covariance of process noise ωk is Qk = diag ([4, 4, 1, 1]). The kinematic state of the object is observed by means of four sensors modeled as

¯ k,i xk + bk,i + vk,i , i = 1, 2, 3, 4. yk,i = H The observation noise [vk,1 , . . . , vk,N ]T is i.i.d. Gaussian with covariance Rk = 4 × I4 . Additionally, the sensor network’s communication topology is assumed to be directed ( and switching, ) 1 0 0 0 0.5 0.5 0 0 , 0 0.5 0.5 0 0 0 0.5 0.5 0.5 0.5 0 0 0 0.5 0.5 0 . And the 0 0 1 0 0.25 0.25 0.25 0.25 k 3 1, where mod(a b) 5

whose adjacency matrix is selected from A1 =

( 0.5 0.5

0 0 0 1 0 0 0 0.3 0.4 0.3 0 0.5 0 0.5

A2 =

)

( and A3 =

)

, topology switching signal σk = mod ⌊ ⌋, + stands for the modulo operation of a by b. In the following, we conduct the numerical simulations through Monte Carlo experiment, in which 500 runs for the considered algorithms are implemented, respectively. The root MSE√(RMSE) averaged over

(

MSEk,i =

1 500

j=1

j xk,i

j xk,i

−ˆ

)T (

j xk,i

1 4

j xk,i

−ˆ

)

∑4

i=1

MSEk,i , where

j xk,i

, and ˆ

is the state

(position or velocity) estimate of the jth run ∑4of sensor i at the kth time instant. Besides, we denote Pk = 14 i=1 Pk,i . The mean estimation error averaged over all sensors is defined by MEk = ) ( 1 4

∑4

1 i=1 500

∑500 1 ∑4 j=1 4

l=1

j xk,i (l)

j xk,i (l)

−ˆ

. It is assumed that the

initial state is a zero-mean random vector with covariance matrix P0 = diag ([10, 10, 1, 1]). The uncertain dynamics and bias are as( the state-correlated ) sumed to be f (xk , k) =

1 3

sin(xk (3))+k sin(xk (4))+k

and bk,i = sat (2 sin

x2k (1) + x2k (2) + b0,i , 2 , respectively, where the initial bias b0,i is generated uniformly within [-2,2]. We assume Bk,i = 4. The observation matrices are supposed to be switching with time, ¯ k,i = H¯ ( k ) , and H¯ 1 = ( 1 0 0 0 ) , H¯ 2 = following H

(

)

stable in the given period and the consistency of each sensor remains.

)

all the sensors is defined as RMSEk =

∑500 (

Fig. 1. RMSE difference between Algorithms 1 and 2.

)

mod i+⌊ 10 ⌋,4 +1

of Algorithms ( ) , H¯ 3 = H¯ 4 = ( 0 0 0 0 ). And the ( parameters ) P0 04×2 ˆ 1 and 2 are set Xi,0 = 06×1 , Pi,0 = 0 , Qk = 10−3 × I2 , 2×4 I2 µk,i = 0.3, τi = 0.001, ∀i = 1, 2, 3, 4. First, we carry out numeri0100

cal simulations for Algorithm 1 (i.e., ESDKF) and Algorithm 2 with results given in Figs. 1 and 2. Fig. 1 shows Algorithm 2 generally have better performance than Algorithm 1. Meanwhile, Fig. 2 shows the triggering time instants of observation update for each sensor. Because of the periodic switching of observation matrices, the triggering time instants of all sensors are also periodic. Thus, compared with Algorithms 1, 2 can reduce the frequency of observation update with competitive estimation √ performance. Fig. 2 also gives the behavior of the RMSE and tr (Pk ) of Algorithm 2, from which one can see the MSE of the proposed ESDKF keeps

6.2. Asymptotically unbiasedness of Algorithms 1 and 2 Next, we give a numerical simulation to verify Theorems 4.2 and 5.2. In this subsection, ( )the uncertain dynamics is assumed to be f (xk , k) =

1 2k

sin(xk (3)) sin(xk (4))

+

(1) 1

, and the state-correlated bias

bk,i has two situations:

• Situation 1: ( ) 1 4 bk,i = sat sk,i , 2 , Bk,i = 2 k

k

(

i = 1, 2, 3, 4

)

where sk,i = 2 sin x2k (1) + x2k (2) + b0,i , and b0,i is generated uniformly within [-2,2]. • Situation 2:

⎧ ⎨ 1 sat (s , 2) , B = 4 , i = 1, 2, k,i k,i bk,i = k k2 ) ⎩ ( sat s˜k,i , 40 , Bk,i = 1600, i = 3, 4, ( ) where s˜k,i = 40 sin x2k (1) + x2k (2) + b0,i , b0,1 and b0,2 are generated uniformly within [-2,2], b0,3 and b0,4 are generated uniformly within [-40,40]. In Situation 2, the biases of sensor 3 and sensor 4 do not tend ¯ k,1 = to zero. The observation matrices are supposed to be H ( 1 0 0 0 ) , H¯ k,2 = ( 0 1 0 0 ) , H¯ k,3 = ( 1 0 0 0 ) , H¯ k,4 = ( 0 1 0 0 ). And the parameters of Algorithms 1 and (2 are set ) Xi,0 =

)T ( √ √ 11P0 04×2 , Qˆ k = − 10 10 10 10 0 0 , Pi,0 = 02×4 I2 −3 10 × I2 , µk,i = 0.3, τi = 0.001, ∀i = 1, 2, 3, 4. Fig. 3 gives the mean estimation error of Algorithms 1 and 2 under Situation 1 and Situation 2. From this figure, one can see, under Situation 1, both Algorithms 1 and 2 are asymptotically unbiased; however, under Situation 2, only Algorithm 2 is asymptotically unbiased. Thus Theorems 4.2 and 5.2 are verified.

10

X. He, W. Xue, X. Zhang et al. / Automatica 114 (2020) 108842

Fig. 2. Triggering times, RMSE and

√

tr(P) of Algorithm 2.

Fig. 4. Communication topology of the sensor network. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

¯ A = ( 1 0 0 0 ) , H¯ B = ( 0 1 0 0 ). The sensor B are supposed to be H distribution of these 3 kinds of nodes is given in Fig. 4, which is undirected and switching between two graphs. Note that a non¯ k,i = 0 sensing node i can implement algorithms by assuming H and yk,i = 0. In the figure, the dotted red lines and blue lines will exist successively for every five time instants. The elements G of the adjacency matrix are set to be ai,js = ⏐⏐ 1Gs ⏐⏐ . The setting of

Fig. 3. Mean estimation error of Algorithms 1 and 2.

6.3. Comparisons with other algorithms To verify that the proposed algorithm can handle singular system matrices, as stated in Theorems 4.1 and 5.1, in this subsection, let us leave out the physical meaning of system (21) ( ) and assume that the system matrix satisfies A¯ k = mod (k, 10) < 8. Otherwise, A¯ k =

(1 0 T 0) 01 0 T 00 1 0 00 1 0

1 0 0 0

0 1 0 0

T 0 1 0

0 T 0 1

if

. Besides, we consider

a network with 20 nodes, where 3 kinds of nodes in this network: sensor A, sensor B and non-sensing node. A non-sensing node has no observation but it is capable to run algorithms and communicate with other nodes. The observation matrices of sensor A and

⏐Ni

⏐

uncertain dynamics and bias is the same as Section 6.1. Then we carry out numerical simulations to compare Algorithm 2 with other three algorithms, namely, distributed state estimation with consensus on the posteriors (DSEA-CP) (Battistelli & Chisci, 2014), centralized Kalman filter (CKF) and centralized extended state based Kalman filter (CESKF) (Bai et al., 2018). For DSEA-CP, the initial estimate is assume to be xˆ 0,i = 04×1 , P0,i = P0 , ∀i = 1, 2, 3, 4. As for CKF, in a data center, the CKF is utilized with the form of standard Kalman filter by employing the collected observations from all sensors. Here, the initial estimate of CKF is xˆ 0 = 04×1 , P0CKF = P0 . Similar to

X. He, W. Xue, X. Zhang et al. / Automatica 114 (2020) 108842

11

Appendix A. Proof of Proposition 3.1

( Recall Ak =

Ak ATk =

(

)

A¯ k 0

¯k G ¯ k G¯ T } < ∞, then we have , and supk∈N {G k Ip

¯ k G¯ T A¯ k A¯ Tk + G k ¯GT k

¯k G Ip

)

.

(A.1)

First, we consider the proof of (1). We start (with the sufficiency ) ¯ G¯ T 0 A¯ A¯ T +G of (1). According to (A.1), we have Ak ATk ≤ 2 k k 0 k k I . Thus, p

¯ k G¯ T } < ∞ lead to supk {Ak AT } < ∞. supk {A¯ k A¯ Tk } < ∞ and supk {G k k We then consider the necessity of (1). If supk {Ak ATk } < ∞, it T T ¯ k G¯ } < ∞. Due to G¯ k G¯ T ≥ follows from (A.1) that supk {A¯ k A¯ k + G k k T 0, then supk {A¯ k A¯ k } < ∞. Next, we consider ( ) (the proof ) ( of (2). )

According to (A.1), it holds that Ak ATk =

Fig. 5. Estimation performance of different algorithms.

CKF, CESKF is utilized with the form of ESKF (Bai et al., 2018) by employing the collected observations from all sensors. ( As for the ) parameter of CESKF are set to be Xˆ 0 = 06×1 , P0CESKF =

P0 04×2 02×4 I2

,

Qˆ k = I2 . The performance comparison result of the above algorithms is shown in Fig. 5. From this figure, one can see that the RMSE of x3 and x4 for DSEA-CP and CKF become unstable, but the estimation errors of CESKF and Algorithm 2 still keep stable. The stability of CESKF and Algorithm 2 lies in their capability in handling with uncertain nonlinear dynamics. Since CESKF is a centralized filter without being affected by the quantized channels and the switching of communication topologies, its estimation error is smaller than ESDKF. For the DSEA-CP, due to the existence of unbounded uncertain dynamics and the switching topologies, both the position estimation error and the velocity estimation error are divergent. As for CKF, since observations (position information) of all sensors are available, the position estimation error is stable. However, the velocity estimation error is divergent since the existence of unbounded uncertain dynamics. Based on the above results, we can see the proposed Algorithms 1 and 2 are effective distributed filtering algorithms for the considered scenarios. 7. Conclusion

A¯ k A¯ Tk 0 0 Ip

¯k G Ip

In 0 ¯ T Ip G k

.

Thus, λmin (A¯ k A¯ Tk ) = min{λmin (A¯ k A¯ Tk ), 1}. We study the sufficiency of (2). If {A¯ k |A¯ k ∈ Rn×n , k ∈ N} has an L-SS {Tl , l ∈ N} which satisfies λmin (A¯ Tl +s A¯ TT +s ) > β, s ∈ [0 : L). Then for the same l time sequence {Tl , l ∈ N}, it holds that λmin (ATl +s ATT +s ) > β ∗ = l min{β, 1}, s ∈ [0 : L). Thus, the time sequence {Tl , l ∈ N} is (n+p)×(n+p) also an L-SS of {Ak |Ak ∈ R , k ∈ N}. We then consider the necessity of (2). If {Ak |Ak ∈ R(n+p)×(n+p) , k ∈ N} has an L¯ s ∈ [0 : L). Recall SS {T¯l , l ∈ N}, then λmin (AT¯l +s ATT¯ +s ) > β, l

λmin (A¯ k A¯ Tk ) = min{λmin (A¯ k A¯ Tk ), 1}., then we have λmin (A¯ T¯l +s A¯ TT¯ +s ) > l ¯ s ∈ [0 : L). As a result, the time sequence {T¯l , l ∈ N} is also an β, L-SS of {A¯ k |A¯ k ∈ Rn×n , k ∈ N}. Appendix B. Proof of Proposition 3.2

Denote Φj,k = Φj,j−1 · · · Φk+1,k (j > (k), Φk,k = )In+p , and ¯ j−1 A¯ j−1 G × ··· × Φk+1,k = Ak , then it holds that Φj,k = 0 Ip ) ) ( ( A¯ k 0

¯k G Ip

¯ j,k Φ

˜ j,k+1 Φ

0

Ip

˜ j,k+1 = , where Φ

∑j

¯ j,i G¯ i−1 . Φ ) Θi1,j,,1k Θi1,j,,2k T T −1 Thus, we obtain Φj,k Hj,i (Rj,i + Bj,i ) Hj,i Φj,k = 1,2 2,2 , (Θi,j,k )T Θi,j,k 1,1 ¯ jT,k H¯ jT,i (Rj,i + Bj,i )−1 H¯ j,i Φ ¯ j,k , Θi1,j,,2k = Φ ¯ jT,k H¯ jT,i (Rj,i + where Θi,j,k = Φ 2 , 2 ¯ j,i Φ ˜ j,k+1 ). ˜ j,k+1 )T (Rj,i + Bj,i )−1 (H¯ j,i Φ ˜ j,k+1 and Θi,j,k = (H¯ j,i Φ Bj,i )−1 H + ¯ For N ∈ N , we denote ( ) ( 1,1 ) N k+N¯ 1,2 ¯ ¯ ¯ 1,2¯ ∑ ∑ Θ 1 ,1 Θ Θ Θ i,j,k i,j,k k,N k,N := , 1,2 2,2 ¯ 1,2¯ )T Θ ¯ 2,2¯ (Θi,j,k )T Θi,j,k (Θ =

i=k+1

(

k,N

i=1 j=k

In this paper, we studied a distributed filtering problem for a class of general uncertain stochastic systems. By treating the nonlinear uncertain dynamics as an extended state, we proposed a novel consistent DKF based on quantized sensor communications. To alleviate the effect of biased observations, the eventtriggered observation update based DKF was presented with a tighter bound of MSE than that of the time-driven one by designing a proper threshold. Based on mild conditions, the boundedness of the MSE and the asymptotic unbiasedness of state estimate for both the proposed two distributed filters were proved.

In 0

k,N

where

¯ 1,1¯ = Θ k,N

N k+N¯ ∑ ∑

Θi1,j,,1k

i=1 j=k

¯ 1,2¯ = Θ k,N

N k+N¯ ∑ ∑

Θi1,j,,2k

(B.1)

i=1 j=k

¯ 2,2¯ = Θ k,N

N k+N¯ ∑ ∑

Θi2,j,,2k

i=1 j=k

Acknowledgments The authors would like to thank the editors and anonymous reviewers for their very constructive comments, which greatly improved the quality of this work.

(1) Necessity. If the reformulated system (5) is uniformly collec+ ¯ steps, then there tively observable ( after M ) exists N¯ ∈ N and

α > 0 such that

¯ 1,1¯ Θ k,N ¯ 1,2¯ )T (Θ k,N

¯ 1,2¯ Θ k,N

¯ 2,2¯ − α Ip Θ k,N

¯ − α In+p > 0, ∀k ≥ M.

12

X. He, W. Xue, X. Zhang et al. / Automatica 114 (2020) 108842

¯ 1,1¯ > 0, then in light of Schur Complement (Seber, 2007), Thus, Θ k,N

Appendix E. Proof of Lemma 4.3

(2) Sufficiency. If there exists N¯ ∈ N+ and α > 0, such that ¯ For k ≥ M, ¯ by (Θ ¯ 1,1¯ − α In > 0, for k ≥ M. ¯ 2,2¯ − α Ip ) − Θ k,N k,N

For the proof of (1), exploiting the matrix inverse formula on P˜ k,i directly yields the conclusion. Next, we consider the proof of (2). First, we prove there exists a constant positive definite ¯ such that Pk,i ≥ P. ¯ Consider P¯ k,i ≥ 1+θk,i Q¯ k−1 + Q˜ k−1 ≥ matrix P, θ

2 ,2

1,2

1 ,1

1,2

¯ (Θ ¯ ¯ − α Ip ) − (Θ ¯ ¯ ) T (Θ ¯ ¯ − α In )−1 Θ ¯ ¯ > 0. for k ≥ M, k,N k,N k,N k,N

1,2

1 ,1

1 ,2

¯ ¯ ) T (Θ ¯ ¯ − α In )−1 Θ ¯ ¯ > 0 and Schur Complement (Seber, (Θ k,N k,N k,N

(

¯ 2007), for k ≥ M,

¯ 1,1¯ Θ k,N ¯ 1,2¯ )T (Θ

¯ 1,2¯ Θ k,N ¯ 2,2¯ Θ

k,N

)

> α In+p holds. Thus, the

k,N

reformulated system (5) is uniformly collectively observable after ¯ steps. M

k,i

Q¯ k−1 + Q˜ k−1 ≥ Q ∗ > 0, where Q ∗ can be obtained by θ2 noting infk Qk ≥ Q > 0 and Assumption 2.3. According to (1) 1+θ2

∗

1 Q of Lemma 4.3, then we have P˜ k−,i1 ≤ 1+µ + HkT,i ∆R− k,i Hk,i ≤ Q∗ , k,i where Q∗ > 0 is obtained ( by employing the ) condition (3) of

−1

˜ −1 j∈N (k) ai,j (k)Pk,j

∑

Assumption 2.1. Recall Pk,i = Appendix C. Proof of Lemma 4.1 Here we utilize an inductive method for the proof. At the initial moment, under Assumption 2.1, E {(Xˆ 0,i − X0 )(Xˆ 0,i − X0 )T } ≤ P0,i . Suppose E {(Xˆ k−1,i − Xk−1 )(Xˆ k−1,i − Xk−1 )T } ≤ Pk−1,i . Recall e¯ k,i = X¯ k,i − Xk = Ak−1 ek−1,i − Duk−1 − ωk−1 . Since the estimation error ek−1,i is measurable to σ {Fk−2 , vk−1,i , i ∈ V }, and ωk−1 is independent from Fk−2 and vk−1,i , i ∈ V , we have E {ek−1,i ωkT√ −1 } = 0. T θk,i x + Similarly, it holds that E { u ω } = 0. Then, due to E { ( k − 1 k − 1 √ √y )( θk,i x + √y )T } ≥ 0, we have the inequality E {xyT + θk,i

θk,i

yxT } ≤ E {θk,i xxT +

1

θk,i

yyT }, ∀θk,i > 0. Then, we have E {¯ek,i e¯ Tk,i } ≤

(1 + θk,i )Ak−1 Pk−1,i ATk−1 + (1 +

)Q¯ k−1 + Q˜ k−1 . According to the T ¯ ¯ definition of Pk,i , E {¯ek,i e¯ k,i } ≤ Pk,i hold. At the update stage, there is e˜ k,i = X˜ k,i − Xk = (I − Kk,i Hk,i )e¯ k,i + Kk,i bk,i + Kk,i vk,i . Under Assumption 2.1 and the fact that e¯ k,i is measurable to Fk−1 , it follows that E {¯ek,i vkT,i } = E {E {¯ek,i vkT,i |Fk−1 }} = E {Ak−1 ek−1,i E {vkT,i | Fk−1 }}+E {E {wk−1 vkT,i |Fk−1 }} = E {wk−1 vkT,i } = 0. Similarly, it holds that E {bk,i vkT,i } = 0 by noting that bk,i is measurable to Fk−1 . Then, ∀µk,i > 0, we have E {˜ek,i e˜ Tk,i } ≤ (1 + µk,i )(I − Kk,i Hk,i )E {¯ek,i e¯ Tk,i }(I − 1+µk,i Kk,i Hk,i )T + Kk,i E {vk,i v T }K T + Kk,i E {bk,i bT }K T ≤ P˜ k,i . k,i

k,i

1

θk,i

µk,i

k,i

k,i

Recall the existence of quantization operation with respect to X˜ k,i + ξk,i , where ξk,i stands for the dithering noise vector. We denote the estimation error eˇ k,i := Xˇ k,i − Xk = e˜ k,i + ξk,i + ϑk,i , where ϑk,i is the quantization error vector of X˜ k,i + ξk,i . By Assumption 2.6, ξk,i + ϑk,i is independent of e˜ k,i . Then E {ˇek,i eˇ Tk,i } ≤ E {˜ek,i e˜ Tk,i } + E {(ξk,i + ϑk,i )(ξk,i + ϑk,i )T } ≤ P˜ k,i +

∆2i n¯ In¯ = Pˇ k,i + Pk∗,i + ∆2i n¯ In¯ , where Pk∗,i ∈ Rn¯ ׯn is the symmetric quantization error matrix of Pˇ k,i . It holds that Pk∗,i ≤ λmax (Pk∗,i )In¯ ≤ √( )2 ) ∑n¯ ∑n¯ ( ∗ ∆ n¯ ∗ ∥Pk,i ∥F In¯ := In¯ ≤ 2i In¯ , where ∥ · ∥F s=1 l=1 Pk,i (s, l) is the Frobenius norm, and Pk∗,i (s, l) is the (s, l)th element of Pk,i .

n¯ ∆i (2∆i +1) In¯ . Notice that ek,i = 2 ∑ −1 ˆXk,i − Xk = Pk,i ( ˜ ˇ a (k) P e + ai,i (k)P˜ k−,i1 e˜ k,i ), where i , j k , j k,j j∈Ni (k),j̸ =i ¯ n ∆ (2 ∆ + 1) i i P˜ k,j = Pˇ k,j + In¯ , if i ̸ = j, and j ∈ Ni . According to the 2

Thus, we have E {ˇek,i eˇ Tk,i } ≤ Pˇ k,i +

consistent estimate of Covariance Intersection (Niehsen, 2002), there is E {ek,i eTk,i } ≤ Pk,i . Therefore, the proof is finished.

1+µk,i

Bk,i

+µ , k,i

Kk∗,i

l

Due to sup k Qk ≤ Q¯ < ∞, there is a scalar ϖ > 0, such that QTl +s ≤ ϖ ATl +s PTl +s,i ATT +s . Then P¯ Tl +s+1,i = ATl +s PTl +s,i ATT +s + l

QTl +s ≤ (1 + ϖ )ATl +s PTl +s,i ATT +s . Let η = l (2) of this lemma holds.

1 1+ϖ

l

, then the conclusion

Appendix F. Proof of Proposition 4.1 Consider tr(P¯ k,i ), then we have tr(P¯ k,i ) = (1 + θk,i ) tr(Ak−1 1+θ Pk−1,i ATk−1 ) + θ k,i tr(Q¯ k−1 ) + tr(Q˜ k−1 ). Hence θk∗,i = arg minθk,i k,i

tr(P¯ k,i ) = arg minθk,i fk (θk,i ), where fk (θk,i ) = θk,i tr(Ak−1 Pk−1,i ATk−1 )

+

tr(Q¯ k−1 )

θk,i

, which is minimized if θk∗,i tr(Ak−1 Pk−1,i ATk−1 ) =

√

As a result, θk∗,i =

tr(Q¯ k−1 ) . tr(Ak−1 Pk−1,i ATk−1 )

and Ak−1 ̸ = 0, we have

θk∗,i

tr(Q¯ k−1 ) . θk∗,i

Since Pk−1,i > 0, Q¯ k−1 > 0

> 0.

Appendix G. Proof of Lemma 4.5 By Theorem 1 of Cattivelli (1) can be proved. ( ρ )(2010), ( ρ )k and ∑Sayed k−i mi k x For (2), we have δkk = δ x0 + i=0 δ . Due to ρ < δ < 1, δi ρ

( ρ )k

m

¯ i = δii = o(1), x0 → 0. Denote ρ¯ = δ ∈ (0, 1) and m ¯ k with then construct a sequence {¯xk } satisfying x¯ k+1 = ∑ ρ¯ x¯ k + m k k−i ¯ i, m x¯ 0 = 0. By (1), we have x¯ k = o(1). In light of x¯ k = ρ) ¯ ( i=0 δ

∑k ( ρ )k−i

mi

∑k

¯ i → 0 as k → ∞. Hence, xk = ¯ k−i m (ρ) ∑k o(δ ). For (3), consider xk kM −1 = ρ k kM −1 x0 + kM −1 i=0 ρ k−i mi . k M −1 Notice ρ k = o(1), then ∑k consider the convergence of the second term, namely, kM −1 i=0 ρ k−i mi . Due to mk = o( k1M ), we have ∑k ∑k ∑k kM −1 k−i kM −1 kM −1 i=0 ρ k−i mi = ) = i=0 o(ρ i=0 o( iM (k−i)M ) = iM ∑k 1 i=0 o( k ) = o(1), where the second equality is obtained by k−i ρ = o( (k−1i)M ) and the third equality is obtained by i(k − i) ≤ 2k . Thus, xk kM −1 = o(1), which means xk = o( kM1−1 ). i=0 k

δ

δi

=

i=0

Appendix H. Proofs of Lemmas 5.1, 5.3 and Proposition 5.1

1+µk,i

P¯ k−,i1 1+µk,i

1 + HkT,i ∆R− k,i Hk,i . where ∆Rk,i = Rk,i +

Bk,i . Substituting P˜ k−,i1 into (17), the conclusion of this lemma holds. µk,i

∆P˜ k,i ≜(I − Kk,i Hk,i )P¯ k,i (I − Kk,i Hk,i )T + Kk,i R˜ k,i KkT,i , =(Kk,i − Kk∗,i )(Hk,i P¯ k,i HkT,i + R˜ k,i )(Kk,i − Kk∗,i )T Rk,i

Q∗ , which means Pk,i ≥ Q∗−1 > 0. Consider the time sequence {Tl , l ∈ N}, which is the L-SS of {Ak , k ∈ N}. Under Assumption 2.2, there exists a scalar β , such that ATl +s PTl +s,i ATT +s ≥ β Q∗−1 > 0.

P˜ k,i yields P˜ k−,i1 =

By Lemma 4.1, it holds that P˜ k,i = (1 + µk,i )∆P˜ k,i , with

where R˜ k,i =

, then Pk−,i1 ≤

Proof of Lemma 5.1. Employing the matrix inverse formula on

Appendix D. Proof of Lemma 4.2

+ (I − Kk∗,i Hk,i )P¯ k,i ,

i

(D.1)

= P¯ k,i HkT,i (Hk,i P¯ k,i HkT,i + R˜ k,i )−1 . Thus,

it is seen from (D.1) that P˜ k,i is minimized in the sense of positive definiteness (i.e., tr(P˜ k,i ) is minimized) when Kk,i = Kk∗,i .

Proof of Lemma 5.3. If (17) is satisfied, we have P˜ k−,i1 = 1 HkT,i ∆R− k,i Hk,i .

P¯ k−,i1 1+µk,i

+

Thus, conclusion of Lemma 5.3 holds in this case. If the update event in (17) is not triggered, then P˜ k,i = P¯ k,i . Besides, according to the scheme, it follows that P˜ k−,i1 ≥ 1 HkT,i ∆R− k,i Hk,i − τ In .

P¯ k−,i1 1+µk,i

+

X. He, W. Xue, X. Zhang et al. / Automatica 114 (2020) 108842

Proof of Proposition 5.1. In light of Lemma 5.3, for τ = 0, P˜ k−,i1 ≥

P¯ k−,i1

1+µk,i

+

1 HkT,i ∆R− k,i Hk,i

= Pˇ k−,i1 , which means P˜ k,i ≤ Pˇ k,i ,

where Pˇ k,i corresponds to the observation update of Algorithm 1. By using the mathematical induction method, the proof of this proposition can be finished. References Bai, W., Xue, W., Huang, Y., & Fang, H. (2018). On extended state based Kalman filter design for a class of nonlinear time-varying uncertain systems. Science China. Information Sciences, 61(4), 042201. Bartholomew-Biggs, M. (2008). Nonlinear optimization with engineering applications. Springer Science & Business Media. Battistelli, G., & Chisci, L. (2014). Kullback–Leibler average, consensus on probability densities, and distributed state estimation with guaranteed stability. Automatica, 50(3), 707–718. Battistelli, G., & Chisci, L. (2016). Stability of consensus extended Kalman filter for distributed state estimation. Automatica, 68, 169–178. Caglayan, A. K., & Lancraft, R. E. (1983). A separated bias identification and state estimation algorithm for nonlinear systems. Automatica, 19(5), 561–570. Cai, Z., Queiroz, M. S. D., & Dawson, D. M. (2006). Robust adaptive asymptotic tracking of nonlinear systems with additive disturbance. IEEE Transactions on Automatic Control, 51(3), 524–529. Cattivelli, F. S., & Sayed, A. H. (2010). Diffusion strategies for distributed Kalman filtering and smoothing. IEEE Transactions on Automatic Control, 55(9), 2069–2084. Chow, Y. S., & Teicher, H. (2012). Probability theory: Independence, interchangeability, martingales. Springer Science & Business Media. Ding, D., Wang, Z., Dong, H., & Shu, H. (2012). Distributed H∞ state estimation with stochastic parameters and nonlinearities through sensor networks: The finite-horizon case. Automatica, 48(8), 1575–1585. Han, D., You, K., Xie, L., Wu, J., & Shi, L. (2015). Optimal parameter estimation under controlled communication over sensor networks. IEEE Transaction on Signal Processing, 63(24), 6473–6485. He, X., Hu, C., Hong, Y., Shi, L., & Fang, H. (2020). Distributed Kalman filters with state equality constraints: Time-based and event-triggered communications. IEEE Transactions on Automatic Control, 65(1), 28–43. He, X., Xue, W., & Fang, H. (2018). Consistent distributed state estimation with global observability over sensor network. Automatica, 92, 162–172. He, X., Xue, W., Zhang, X., & Fang, H. (2019). Distributed filtering for uncertain systems under switching sensor networks and quantized communications. arXiv https://arxiv.org/abs/1811.03438v3. Jazwinski, A. H. (1970). Stochastic processes and filtering theory (pp. 231–243). Academic Press. Julier, S. J., & Uhlmann, J. K. (1997). A non-divergent estimation algorithm in the presence of unknown correlations. In American control conference (pp. 2369–2373). Kar, S., & Moura, J. M. (2010). Distributed consensus algorithms in sensor networks: Quantized data and random link failures. IEEE Transactions on Signal Processing, 58(3), 1383–1400. Keller, J. Y., & Darouach, M. (1997). Optimal two-stage Kalman filter in the presence of random bias. Automatica, 33(9), 1745–1748. Khan, U. A., & Jadbabaie, A. (2014). Collaborative scalar-gain estimators for potentially unstable social dynamics with limited communication. Automatica, 50(7), 1909–1914. Khan, U. A., Kar, S., Jadbabaie, A., & Moura, J. M. F. (2010). On connectivity, observability, and stability in distributed estimation. In IEEE conference on decision and control (pp. 6639–6644). Liu, P., Tian, Y., & Zhang, Y. (2018). Distributed Kalman filtering with finite-time max-consensus protocol. IEEE Access, 6, 10795–10802. Liu, Q., Wang, Z., He, X., & Zhou, D. (2017). On Kalman-consensus filtering with random link failures over sensor networks. IEEE Transactions on Automatic Control, 63(8), 2701–2708. Niehsen, W. (2002). Information fusion based on fast covariance intersection filtering. In International conference on information fusion (pp. 901–904). Olfati-Saber, R. (2007). Distributed Kalman filtering for sensor networks. In Proceedings of the IEEE conference on decision and control (pp. 5492–5498). Orihuela, L., Millán, P., Vivas, C., & Rubio, F. R. (2016). Suboptimal distributed control and estimation: Application to a four coupled tanks system. International Journal of Systems Science, 47(8), 1755–1771. Reif, K., Gunther, S., Yaz, E., & Unbehauen, R. (1999). Stochastic stability of the discrete-time extended Kalman filter. IEEE Transactions on Automatic control, 44(4), 714–728.

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Xingkang He received the B.S. degree in School of Mathematics from Hefei University of Technology in 2013, and the Ph.D. degree in Academy of Mathematics and Systems Science, Chinese Academy of Sciences at Beijing in 2018. Since Oct. 2018, he has been a postdoctoral researcher at the Division of Decision and Control Systems in KTH Royal Institute of Technology. His research interests include estimation and control of networked systems, security of cyber–physical systems. He is a recipient of Best Paper Award in 2018 IEEE Data Driven Control and Learning Systems Conference.

Wenchao Xue received the B.S. degree in applied mathematics from Nankai University, Tianjin, China, in 2007, and the Ph.D. degree in control theory from the Academy of Mathematics and Systems Science (AMSS), Chinese Academy of Sciences (CAS), Beijing, in 2012. He is now an Associate Professor of Key Lab of System and Control, AMSS, CAS. He is an Associate Editor for IFAC journal of Control Engineering Practice. His research interests include nonlinear uncertain systems control, nonlinear uncertain systems filter and distributed filter.

Xiaocheng Zhang received the B.S. degree from Xiamen University, Xiamen, China, in 2016. He is currently working toward the Ph.D. degree in the Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China. His research interests include estimation of uncertain systems, bias estimation, and distributed state estimation.

Haitao Fang received his B.S. degree in probability and statistics and M.S. degree in applied mathematics in 1990 from the Peking University and 1993 from Tsinghua University, respectively. He received Ph.D. degree in applied mathematics in 1996 from the Peking University. From 1996–1998 he was a Postdoc at the Institute of Systems Science, Chinese Academy of Sciences (CAS), and joined the Institute as an Assistant Professor in 1998. He now is with the Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, CAS as a Full Professor. He currently serves as Deputy Editor-in-Chief of Journal of Control Theory and Applications and Associate Editor of Acta Mathematicae Applicatae Sinica. His research interests include dynamical behavior analysis of networked systems and distributed system estimation, optimization and control.