Distributed variance-constrained robust filtering with randomly occurring nonlinearities and missing measurements over sensor networks

Communicated by Dr. Derui Ding

Accepted Manuscript

Distributed variance-constrained robust filtering with randomly occurring nonlinearities and missing measurements over sensor networks Zhigong Wang, Dongyan Chen, Junhua Du PII: DOI: Reference:

S0925-2312(18)31213-X https://doi.org/10.1016/j.neucom.2018.10.025 NEUCOM 20047

To appear in:

Neurocomputing

Received date: Revised date: Accepted date:

28 May 2018 16 September 2018 7 October 2018

Please cite this article as: Zhigong Wang, Dongyan Chen, Junhua Du, Distributed variance-constrained robust filtering with randomly occurring nonlinearities and missing measurements over sensor networks, Neurocomputing (2018), doi: https://doi.org/10.1016/j.neucom.2018.10.025

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ACCEPTED MANUSCRIPT

Distributed variance-constrained robust filtering with randomly occurring nonlinearities and missing measurements over sensor networks

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Zhigong Wang a , Dongyan Chen a,∗ , Junhua Du b Department of Mathematics, Harbin University of Science and Technology, Harbin 150080, China College of Science, Qiqihar University, Qiqihar 161006, China

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Abstract

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This paper is concerned with the distributed variance-constrained robust filtering problem for a class of time-varying stochastic systems subject to both randomly occurring nonlinearities and missing measurements. The target plant is disturbed by the multiplicative noises, randomly occurring nonlinearities as well as additive noises. The phenomena of the randomly occurring nonlinearities and missing measurements are modeled by the Bernoulli distributed random variables with known occurrence probabilities. The available measurements of each sensor node and its neighbor nodes can be communicated based on the network topology structure. Attention is focused on the design of a new distributed variance-constrained robust filtering algorithm such that, in the simultaneous presence of the missing measurements, multiplicative noises and randomly occurring nonlinearities, an upper bound of the filtering error covariance is obtained via the solutions to two recursive matrix equations. Subsequently, the filter parameters are designed to minimize the obtained upper bound of the filtering error covariance. Furthermore, by utilizing the mathematical induction method, a sufficient condition is provided to guarantee the boundedness of the upper bound of the filtering error covariance. At last, we provide a numerical simulation to illustrate the effectiveness of distributed variance-constrained robust filtering method.

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Key words: Time-varying networked systems, Sensor networks, Distributed variance-constrained robust filtering, Missing measurements, Randomly occurring nonlinearities.

Introduction

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In the past few decades, the utilizations of sensor networks have been greatly improved due to the rapid developments of microelectronic technology. Accordingly, sensor network has been widely used in a great number of areas, including environmental monitoring, health care, military industry, power systems and wireless networks, see [1, 2]. As such, the distributed filtering and state estimation problems over sensor network have re-

Preprint submitted to Neurocomputing

ceived growing research attention, see [3, 4] and the references therein. Different from single sensor, the available measurements regarding to sensor networks at each sensor node can be communicated with its neighboring nodes according to certain topology structure. Hence, there is a need to better utilize the available information when designing the filtering algorithms [5,6]. So far, the distributed H∞ filtering algorithms have been designed in [7] for nonlinear systems by using a stochastic sampled-data approach and in [8] for polynomial

16 October 2018

ACCEPTED MANUSCRIPT where variance-constrained filtering schemes have been developed via the recursive matrix equation method. On the other hand, the nonlinearity is an essential characteristic in practical systems and there is a need to cope with the effects of the nonlinear disturbances [26–29]. In the networked environment, the nonlinear disturbances may undergo the random changes under certain form and intensity [30]. It is customary to call this network-induced nonlinear disturbance as randomly occurring nonlinearities [31–34]. For example, the phenomenon of the randomly occurring nonlinearities has been firstly introduced and modeled in [32] by the Bernoulli distributed random variable, and the synchronization method has been provided for complex networks with randomly occurring nonlinearities and mixed time-delays. In [33], the H∞ control method has been given for discrete time-varying systems with both randomly occurring nonlinearities and fading measurements. In [34], a new distributed state estimation algorithm has been developed for a class of dynamical systems over sensor networks with randomly occurring nonlinearities, where sufficient condition has been proposed to ensure the globally asymptotical stability of the estimation error in the sense of mean square. Nevertheless, it should be mentioned that most of the existing results concerning on the topic of distributed variance-constrained filtering can be applicable for linear time-varying systems or nonlinear time-invariant systems only. In addition, we need to face the limitation that effective yet proper performance analysis methods for the developed filtering algorithm are insufficient, which motivate the current research topic.

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nonlinear stochastic systems in terms of the parameterdependent linear matrix inequality mehtod. In [9], a distributed H∞ filtering method has been developed for linear system subject to intercommunication delays. To better characterize the parameter uncertainties and compensate induced effects, a new distributed state estimation scheme has been proposed in [10] for timevarying nonlinear systems with stochastic parameters, where the expressions of the estimator parameters have been obtained via the solutions to coupled backward recursive Riccati difference equations. Recently, a locally optimal distributed filtering method has been presented in [11] for time-varying stochastic systems with multiplicative noises and deception attacks over sensor networks. In addition, some new distributed event-triggered H∞ filtering and set-membership estimation methods have been given in [12–14] for complex systems over sensor networks.

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It is well known that the network-induced phenomena are inevitable in the networked communication transmissions [15–17]. Due to limited capacity of sensor nodes, the phenomenon of the missing measurements would be occurred when the measurement signals are collected. Thus, the filtering problems for dynamics systems with missing measurements have been widely investigated, see [18, 19]. To be specific, a distributed filtering problem has been studied in [20] for switched positive systems with missing measurements and randomly varying nonlinearities over sensor networks, where sufficient conditions have been given to attain the positivity of addressed system and the existence of desired filtering approach. In [21], a new distributed filtering method has been given for time-invariant nonlinear systems with uncertain observations including randomly occurring sensor saturations and successive packet dropouts. Regarding to the time-varying case, new distributed filtering algorithms have been developed in [22, 23] for time-varying stochastic systems, where the effects of randomly varying nonlinearities, quantization errors, successive packet dropouts and event-triggered mechanism onto the distributed filtering algorithm accuracy have been examined. Very recently, new distributed filtering problems have been addressed in [24,25] for linear time-varying stochastic systems subject to incomplete measurements over sensor networks,

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In view of the above discussions, the purpose of this paper is to propose a distributed robust filtering method for addressed time-varying systems with missing measurements, multiplicative noises and randomly occurring nonlinearities over sensor networks. The phenomena of missing measurements and randomly occurring nonlinearities are characterized by employing some Bernoulli distributed random variables. For the conducted topic, there are mainly three aspects of difficulties: i) we need to propose the recursion of the state covariance in the presence of the randomly occurring nonlinearities under certain assumption; ii) we need to comprehensively reveal

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ACCEPTED MANUSCRIPT matical induction method. The performance evaluation problem is discussed in section 4. A numerical example is provided in section 5 and conclusions are given in Section 6.

the effect from the missing measurements onto the estimation accuracy and depict the involved relationship; and iii) we need to provide proper performance analysis result to clearly reflect the proposed filtering method under minimization criterion. To deal with the above three difficulties/challenges, 1) the extensive stochastic analysis technique together with the assumption on the randomly occurring nonlinearities are used to provide the proper recursion of the state covariance; 2) the measurement information from adjacent nodes is used to design the distributed variance-constrained robust filter, and new upper bound of the filtering error covariance involved the occurrence probabilities of the missing measurements is given in terms of the solutions two recursive matrix equations; and 3) a boundedness analysis condition with respect to the upper bound of the filtering error covariance is given based on the mathematical induction approach.

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Notations. The notations used throughout the paper are standard unless otherwise noted. Rn denotes the n dimensional Euclidean space and Rm×n represents the set of all n × m real matrices. I denotes the identity matrix of compatible dimension. The notation X ≥ Y (X > Y ) means that X − Y is positive semi-definite (positive definite), where X and Y are symmetric matrices. X T represents the transpose of X and E{x} stands for the expectation of stochastic variable x. The shorthand diagn {Mk } denotes a block diagonal matrix with diagonal blocks being the matrices Mk . 1n ∈ Rn×n is a square matrix with all the elements being one. The symbol ⊗ denotes the Kronecker product. The symbol ◦ denotes the Hadamard product.

The main contributions of this paper can be highlighted as follows: (1) the system model considered is more comprehensive that includes the missing measurements, multiplicative noises as well as randomly occurring nonlinearities simultaneously; (2) a proper method is presented to deal with randomly occurring nonlinearities, then an upper bound for the filtering error covariance is obtained by solving two recursive matrix equations and the distributed filter parameters are designed to minimize the trace of such an upper bound at each time step, where a new matrix simplification technique is used to deal with the sparse matrix; and (3) the performance analysis of the proposed distributed robust filtering algorithm is carried out, i.e., a sufficient criterion is given to ensure the boundedness of the obtained upper bound, where certain assumptions about the system parameters are made. Finally, we provide the simulations to illustrate the usefulness of the proposed distributed variance-constrained robust filtering scheme.

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2

Problem Formulation

In the paper, we consider a sensor network with n sensor nodes. The topology is described by a given directed graph G = (V, E, H) of order n having the set of nodes V = {1, 2 . . . , n}, the set of edges E ∈ V × V, and a weighted adjacency matrix H = [hij ] with nonnegative adjacency elements hij . An edge of G is denoted by the pair (i, j). The adjacency elements associated with the edges of the graph are positive, i.e., hij > 0 ⇐⇒ (i, j) ∈ E, which represents that there exists the information transmission from sensors j to i. Furthermore, suppose that hii = 1 for all i ∈ V. The set of neighbors of node i plus the node itself are denoted by Ni = {j ∈ V : hij > 0}.

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In this paper, we consider the following time-varying stochastic systems with randomly occurring nonlinearities

The rest of this paper is organized as follows: Section 2 formulates the distributed variance-constrained robust filtering problem for the nonlinear time-varying system subject to missing measurements, multiplicative noises as well as randomly occurring nonlinearities. In section 3, a sufficient condition for the existence of the desired robust filtering scheme is given in terms of the mathe-

xk+1 = (Ak + αk A¯k )xk + ξk f (xk ) + Bk ωk ,

(1)

measured by n sensors, the output of i-th sensor node is described by: yi,k = γi,k (Ci,k + βi,k C¯i,k )xk + νi,k , i = 1, 2, · · · , n (2) where xk ∈ Rnx is the system state, the initial value is x0 3

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¯ (xk ). +ξ¯ f (xk ) − f (ˆ xi,k|k ) + (ξk − ξ)f

with mean-value x ¯0 , yi,k ∈ Rny is the measurement output by the sensor i, ωk ∈ Rnω is zero mean process noise with covariance Qk > 0, and νi,k ∈ Rnν is zero mean measurement noise with covariance Ri,k > 0. The multiplicative noises αk ∈ R and βi,k ∈ R (i = 1, 2, · · · , n) have zero mean and unity variances. Ak , A¯k Bk , Ci,k and C¯i,k are real-valued time-varying matrices of appropriate dimensions.

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Similarly, for node i (i = 1, 2, · · · , N ), the filtering error can be obtained x ˜i,k+1|k+1 = x ˜i,k+1|k −

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hij Kij,k+1 (˜ γj,k+1 Cj,k+1 xk+1

j∈Ni

+¯ γ Cj,k+1 x ˜j,k+1|k + γj,k+1 βj,k+1 C¯j,k+1 xk+1 +νj,k+1 ),

The Bernoulli distributed random variables γi,k and ξk , which are employed to characterize the phenomena of the missing measurements and randomly occurring nonlinearities, have the following statistical properties:

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where γ˜j,k+1 = γj,k+1 − γ¯ .

For simplicity, we introduce the following notations: x ˜k+1|k = coln {˜ xi,k+1|k }, x ˜k|k = coln {˜ xi,k|k },

Prob{γi,k = 1} = E{γi,k } = γ¯ ,

x ˆk|k = coln {ˆ xi,k|k }, ~xk = coln {xk }, Ak = diagn {Ak }, A¯k = diagn {A¯k },

Prob{γi,k = 0} = 1 − γ¯ ,

Bk = diagn {Bk }, Ck = diagn {Ci,k }, C¯k = diagn {C¯i,k }, α ~ k = diagn {αk I},

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¯ Prob{ξk = 1} = E{ξk } = ξ, ¯ Prob{ξk = 0} = 1 − ξ,

β~k = diagn {βi,k I}, Γk = diagn {γi,k I}, ˜ k = diagn {˜ Γ γi,k I}, ω ~ k = coln {ωk },

where γ¯ ∈ [0, 1] and ξ¯ ∈ [0, 1] are known constants. In the sequel, we assume that αk , ξk , γi,k , βi,k , ωk and νi,k are mutually independent.

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Assumption 1 The nonlinear function f (xk ) satisfies the following Lipschitz condition: kf (x) − f (y)k ≤ lkx − yk,

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where l is a known constant.

In this paper, we design the following robust distributed filter:

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¯ (ˆ x ˆi,k+1|k = Ak x ˆi,k|k + ξf x ), X i,k|k x ˆi,k+1|k+1 = x ˆi,k+1|k + hij Kij,k+1 (yj,k+1

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Kk = {Kij,k }n×n , Hi = diag{hi1 I, . . . , hin I}.

Then, it follows from the above notations that we can obtain the one-step prediction error x ˜k+1|k = xk+1 − x ˆk+1|k and the filtering error x ˜k+1|k+1 = xk+1 −ˆ xk+1|k+1 by the following compact form: ¯ k − fˆk ) x ˜k+1|k = Ak x ˜k|k + α ~ k A¯k ~xk + Bk ω ~ k + ξ(f ¯ k, +(ξk − ξ)f (7) n X ˜ k+1 Ck+1 ~xk+1 x ˜k+1|k+1 = x ˜k+1|k − Ei Kk+1 Hi Γ

(3)

j∈Ni

−¯ γ Cj,k+1 x ˆj,k+1|k ),

~νk = coln {νi,k }, fk = coln {f (xk )}, fˆk = coln {f (ˆ xi,k|k )}, Ei = diag{0, . . . , 0, I, 0, . . . , 0}, | {z } | {z }

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i=1

~k+1 C¯k+1 ~xk+1 +¯ γ Ck+1 x ˜k+1|k + Γk+1 β
+~νk+1 .

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where x ˆi,k|k is the filter at time k with the initial value x ˆi,0|0 , x ˆi,k+1|k is the one-step prediction at time k, and Kij,k+1 is the filter parameter to be determined.

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Remark 1 For the addressed distributed filtering problem, the information available on each node is not only from itself but also from its neighbors according to the given topology. As such, the innovation P ¯ Cj,k+1 x ˆj,k+1|k ) is employed j∈Ni hij Kij,k+1 (yj,k+1 − γ in (4), where the information from the neighboring nodes are utilized with hope to improve the filtering algorithm accuracy.

Let us denote the one-step prediction error and the filtering error about node i as x ˜i,k+1|k = xk+1 − x ˆi,k+1|k and x ˜i,k+1|k+1 = xk+1 − x ˆi,k+1|k+1 , respectively. According to (1) and (3) for node i (i = 1, 2, · · · , N ), the one-step prediction error is calculated by x ˜i,k+1|k = Ak x ˜i,k|k + αk A¯k xk + Bk ωk

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ACCEPTED MANUSCRIPT Lemma 2 For matrices M , N , X and L with appropriate dimensions, one has:

Remark 2 As it is well known, many practical engineering systems are commonly influenced by the additive nonlinear disturbances induced by the environmental circumstances. Such unpredictable disturbance may be subject to random abrupt variations. Hence, in this paper, the phenomenon of the randomly occurring nonlinearities is considered during the system modeling, and the available probability information ξ¯ is utilized in (3).

∂tr{M X T } ∂tr{XM } = M, = MT , ∂X ∂X ∂tr{M XN } ∂{M X T N } = MT NT , = N M, ∂X ∂X T ∂tr{XM X } = 2XM, ∂X ∂tr{M XN X T L} = M T N T XLT + LM XN. ∂X

Remark 3 As justified in the introduction, the missing measurements, randomly occurring nonlinearities and different sources of noises exist commonly in real-world networked systems, the addressed time-varying systems over sensor networks are of practical significance and the corresponding distributed filtering problem is worth investigating. The main objective of this paper are: 1) for the addressed networked system over sensor networks, which includes the missing measurements, randomly occurring nonlinearities and different sources of noises in a unified framework, we endeavour to propose a new distributed robust filter; 2) based on the available information of the adjacent nodes, we aim to propose a new filtering method to deal with the aforementioned phenomena and the variance constraint; and 3) under certain assumption, we are looking for proposing a proper criterion to guarantee the boundedness of the upper bound of the filtering error covariance.

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Moreover, we have

∂tr{(M XN )P (M XN )T } = 2M T M XN P N T , ∂X with P being any symmetric matrix.

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Lemma 3 For the addressed system (1) and ε > 0, the state covariance matrix Xk+1 = E{~xk+1 ~xTk+1 } satisfies the following inequality: Xk+1 ≤ (1 + ε)Ak Xk ATk + A¯k Xk A¯Tk + Bk Qk BkT ¯ 2 tr(Xk )I, +(1 + ε−1 )ξl

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Design of Distributed Filtering Algorithm

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In this section, we aim to propose a new distributed variance-constrained robust filtering algorithm via the mathematical induction method and provide the expression form of the filter parameters. To begin, let us introduce the following useful Lemmas.

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2  E{b1 } E{b1 b2 }    E{b2 b1 } E{b22 } T E{BAB } =   .. ..  . .   E{bn b1 } E{bn b2 }

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 Qk    Qk Qk =   .  ..   Qk

 Qk . . . Qk    Qk . . . Qk   .. . . ..  . .  .   Qk . . . Qk

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Proof : It is easy to verify that the system (1) can be rewritten as the following compact form: ~xk+1 = (Ak + α ~ k A¯k )~xk + ξk fk + Bk ω ~ k.

Lemma 1 [35] Letting A = [aij ]n×n be a real-valued matrix and B = diag{b1 , b2 , . . . , bn } be a diagonal stochastic matrix, then we have 

(9)

(10)

Then, it follows from the definition of the state covariance matrix that

 · · · E{b1 bn }    · · · E{b2 bn }   ◦ A,  .. ..  . .   · · · E{bn bn }

 Xk+1 = E Ak ~xk ~xTk ATk + α ~ k A¯k ~xk ~xTk A¯Tk α ~ kT + ξk2 fk fkT +Bk ω ~ kω ~ kT BkT + Ak ~xk fkT ξk + ξk fk ~xTk ATk . (11)

By utilizing the elementary inequality xy T + yxT ≤ εxxT + ε−1 yy T , we obtain

where ◦ is the Hadamard product.

Ak ~xk fkT ξk + ξk fk ~xTk ATk 5

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(12)

the filtering error covariance Pk+1|k+1 satisfy

with ε being a positive scalar. Then, it follows from (11)(12) that

Pk+1|k ≤ Ξk+1|k , Pk+1|k+1 ≤ Ξk+1|k+1 .

 Xk+1 ≤ E (1 + ε)Ak ~xk ~xTk ATk + α ~ k A¯k ~xk ~xTk A¯Tk α ~ kT +(1 + ε−1 )ξk2 fk fkT + Bk ω ~ kω ~ kT BkT = (1 + ε)Ak Xk ATk + A¯k Xk A¯Tk  ¯ fk f T + Bk Qk B T . +(1 + ε−1 )ξE k k

Proof : To prove this theorem, we use the mathematical induction approach. By considering the initial condition, we have P0|0 ≤ Ξ0|0 . Assuming Pk|k ≤ Ξk|k , then we need to prove that Pk+1|k+1 ≤ Ξk+1|k+1 .

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The one-step prediction error covariance Pk+1|k can be calculated as follows:

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≤ kfk k I =

fkT fk I.

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Next, it is not difficult to verify that fk fkT

 Pk+1|k = E x ˜k+1|k x ˜Tk+1|k  = E Ak x ˜k|k x ˜Tk|k ATk + α ~ k A¯k ~xk ~xTk A¯Tk α ~ kT +Bk ω ~ kω ~ T B T + ξ¯2 (fk − fˆk )(fk − fˆk )T

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By using the Assumption 1, we obtain

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  E fkT fk ≤ l2 E ~xTk ~xk = l2 tr(Xk ).

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¯ 2 fk f T + Ak x +(ξk − ξ) ˜k|k (fk − fˆk )T ξ¯ k ¯ k − fˆk )˜ +ξ(f xT AT .

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k|k

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Thus, the state covariance matrix Xk+1 satisfies (9), which completes the proof.

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By utilizing the elementary inequality, we can obtain ¯ k − fˆk )˜ Ak x ˜k|k (fk − fˆk )T ξ¯ + ξ(f xTk|k ATk ≤ εAk x ˜k|k x ˜T AT + ε−1 ξ¯2 (fk − fˆk )(fk − fˆk )T .

Subsequently, based on the one-step prediction error (7), the filtering error (8) and Lemma 3, the upper bounds of the one-step prediction error covariance Pk+1|k and the filtering error covariance Pk+1|k+1 can be given.

k|k

k

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It follows from (20) and (21) that

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Theorem 1 For the addressed system (1) with (2), if the following two recursive matrix equations with the initial condition P0|0 ≤ Ξ0|0 Ξk+1|k = (1 + ε)Ak Ξk|k ATk + A¯k Xk A¯Tk +(1 + ε−1 )ξ¯2 l2 tr(Ξk|k )I + Bk Qk B T

(19)

Pk+1|k ≤ (1 + ε)Ak Pk|k ATk + A¯k Xk A¯Tk + Bk Qk BkT  +(1 + ε−1 )ξ¯2 E (fk − fˆk )(fk − fˆk )T  ¯ − ξ)E ¯ +ξ(1 fk fkT , (22) where Pk|k = E{˜ xk|k x ˜Tk|k }. Applying the properties of matrix operation, we obtain

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k

¯ − ξ)l ¯ 2 tr(Xk )I, +ξ(1

+

(16)

(fk − fˆk )(fk − fˆk )T ≤ k(fk − fˆk )k2 I = (fk − fˆk )T (fk − fˆk )I,

Gk+1 Ψ1k+1 GTk+1 ,(17)

fk fkT

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Ξk+1|k+1 = Ψ0k+1 Ξk+1|k (Ψ0k+1 )T

have the positive-definite solutions, where

 E (fk − fˆk )T (fk − fˆk )  T ≤ l2 E x ˜k|k x ˜k|k = l2 tr(Pk|k ),  E fkT fk  ≤ l2 E ~xTk ~xk = l2 tr(Xk ).

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T Ψ1k+1 = Υ1 ◦ (Ck+1 Xk+1 Ck+1 ) + Rk+1 T +Υ2 ◦ (C¯k+1 Xk+1 C¯ ),

Gk+1 = −

k+1

Ei Kk+1 Hi ,

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Υ1 = I ⊗ (¯ γ (1 − γ¯ )1n ), Υ2 = I ⊗ (¯ γ 1n ),

Rk+1 = diag{R1,k+1 , R2,k+1 , . . . , Rn,k+1 },

≤ kfk k I =

fkT fk I.

(23) (24)

According to Assumption 1, we have

Ψ0k+1 = I + γ¯ Gk+1 Ck+1 ,

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Substituting (23)-(26) into (22) leads to

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Pk+1|k ≤ (1 + ε)Ak Pk|k ATk + A¯k Xk A¯Tk + Bk Qk BkT

then the one-step prediction error covariance Pk+1|k and

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ACCEPTED MANUSCRIPT ¯ − ξ)l ¯ 2 tr(Xk )I. +(1 + ε−1 )ξ¯2 l2 tr(Pk|k )I + ξ(1

and Ξi,k+1|k to be the i-th row of the block matrix Ξk+1|k . Based on the method in [23], define

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T Ni,k+1 = γ¯ Ξi,k+1|k Ck+1 Hi ,

Then, we can conclude that Pk+1|k ≤ Ξk+1|k .

T Mi,k+1 = Hi (¯ γ 2 Ck+1 Ξk+1|k Ck+1 + Ψ1k+1 )Hi .

On the other hand, by noticing the compact form Gk+1 in (18), we have

¯i,k+1 and K ¯ i,k+1 by Subsequently, it is easy to obtain N simply removing the b-th (b ∈ / Ni ) column block from the matrices Ni,k+1 and Ki,k+1 , respectively. Moreover, we ¯ i,k+1 by simply removing both the b-th row can have M and b-th column block from Mi,k+1 when b ∈ / Ni . Based on the above notations, the obtained upper bound of the filtering error covariance can be minimized by properly designing the filter parameters.

x ˜k+1|k+1 (28)

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˜ k+1 Ck+1 ~xk+1 = (I + γ¯ Gk+1 Ck+1 )˜ xk+1|k + Gk+1 Γ +Gk+1 Γk+1 β~k+1 C¯k+1 ~xk+1 + Gk+1~νk+1 .

Now, according to the properties of matrix operation, let us calculate the filtering error covariance Pk+1|k+1 as follows:

Theorem 2 For the addressed system (1) with measurements (2), the upper bound of the filtering error covariance Ξk+1|k+1 can be minimized at each time step if the gain matrix of the filter (3) and (4) is given as follows:

k+1 k+1

k+1

k+1

k+1

From Lemma 1, one has

Kij,k+1 =

   0

, hij = 0

  (N ¯i M ¯ −1 )] , hij 6= 0 i

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where (∗)] extracts the corresponding submatrix from matrix ‘∗’ associated with the parameter Kij,k+1 .

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Pk+1|k+1  =E x ˜k+1|k+1 x ˜Tk+1|k+1  = (I + γ¯ Gk+1 Ck+1 )E x ˜k+1|k x ˜Tk+1|k (I + γ¯ Gk+1 Ck+1 )T  T ˜T ˜ k+1 Ck+1 ~xk+1 ~xTk+1 Ck+1 +Gk+1 E Γ Γk+1 GTk+1  T +Gk+1 E Γk+1 β~k+1 C¯k+1 ~xk+1 ~xTk+1 C¯k+1  T ×β~ T ΓT GT + Gk+1 E ~νk+1~ν T G . (29)

= (I + γ¯ Gk+1 Ck+1 )Pk+1|k (I + γ¯ Gk+1 Ck+1 )T  +Gk+1 (I ⊗ (¯ γ (1 − γ¯ )1n )) ◦ (Ck+1 E{~xk+1 ~xTk+1 }   T ×Ck+1 ) GTk+1 + Gk+1 (I ⊗ (¯ γ 1n )) ◦ (C¯k+1  T T T ×E{~xk+1 ~xk+1 }C¯k+1 ) Gk+1 + Gk+1 Rk+1 GTk+1

Proof : Taking the trace for the both sides of (17) yields

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 tr Ξk+1|k+1  = tr (I + γ¯ Gk+1 Ck+1 )Ξk+1|k (I + γ¯ Gk+1 Ck+1 )T  +tr Gk+1 Ψ1k+1 GTk+1 . (32)

PT

= (I + γ¯ Gk+1 Ck+1 )Pk+1|k (I + γ¯ Gk+1 Ck+1 )T  T +Gk+1 (I ⊗ (¯ γ (1 − γ¯ )1n )) ◦ (Ck+1 Xk+1 Ck+1 )  T T +(I ⊗ (¯ γ 1n )) ◦ (C¯k+1 Xk+1 C¯k+1 ) + Rk+1 Gk+1 , (30)

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The first term in the right-hand side of (32) can be rewritten into the following expression:

 tr (I + γ¯ Gk+1 Ck+1 )Ξk+1|k (I + γ¯ Gk+1 Ck+1 )T   = tr Ξk+1|k + 2tr γ¯ Gk+1 Ck+1 Ξk+1|k  T +tr γ¯ 2 Gk+1 Ck+1 Ξk+1|k Ck+1 GTk+1 . (33)

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where Rk+1 = diag{R1,k+1 , R2,k+1 , . . . , Rn,k+1 }. Finally, from (27) and (30), it is easy to verify that Pk+1|k+1 ≤ Ξk+1|k+1 , then the proof is now complete. In the sequel, we will design the proper filter parameters and then minimize the upper bound Ξk+1|k+1 at each step. It should be noted that we need to face the sparsity problem of the matrix because of the characteristics of sensor networks. To proceed, let us define Ki,k+1 to be the i-th row of the block matrix Kk+1 , i.e.,

Resorting to the properties of trace, we have  tr Ei Kk Hi ΦHjT KkT EjT  = tr EjT Ei Kk Hi ΦHjT KkT = 0, i 6= j

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for an arbitrary matrix Φ with appropriate dimension. Noticing the definition of Gk+1 and using (34), it is ob-

Ki,k+1 = [Ki1,k+1 , . . . , Kin,k+1 ],

7

ACCEPTED MANUSCRIPT vious that

Subsequently, it is easy to see that

 T tr γ¯ 2 Gk+1 Ck+1 Ξk+1|k Ck+1 GTk+1 n  X T T = tr γ¯ 2 Ei Kk+1 Hi Ck+1 Ξk+1|k Ck+1 HiT Kk+1 EiT .

n X j=1

If b ∈ / Ni , we have Mib,k+1 = 0 and Nb,k+1 = 0, thus the above equation is always true. Furthermore, for b ∈ / Ni , we can choose the filter parameter Kib,k+1 = 0 because the local sensor cannot receive any information from its non-neighbor nodes. Then, we obtain

i=1

With regard to the second term in the right-hand side of (32), we have

¯ i,k+1 M ¯ i,k+1 = N ¯i,k+1 , for i = 1, 2, . . . , n K

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 tr Gk+1 Ψ1k+1 GTk+1 n X T = tr Ei Kk+1 Hi Ψ1k+1 HiT Kk+1 EiT .

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i=1

¯ i,k+1 is positive definite, we Noticing that the matrix M have

Further, taking the partial derivation of the trace of the matrix Ξk+1|k+1 with respect to the gain parameters Kk+1 and letting the partial derivation be zero, we arrive at

i=1

i=1

T ×Ξk+1|k Ck+1 + Ψ1k+1 )Hi

(36)

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Then, we can obtain

T Ei Kk+1 Hi (¯ γ 2 Ck+1 Ξk+1|k Ck+1 + Ψ1k+1 )Hi

i=1 n X

= γ¯

i=1

T Ei Ξk+1|k Ck+1 Hi ,

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that is,

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n X

Therefore, the parameter Kij,k+1 can be obtained by selecting the corresponding column block matrix in the ¯i,k+1 M ¯ −1 , which completes the proof of this matrix N i,k+1 theorem.

M

= 0.

¯ i,k+1 = N ¯i,k+1 M ¯ −1 , for i = 1, 2, . . . , n. K i,k+1

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∂tr{Ξk+1|k+1 } ∂Kk+1 n n X X T Ei Ξk+1|k Ck+1 Ei Kk+1 Hi (¯ γ 2 Ck+1 = −2¯ γ Hi + 2

Kij,k+1 Mib,k+1 = Nb,k+1 , for b ∈ Ni .

Ki,k+1 Mi,k+1 = Ni,k+1 , i = 1, 2, . . . , n.

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Now, we aim to obtain the Ki,k+1 by solving the above equation. Note that

Remark 4 Noting the sparsity of the communication topology, i.e. some rows or columns of H are zero, we cannot ensure that Hi is positive definite. Hence, there is a noticeable difficulty to obtain the value of filter parameters Kij,k+1 . In fact, the diagonal entries of matrix Hi are nonzero when the corresponding sensor is within the neighbor set of sensor i, otherwise the diagonal entries of matrix Hi are zero when the corresponding sensor is not in the neighboring set of sensor i. In other words, hij > 0 if j ∈ Ni , otherwise hij = 0 if j ∈ / Ni . Hence, we can see that Hi will be likely to rank deficient, which means that Mi,k+1 maybe also rank deficient. Motivated by the matrix simplification technique in [23], we remove the zero rows and columns to guarantee the positive definiteness ¯ i,k+1 , and present the explicit expresof the simplified M sion of Kij,k+1 . 4

Boundedness Analysis

In this section, we are ready to provide a sufficient condition to ensure the boundedness of the obtained upper bound. In what follows, we introduce the following Assumption and Lemma to facilitate further derivations.

Hi = diag{hi1 I, hi2 I, . . . , hin I} with hij = 0 if j ∈ / Ni . Then, we can obtain the simpli¯ i,k+1 and N ¯i,k+1 directly based on above fied matrices M mentioned method. Then, we can partition the matrices by Mi,k+1 = {Mab,k+1 }n×n and Ni,k+1 = {Nb,k+1 }1×n .

Assumption 2 There are positive real constants a, a ¯, ad , a ¯d , b, ¯b, c, c¯, cd , c¯d , q, q¯ and r, r¯ such that the ¯ ¯ ¯ ¯ ¯ ¯ ¯ 8

ACCEPTED MANUSCRIPT we can obtain from (16) that

following conditions:

κI ≤ Ξk+1|k ≤ κ ¯ I, ¯

aI ≤ Ak ATk ≤ a ¯I, ad I ≤ A¯k A¯Tk ≤ a ¯d I, ¯ ¯ bI ≤ Bk BkT ≤ ¯bI, cI ≤ Ck CkT ≤ c¯I, ¯ ¯ cd I ≤ C¯k C¯kT ≤ c¯d I, qI ≤ Qk ≤ q¯I, ¯ ¯ rI ≤ Rk ≤ r¯I. ¯

where κ and κ ¯ are defined below (39). In the following, ¯ it is not difficult to see that Ψ0k+1 Ξk+1|k (Ψ0k+1 )T

Remark 5 The Assumption 2 can be found in [25], which is utilized to depict the boundedness requirements of the system parameters. Based on this assumption and the following Lemma on the state covariance, we can obtain the desired boundedness analysis results with respect to the upper bound of the filtering error covariance.

= (I + γ¯ Gk+1 Ck+1 )Ξk+1|k (I + γ¯ Gk+1 Ck+1 )T T ≤ 2Ξk+1|k + 2¯ γ 2 Gk+1 Ck+1 Ξk+1|k Ck+1 GTk+1 .

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T Ξk+1|k+1 ≤ 2Ξk+1|k + 2¯ γ 2 Gk+1 Ck+1 Ξk+1|k Ck+1 GTk+1

+Gk+1 Ψ1k+1 GTk+1 .

(1 + ε)¯ a+a ¯ + (1 + ε

−1

 )ξl p¯x + ¯b¯ q ≤ p¯x ¯2

T Ξck+1|k = γ¯ 2 Ck+1 Ξk+1|k Ck+1

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then we can know

is true, then we can verify that px I ≤ Xk ≤ tr(Xk )I ≤ ¯ p¯x I holds for each k > 0 with px > 0 satisfying px ≤ bq. ¯¯ ¯ ¯ Motivated by the results in [25], we have the following condition which can guarantee the boundedness of the obtained upper bound.

M

Ξck+1|k ≥ γ¯ 2 cκI. ¯¯

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Theorem 3 For the system (1) and (2) with filter (3)(4), according to Assumption 2, if there exists a scalar ε > 0 satisfying

where

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κ = (1 + ε)apΞ + a px + bq + (1 + ε ¯ ¯¯ ¯ ¯¯ ¯ − ξ)l ¯ 2 px , ¯ +ξ(1   ¯ λ = γ¯ (1 − γ¯ )c + γ¯ cd px + r, ¯ ¯ ¯ ¯

−1

Taking the partial derivation of the trace of the matrix Ξk+1|k+1 with respect to the gain Gk+1 and setting the partial derivation be zero, we have ∂tr{Ξk+1|k+1 } ∂Gk+1 T T = 2¯ γ Ξk+1|k Ck+1 + 2¯ γ 2 Gk+1 Ck+1 Ξk+1|k Ck+1 = 0.

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It is easy to obtain that

2¯ cκ ¯ γ¯ 2 c¯κ ¯ , + ρ= λ cκ ¯¯ κ ¯ = (1 + ε)¯ ap¯Ξ + a ¯d p¯x + ¯b¯ q + (1 + ε−1 )ξ¯2 l2 p¯Ξ ¯ − ξ)l ¯ 2 p¯x , +ξ(1 d

(43)

+2Gk+1 Ψ1k+1

(39)

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(2 + ρ)¯ κ ≤ p¯Ξ ,

(42)

For the sake of simplicity, denoting

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d

(41)

Substituting (41) into (17) leads to

Lemma 4 Let X0 ≤ tr(X0 )I ≤ p¯x I, where p¯x > 0 is a given scalar. For the addressed system (1), if there exists a scalar ε > 0 and 

(40)

T Gk+1 = (−¯ γ Ξk+1|k Ck+1 )(Ξck+1|k + Ψ1k+1 )−1 .

(45)

Due to the fact that Ξck+1|k and Ψ1k+1 are positive definite matrices, it can be derived that

¯2 2

)ξ l pΞ ¯

(Ξck+1|k + Ψ1k+1 )−1 Ξck+1|k (Ξck+1|k + Ψ1k+1 )−1 = [Ξck+1|k + 2Ψ1k+1 + Ψ1k+1 (Ξck+1|k )−1 Ψ1k+1 ]−1 ≤ (Ξck+1|k )−1 ,

and p¯Ξ is a known positive scalar, then Ξk|k ≤ p¯Ξ I holds for each k ≥ 1.

and

Proof : In order to prove the boundedness, we suppose that pΞ I ≤ Ξk|k ≤ tr(Ξk|k )I ≤ p¯Ξ I holds. Then, ¯

(Ψ1k+1 + Ξck+1|k )−1 Ψ1k+1 (Ψ1k+1 + Ξck+1|k )−1

9

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According to Assumption 2, we know Ψ1k+1 ≥ λI. Substituting (45)-(47) into (42) results in Ξk+1|k+1 ≤ 2Ξk+1|k + 2Gk+1 Ξck+1|k GTk+1 + Gk+1 Ψ1k+1 GTk+1

T ≤ 2Ξk+1|k + 2¯ γ 2 Ξk+1|k Ck+1 (Ξck+1|k )−1 Ck+1 Ξk+1|k T +¯ γ 2 Ξk+1|k Ck+1 (Ψ1k+1 )−1 Ck+1 Ξk+1|k .

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≤ (Ψ1k+1 )−1 .

the distributed variance-constrained robust filtering problems with Round-Robin protocol and deception attacks become concern, the main difficulties induced by the RoundRobin protocol and deception attacks are how to obtain the explicit forms of the recursion of the filtering error covariance and its upper bound, which would be appeared in the near future. On the other hand, we should point out that a new boundedness analysis method with respect to the obtained upper bound of estimation covariance has been given in Theorem 3, which depends on certain assumptions concerning on the system parameters. Additional efforts will be made to further reduce the conservatism of the proposed analysis approach.

Finally, based on the above conclusions, we can obtain 5 Ξk+1|k+1 ≤ (2 + ρ)Ξk+1|k ≤ (2 + ρ)¯ κI ≤ p¯Ξ I.

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An Illustrative Example

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In this section, a simulation example is presented to illustrate the effectiveness and feasibility of the proposed distributed robust filtering method.

Therefore, via the mathematical induction approach, it can be concluded that Ξk|k < p¯Ξ I holds for each k ≥ 1, which completes the proof.

1

2

4

3

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M

Remark 6 It is worthwhile to mention that the available information concerning on the addressed system, filtering method and given communication topology has been clearly reflected in Theorem 3, where an analysis method with respect to the existence of ε and p¯Ξ guaranteeing (39) has been provided. During the implementation, the corresponding bounds of the system parameters, the occurrence probabilities of randomly occurring nonlinearities and missing measurements, the communication topology and the covariances of different sources noises could be available to the time-varying filter design by means of the parameter identifications. Hence, it would be easy to check the existence of (ε, p¯Ξ ) satisfying (39) by using the Matlab software tools. Thus, it’s very convenient to test the boundedness of the obtained upper bound of the filtering error covariance.

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Fig. 1. Topology of sensor network.

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We consider a sensor network with n = 4 sensor nodes, the network topology is depicted in Fig. 1. To be more specific, the directed graph can be described as G = (V, E, H). In addition, the set of nodes V, the set of edges E and the weighted adjacency matrix H are represented as follows:

Remark 7 It should be noted that the network-induced phenomena are commonly occurred when the information is transmitted by the communication networks. Recently, some newly efficient control methods have been given in [36, 37] to handle the phenomena of the RoundRobin scheduling protocol and deception attacks, where the characteristics of the involved two mentioned factors have been comprehensively taken into account. When

V = {1, 2, 3, 4},

E = {(1, 1), (2, 1), (2, 2), (3, 1), (3, 2), (3, 3), (4, 3), (4, 4)},

10

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0 0 0   1 0 0 .  1 1 0   0 11

    2 2.8     x ¯0 =   , x ˆ10 =  , 2 1.5     3 3 ˆ30 =   , x ˆ20 =   , x 3 2    2  x ˆ40 =   , Ξ1,0|0 = 20I2 , 2.5

Consider the time-varying nonlinear system (1)-(2) with following parameters: 

Ξ2,0|0 = 25I2 , Ξ3,0|0 = 30I2 , Ξ4,0|0 = 35I2 .

To quantify the estimation accuracy, we use the mean square error (MSE) given as follows: MSE(k) =

AN US M

where S is the number of runs. Note that the argumentation method has been used in the paper, hence the above MSE formula has been employed to quantify the whole filtering algorithm accuracy.

f (xk ) = 

+



1.5x2k

−

sin(x1k x2k )

10

Actual state Node 1 Node 2 Node 3 Node 4

8 6 4 2 0 −2 −4 −6 −8

sin(x1k x2k ) 

−10



CE

1  0.8xk

PT

The nonlinear function f (xk ) is chosen as 

S n 1 XX E{˜ xTi,k|k x ˜i,k|k }, S i=1 k=1

ED

 −0.44 + 0.09sin(0.12πk) −0.05   Ak =  , 0.3 −0.35    0.5sin(0.65k) −0.05  A¯k =  , −0.3 −0.15   1.4   Bk =  , 1.1     C1,k = 0.32 0.42 , C2,k = 0.35 0.4 ,     C3,k = 0.34 0.45 , C4,k = 0.35 0.4 ,     C¯1,k = 0.83 0.62 , C¯2,k = 0.83 0.75 ,     ¯ ¯ C3,k = 0.85 0.74 , C4,k = 0.85 0.73 .

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1   1 H=  1   0



The state x1k and its estimates



5

10

15

20

25

Time(k)

Fig. 2. The state x1k and its estimates.

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 T where xk = x1k x2k is the system state and Lipschitz constant l = 1.5. Suppose that the process noise ωk obeys the Gaussian distribution with zero mean value and covariance Qk = 0.3, and the measurement noise νi,k (i = 1, 2, 3, 4) obeys the Gaussian distribution with zero mean value and covariance R1,k = 0.2, R2,k = 0.4, R3,k = 0.3 and R4,k = 0.1. For simulation, let ε = 0.6, ξk = 0.11, γ1,k = 0.9, γ2,k = 0.8, γ3,k = 0.83 and γ4,k =

Based on the Theorem 1, the upper bound of filtering error covariance can be obtained and the filter parameters can be calculated. The corresponding simulation results are presented in Figs. 2-7. It can be seen from the simulation results that the performance of the algorithm is quite good. Among them, Figs. 2-3 describe the trajectories of the system state xk and the corresponding

11

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10

5 Actual state Node 1 Node 2 Node 3 Node 4

6 4 2 0 −2 −4 −6

3

1 0 −1 −2 −3 −4

5

10

15

20

−5

25

Time(k)

5

3

1 2 3 4

The estimation error variances of x1k

The estimation error of x1k

15

20

2 1 0 −1 −2 −3

20

25

4

Node Node Node Node

1 2 3 4

3.5 3

2.5 2

1.5 1

0.5 0

25

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Time(k)

M

−4 15

4.5

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Node Node Node Node

4

10

10

Fig. 5. The estimation error of x2k .

5

5

5

Time(k)

Fig. 3. The state x2k and its estimates.

−5

1 2 3 4

2

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−8 −10

Node Node Node Node

4

The estimation error of x2k

The state x2k and its estimates

8

5

10

15

20

25

Time(k)

Fig. 6. The estimation error variances of x1k .

estimates at each node. Figs. 4-5 depict the trajectories of estimation errors at each node. The estimation error covariances of all sensor nodes are presented in Figs. 67, which manifest that the fluctuate of estimation error is very small, so the performance of the proposed distributed robust filtering algorithm is good.

vided in Figs. 2-9, we can conclude that the proposed distributed variance-constrained robust filtering algorithm is effective because we have made great efforts to compensate the effects from the missing measurements and randomly occurring nonlinearities onto the whole estimation performance.

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Fig. 4. The estimation error of x1k .

In addition, in order to better illustrate the estimation accuracy of the proposed robust filtering approach, we have plotted the relationship between the upper bound of filtering error covariance and the MSE in Figs. 8-9 under different missing probabilities (Case I: γ1,k = 0.9, γ2,k = 0.8, γ3,k = 0.83 and γ4,k = 0.87; Case II: γ1,k = 0.23, γ2,k = 0.34, γ3,k = 0.15 and γ4,k = 0.26), where the upper bound keeps above the MSE at each time instant. Based on the simulations pro-

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6

Conclusions

In this paper, the distributed variance-constrained robust filtering problem has been investigated for a class of time-varying stochastic system subject to missing measurements, multiplicative noises and randomly occurring nonlinearities. A distributed robust filter has been designed and an optimal upper bound of the filtering error covariance has been constructed, and

12

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5 Node Node Node Node

4

1 2 3 4

MSE(Case I) Upper bound(Case I) MSE(Case II) Upper bound(Case II)

4.5 4

3.5

3.5

3

3

2.5

2.5

2

2

1.5

1.5

1

1

0.5

0.5

0

5

10

15

20

0

25

0

Time(k)

5

10

15

20

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The estimation error variances of x2k

5 4.5

25

Time(k)

Fig. 7. The estimation error variances of x2k .

Fig. 9. MSE of x2k and the upper bound.

Acknowledgments 5

4 3.5 3 2.5 2 1.5

0

0

5

10

15

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25

References [1] P. Braca, R. Goldhahn, G. Ferri, K. D. Lepage, Distributed

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Time(k)

M

1 0.5

This work was supported in part by the Natural Science Foundation of Heilongjiang Province of China under Grant A2018007, the University Nursing Program for Young Scholars with Creative Talents in Heilongjiang Province of China under grant UNPYSCT-2016029, the Fundamental Research Funds in Heilongjiang Provincial Universities of China under Grant 135209250, and the Educational Research Project of Qiqihar University of China under Grant 2017028.

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ACCEPTED MANUSCRIPT Junhua Du received the B.Sc. degree in Mathematics from Qiqihar University, Qiqihar, China, in 1999, and M.Sc. degree in Fundamental Mathematics from Harbin University of Science and Technology, Harbin, China, in 2009. She is an Associate Professor with the Department of Mathematics of Qiqihar University, Qiqihar, China. She has published around 20 journal papers. Her current research interests include modeling and analysis of nonlinear stochastic systems, robust control and filtering, and algebra.

nonlinearities and multiple stochastic disturbances, Journal of Physics A, 42 (13), article135101, 2009. [33] D. Ding, Z. Wang, J. Lam, B. Shen, Finite-horizon H∞

control for discrete time-varying systems with randomly occurring nonlinearities and fading measurements, IEEE Transactions on Automatic Control, 60 (9) (2015) 2488–2493.

[34] J. Liang, Z. Wang, B. Shen, X. Liu, Distributed state estimation in sensor networks with randomly occurring nonlinearities subject to time-delays, ACM Transactions on Sensor Networks, 9 (1) (2012) 1–20. [35] Y. Theodor, U. Shaked, Robust discrete-time minimum-

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variance filtering, IEEE Transactions on Signal Processing, 44 (2) (1996) 181–189. [36] D. Ding, Z. Wang, Q-L. Han, G. Wei, Neural-networkbased output-feedback control under round-robin scheduling protocels,

IEEE

Transactions

on

Cybernetics,

2018,

doi:10.1109/TCYB.2018.2827037. [37] D. Ding, Z. Wang, Q-L. Han, G. Wei, Security control for discrete-time stochastic nonlinear systems subject to Cybernetics:

systems,

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deception attacks, IEEE Transactions on systems, Man, and 2018,

doi:10.1109/TCYB.2018.2827037.

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Zhigong Wang received the B.Sc. degree in Departments of Mathematics from Jiamusi University, Jiamusi, China, in 2016. He is working toward the M.Sc. degree in Operations Science and Control Theory with Department of Mathematics, Harbin University of Science and Technology, Harbin, China. His current research interests include distributed filtering for time-varying systems over sensor networks.

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Dongyan Chen received the B.Sc. degree in Mathematics from Northeast Normal University, Changchun, China, in 1985, M.Sc. degree in Operational Research from Jilin University, Changchun, China, in 1988, and the Ph.D. degree in Aerocraft Design from Harbin Institute of Technology, Harbin, China, in 2000. She is now a Professor and PhD Supervisor with the Department of Mathematics, Harbin University of Science and Technology, Harbin, China. Her current research interests include robust control, time-delay systems, optimization approach, system optimization and supply chain management.

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