Recursive resilient filtering for nonlinear stochastic systems with packet disorders

Recursive Resilient Filtering for Nonlinear Stochastic Systems with Packet Disorders

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Recursive Resilient Filtering for Nonlinear Stochastic Systems with Packet Disorders Dan Liu, Zidong Wang, Yurong Liu, Fuad E. Alsaadi PII: DOI: Reference:

S0016-0032(20)30105-8 https://doi.org/10.1016/j.jfranklin.2020.02.021 FI 4433

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

25 December 2019 28 January 2020 15 February 2020

Please cite this article as: Dan Liu, Zidong Wang, Yurong Liu, Fuad E. Alsaadi, Recursive Resilient Filtering for Nonlinear Stochastic Systems with Packet Disorders, Journal of the Franklin Institute (2020), doi: https://doi.org/10.1016/j.jfranklin.2020.02.021

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Recursive Resilient Filtering for Nonlinear Stochastic Systems with Packet Disorders Dan Liua , Zidong Wangb , Yurong Liua,c,∗ and Fuad E. Alsaadid

Abstract In this paper, a new resilient filtering problem is studied for a class of nonlinear stochastic systems with packet disorders. The underlying system is quite comprehensive that considers both deterministic and stochastic nonlinearities. The phenomenon of packet disorders takes place in the sensor-to-filter channel as a result of the limited capability of the communication network. The random transmission delay, which is the main cause for the packet disorders, is modeled as a bounded random variable obeying a known probability distribution and its influence on the filter performance is examined. Furthermore, the resilient issue of the proposed recursive filter against random fluctuations of the filter gain is thoroughly studied. The purpose of this paper is to develop a resilient filter for the addressed nonlinear stochastic systems such that, in the presence of both stochastic nonlinearities and packet disorders, an upper bound of the filtering error covariance is ensured and then locally minimized through adequate design of the filter gains. Finally, a simulation example is given to illustrate the usefulness of the theoretical results. Index Terms Recursive filtering, resilient filtering, random transmission delays, nonlinear stochastic systems, packet disorders.

I. I NTRODUCTION For decades, the Kalman filtering (KF) scheme has proven to be one of the most popular state estimation algorithms with successful applications in many fields such as signal processing and control engineering, see [1], [24], [29], [33], [34], [48] and the references therein. Note that the traditional KF algorithm works well for linear systems with exactly known system parameters and noise statistics. In the real world, however, the system model might be subject to nonlinearities/uncertainties and, in this case, the KF algorithm might not be effective. As such, considerable effort has been devoted to develop alternative filtering techniques with specific aims to accommodate the system nonlinearities and/or parameter uncertainties. Examples include, but not limited to, H∞ filtering [3], [36], [40], mixed H2 /H∞ This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (DF-199135-1441). The authors, therefore, gratefully acknowledge DSR technical and financial support. The work was also supported by National Natural Science Foundation of China under Grants 61773017, 61873148, 61873230, 61933007 and 11671008, the Yangzhou University International Academic Exchange Fund of China, the Royal Society of the UK, and the Alexander von Humboldt Foundation of Germany. a Department of Mathematics, Yangzhou University, Yangzhou 225002, China. b Department of Computer Science, Brunel University London, Uxbridge, Middlesex, UB8 3PH, United Kingdom. c School of Mathematics and Physics, Yancheng Institute of Technology, Yancheng 224051, China. d Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia. ∗ Corresponding author at: Department of Mathematics, Yangzhou University, Yangzhou 225002, China. E-mail addresses: [email protected] (Z. Wang), [email protected] (Y. Liu)

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filtering [14], [41], set-membership fuzzy filtering [43], extended Kalman filtering (EKF) [7], [15], [39], and unscented Kalman filtering algorithms [16], [32]. Among others, the linearization-based EKF algorithm has been recognized to be particularly effective in handling nonlinear filtering problems as long as the degree of nonlinearities is not high. Also, the EKF is notably convenient-to-design and easy-to-implement, and has therefore attracted an ever-increasing research interest in practical engineering. For example, the dynamical behavior of EKF has been analyzed in [31] for the state estimation problem of nonlinear deterministic systems. A robust EKF has been proposed in [12] for uncertain systems with sum quadratic constraints. In [8], the EKF problem has been revisited to account for stochastic nonlinearities as well as multiple missing measurements. Communication delays are known to be unavoidable during data transmissions owing mainly to the limited bandwidth and the communication capacity of the network channels. Transmission delays, which are typically random as a result from the unpredictable fluctuations of the networked environment, are likely to cause performance degradation or even undesirable divergence, see e.g. [5], [6], [17], [26], [27]. Consequently, some research attention has been attracted to the investigation problem on random transmission delays (RTDs) with hope to attenuate their negative impact on the overall system performance, see e.g. [21], [30], [37]. A straightforward consequence of the RTDs is the phenomenon of packet disorders that arises quite frequently in engineering practice, where a data packet sent earlier (later) might be arrived later (earlier) at the received end. In the presence of packet disorders, the out-of-order packets may carry outdated information which, if directly utilized as the input to the corresponding filter, would give rise to misleading (biased or even erroneous) state estimates that jeopardize the overall system performance. Until now, only a limited number of results have been reported in the literature concerning the packet disorder phenomenon, see e.g. [19], [20], [47], where the RTDs have been described by a Markov chain with precisely known transition probability matrix (TPM). In engineering practice, however, it is by no means a trivial task to acquire/identify the transition probabilities (between different modes) due to unaffordable cost in a typical resource-constrained environment. As such, it is of primary importance to look at alternative characterization of the RTDs that would then facilitate the design of the filters capable of rejecting/attenuating the effects from the RTD-induced packet disorders. Nonlinearities are ubiquitous in almost all practical systems that might result in undesirable system dynamics (e.g. oscillation, chaos or even instability), and much work has therefore been done on the filtering problems for various nonlinear systems, see e.g. [10], [13], [28], [46] and references therein. Most nonlinearity in existing literature has been implicitly assumed to occur in a deterministic way. This assumption is, however, not always true as the nonlinearity may occur randomly due to reasons like sudden environmental changes and intermittent network congestions, which brings about the so-called stochastic nonlinearity. Up to now, plenty of results have been available on the filtering/control problems with respect to stochastic nonlinearities (see e.g. [11], [23]). Unfortunately, when it comes to the packet disorders, the research results have been scattered, and this motivates us to shorten such a gap in this paper. On another research frontier, the resilience issue of filters has received an ever-increasing research interest, where the so-called resilience is actually referred to as the insensitivity to the possible variations/drifts of the filter parameters during the implementation. It is well recognized that controllers/filters are fragile

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to even tiny parameter perturbations that are often inevitable for various reasons such as the rounding error caused by numerical calculations. To date, considerable research effort has been made on the design of resilient filters and a rich body of results have been obtained, see e.g. [22], [25], [44], [45]. For instance, a robust non-fragile filter has been designed in [44] for linear systems with norm-bounded uncertainties by solving certain algebraic Riccati equations. A resilient approach has been put forward in [22] to design the distributed filter for time-varying systems subject to stochastic nonlinearities and sensor degradation. Summarizing the above discussion, in this paper, we endeavor to design a resilient filter for a class of nonlinear stochastic systems with packet disorders. The addressed problem seems to be nontrivial due primarily to the following substantial difficulties: 1) how to select an appropriate way to characterize the RTDs; 2) how to design a proper filter that takes into account the influences brought by the gain variations and RTD-induced packet disorders; 3) how to obtain a local minimized upper bound for the filtering error covariance for a class of nonlinear stochastic systems subject to the stochastic nonlinearities and packet disorders. The main contributions of this paper are that: 1) the system model under consideration is quite general that takes into account stochastic nonlinearity and packet disorders; 2) the RTDs in the sensor-to-filter channel are characterized by a random variable with known probability distribution that can be acquired from statistical tests; 3) a novel resilient filter is, for the first time, purposely constructed that is capable of resisting both gain variations and packet disorders; and 4) the proposed resilient filter is ensured to have an upper bound (on the filtering error covariance) that is subsequently minimized by properly choosing the filter gain. Notation: The notations are quite standard throughout this paper. For a matrix A, AT , A−1 , tr{A} and λmax (A) denote, respectively, the transpose, the inverse, the trace and the maximum eigenvalue of the matrix A. I represents the identity matrix of compatible dimension. E{x} denotes the expectation of stochastic variable x. For an arbitrary real number a, the floor function bac equals the largest integer no larger than a. II. P ROBLEM F ORMULATION Consider a class of discrete-time systems with stochastic nonlinearity: ( xh+1 = f (xh ) + g(xh , ξh ) + Bh wh , yh = Ch xh + Dh vh ,

(1)

where xh ∈ Rnx is the state of system, yh ∈ Rny is the measurement output, ξh is a zero-mean Gaussian noise, Bh , Ch and Dh are known matrices with appropriate dimensions, wh ∈ Rnw and vh ∈ Rny are the zero-mean white noise sequences with covariances Qh > 0 and Rh > 0, respectively. Both wh and vh are assumed to be distributed in bounded domains. f (xh ) : Rnx → Rnx is a known nonlinear function that is continuously differentiable. The initial value x0 of the system state is a random variable with mean x¯0 and covariance P0|0 . Throughout the paper, the random variables x0 , ξh , wh and vh are mutually independent. The stochastic nonlinearity g(xh , ξh ) : Rnx × Rnx → Rnx with g(0, ξh ) = 0 satisfies  E g(xh , ξh ) xh = 0,  E g(xh , ξh )g(xa , ξa )T xh = 0, h 6= a,

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(2)

m=1

where r > 0 is a known positive integer, dimensions.

Sensor

yh

Ψm h

and

Γm h

(m = 1, 2, . . . , r) are known matrices with appropriate

Communication Network

y% h

Filter

Fig. 1: Diagram of filtering problem. The system structure is shown in Fig. 1 where, at the time instant h, the measurement from the sensor is yh while the measurement received by the filter is y˜h . Such a difference is due mainly to the transmission delay which occurs randomly during the transmission of the measurement through a communication network with limited bandwidth. In this paper, a random variable τh is introduced to represent the RTD which satisfies Prob{τh = j} = pj , j = 0, 1, · · · , q, at the time instant h, where q is a known positive integer and pj (j = 0, 1, · · · , q) are known constants P satisfying 0 ≤ pj ≤ 1 and qj=0 pj = 1. Accordingly, the actual measurement y˜h received by the filter at the time instant h is ( y˜h = yh−τh , (3) y˜s = φs , s = −q, −q + 1, · · · , 0, where φs (s = −q, −q + 1, · · · , 0) are the initial values. Defining the Kronecker delta function δ(·, ·) as ( 1, τh = j, δ(τh , j) , 0, τh = 6 j,

the actual measurement y˜h received by the filter can be rewritten as y˜h =

q X

δ(τh , j)yh−j .

(4)

j=0

Remark 1: Generally, a data packet transmitted through the communication network consists of the measurement and the time stamp for signal processing purposes. Note that the packets might not be timestamped so as to save limited bandwidth [35] or simply because the network is unacknowledged. In this paper, the time-stamp information is not required for dealing with the RTD-based packet disorders. Instead,

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the transmission delay is treated as a random delay whose occurrence obeys a probability distribution law that can be determined by statistical experiments. In addition, we only consider the case where only one packet can be received by the filter at each time instant in this paper. Remark 2: In most literature concerning the filtering problem with RTDs [18], the Markov process has been used to characterize the RTDs whose transition probabilities are assumed to be known beforehand. Such an assumption is rather restrictive as the computation of the transition probabilities requires a great deal of resources and this limits the application scope. In this paper, the transmission delays have been naturally modeled as independent and identically distributed random variables obeying probability distributions that can be acquired via relatively convenient statistical experiments. Remark 3: To facilitate the later analysis, a novel stochastic Kronecker delta function has been introduced to express the actual filter input (3) as (4), which is vital in the subsequent derivation of the main results. It is known from (4) that the measurement received by the filter is effectively yh−j for τh = j (j = 0, 1, . . . , q). The recursive filter is designed in the following form: (5)

xˆh+1|h = f (ˆ xh|h ), xˆh+1|h+1 = xˆh+1|h + (Gh+1 + Λh+1 )


× y˜h+1 − Ch+1−d xˆh+1−d|h−d ,

(6)

where xˆh|h ∈ Rnx is the estimate of the state xh at the time instant h with xˆ0|0 = x¯0 and xˆh+1|h ∈ Rnx is the one-step prediction of xh at the time instant h. Gh+1 is the filter gain to be designed. Λh+1 reflects the random perturbation on the estimator gain that results possibly from computation/implementation error. Furthermore, Λh+1 is assumed to have zero mean and a bounded second moment, namely, E{Λh } = 0,

E{Λh ΛTh } ≤ σI,

where σ is a positive scalar. Moreover, d is an integer satisfying  1  b¯ τ c, if τ¯ − b¯ τc < , 2 d=  b¯ τ c + 1, otherwise, where τ¯ , E{τh+1 } =

q P

j=0

jpj and b¯ τ c represents the biggest integer no bigger than τ¯.

(7)

Remark 4: In this paper, the data packets are transmitted through the communication network without having to be equipped with time stamps. In the presence of RTD-induced packet disorders, the actual sent time-instant for the measurement received by the filter at the time instant h cannot be exactly determined. An effective way that has been adopted in this paper is to characterize the filter input based on the probability distribution of RTDs. Clearly, for its realizability, a filter should not contain any random variable (e.g. τh ), and therefore we utilize the expectation of the random variable, i.e. τ¯. As a subscript representing time instant in a discrete-time system, τ¯ must be an integer and, for this reason, τ¯ is rounded to d as shown in (7) by utilizing the floor function. On the basis of the above discussion, the innovation of the filter is constructed as the form of (6) which will be shown to compensate the impacts from packet disorders.

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The aim of this paper is to design a resilient filter of the form (5)-(6) such that an upper bound of the filtering error covariance is obtained, namely, there exists a sequence of positive-definite matrices Πh+1|h+1 satisfying E{(xh+1 − xˆh+1|h+1 )(xh+1 − xˆh+1|h+1 )T } ≤ Πh+1|h+1 . Moreover, such an upper bound is locally minimized at each time instant by appropriately designing the filter gain Gh+1 . III. M AIN R ESULTS In this section, we aim to present an upper bound of the filtering error covariance and then minimize such an upper bound by designing a proper filter gain. To begin with, we first give the following lemmas. Lemma 1: For two real vectors a, b and a scalar ε > 0, the following inequality holds: abT + baT ≤ εaaT + ε−1 bbT . Lemma 2: [38] For given matrices L, B, C and D with appropriate dimensions satisfying CC T ≤ I, let M be a symmetric positive definite matrix and  be an arbitrary positive scalar such that −1 I −DM DT > 0. Then, the following inequality holds: (L + BCD)M (L + BCD)T ≤L(M −1 − DT D)−1 LT + −1 BB T . In what follows, we denote the one-step prediction error and the filtering error as eh+1|h , xh+1 − xˆh+1|h , eh+1|h+1 , xh+1 − xˆh+1|h+1 , respectively. Subtracting (5) from (1) yields eh+1|h = f (xh ) − f (ˆ xh|h ) + g(xh , ξh ) + Bh wh . Expanding f (xh ) in a Taylor series around xˆh|h , one has f (xh ) = f (ˆ xh|h ) + Fh eh|h + o(|eh|h |),

(8)

where

∂f (x) |x=ˆxh|h ∂x and o(|eh|h |) represents the high-order terms of the Taylor series expansion, which can be written as the following form [2], [42]: Fh ,

o(|eh|h |) = Sh Θh Lh eh|h ,

(9)

where Sh is a problem-dependent scaling matrix, Lh is adopted to tune the filter with an extra degree of freedom, and Θh is an unknown time-varying matrix accounting for the linearization error that satisfies Θh ΘTh ≤ I. In view of (8)-(9), the prediction error can be rewritten as follows: eh+1|h = (Fh + Sh Θh Lh )eh|h + g(xh , ξh ) + Bh wh .

(10)

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Subtracting (6) from (1) leads to eh+1|h+1 = eh+1|h − (Gh+1 + Λh+1 )[˜ yh+1 − Ch+1−d xˆh+1−d|h−d ]   = I − δ(τh+1 )(Gh+1 + Λh+1 )Ch+1 eh+1|h − (Gh+1 X q + Λh+1 ) δ(τh+1 , j)Ch+1−j eh+1−j|h−j j=1

q

+

X

δ(τh+1 , j)Ch+1−j xˆh+1−j|h−j +

j=0

q X

δ(τh+1 , j)

j=0



× Dh+1−j vh+1−j − Ch+1−d xˆh+1−d|h−d .

(11)

Next, on the basis of (10) and (11), the one-step prediction error covariance and the filtering error covariance is given in the following lemmas. Lemma 3: The one-step prediction error covariance Ph+1|h can be recursively computed as Ph+1|h = (Fh + Sh Θh Lh )Ph|h (Fh + Sh Θh Lh )T r X
T m T + Ψm h tr E{xh xh }Γh + Bh Qh Bh . m=1

Proof: In light of (2) and (10), we have

Ph+1|h = E{eh+1|h eTh+1|h } n  = E (Fh + Sh Θh Lh )eh|h + g(xh , ξh ) + Bh wh  T o × (Fh + Sh Θh Lh )eh|h + g(xh , ξh ) + Bh wh = (Fh + Sh Θh Lh )Ph|h (Fh + Sh Θh Lh )T r X
T m T Ψm + h tr E{xh xh }Γh + Bh Qh Bh . m=1

The proof is complete. Lemma 4: The filtering error covariance Ph+1|h+1 is given by

P h+1|h+1 n  = E I − δ(τh+1 )(Gh+1 + Λh+1 )Ch+1 Ph+1|h  T o × I − δ(τh+1 )(Gh+1 + Λh+1 )Ch+1 n + E (Gh+1 + Λh+1 )(A1,h+1 + A2,h+1 + A3,h+1 o + A4,h+1 )(Gh+1 + Λh+1 )T − C1,h+1 − CT1,h+1 − C2,h+1 − CT2,h+1 − C3,h+1 − CT3,h+1 + C4,h+1 n + CT4,h+1 + E (Gh+1 + Λh+1 )(C5,h+1 + CT5,h+1 + C6,h+1 + CT6,h+1 − C7,h+1 − CT7,h+1 + C8,h+1

(12)

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where

+ CT8h+1 − C9,h+1 − CT9,h+1 − C10,h+1 − CT10,h+1 ) o × (Gh+1 + Λh+1 )T , A1,h+1 ,

q q X X

δ(τh+1 , j)δ(τh+1 , l)Ch+1−j

j=1 l=1

A2,h+1

T × eh+1−j|h−j eTh+1−l|h−l Ch+1−l , q q XX , δ(τh+1 , j)δ(τh+1 , l)Ch+1−j j=0 l=0

A3,h+1

T × xˆh+1−j|h−j xˆTh+1−l|h−l Ch+1−l , q q XX , δ(τh+1 , j)δ(τh+1 , l)Dh+1−j j=0 l=0

T T × vh+1−j vh+1−l Dh+1−l ,

T A4,h+1 , Ch+1−d xˆh+1−d|h−d xˆTh+1−d|h−d Ch+1−d , X q  C1,h+1 , E δ(τh+1 , j) I − δ(τh+1 )(Gh+1 j=1

 T + Λh+1 )Ch+1 eh+1|h eTh+1−j|h−j Ch+1−j  × (Gh+1 + Λh+1 )T ,

C2,h+1

X q  δ(τh+1 , j) I − δ(τh+1 )(Gh+1 ,E j=0

 T + Λh+1 )Ch+1 eh+1|h xˆTh+1−j|h−j Ch+1−j  T × (Gh+1 + Λh+1 ) ,

C3,h+1 , E

X q j=0

C4,h+1

 δ(τh+1 , j) I − δ(τh+1 )(Gh+1

 T T + Λh+1 )Ch+1 eh+1|h vh+1−j Dh+1−j  × (Gh+1 + Λh+1 )T ,    , E I − δ(τh+1 )(Gh+1 + Λh+1 )Ch+1 eh+1|h  T T T × xˆh+1−d|h−d Ch+1−d (Gh+1 + Λh+1 ) ,

C5,h+1 ,

q q X X

δ(τh+1 , j)δ(τh+1 , l)Ch+1−j

j=1 l=0

C6,h+1

T × eh+1−j|h−j xˆTh+1−l|h−l Ch+1−l , q q XX , δ(τh+1 , j)δ(τh+1 , l)Ch+1−j j=1 l=0

(13)

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C7,h+1

T T × eh+1−j|h−j vh+1−l Dh+1−l , q X δ(τh+1 , j)Ch+1−j eh+1−j|h−j , j=1

C8,h+1

T × xˆTh+1−d|h−d Ch+1−d , q q XX , δ(τh+1 , j)δ(τh+1 , l)Ch+1−j j=0 l=0

C9,h+1

T T × xˆh+1−j|h−j vh+1−l Dh+1−l , q X , δ(τh+1 , j)Ch+1−j xˆh+1−j|h−j j=0

C10,h+1

T , × xˆTh+1−d|h−d Ch+1−d q X δ(τh+1 , j)Dh+1−j vh+1−j , j=0

T × xˆTh+1−d|h−d Ch+1−d .

Proof: According to (11), (13) can be readily derived and the proof is therefore omitted. Subsequently, the following theorem provides an upper bound of the filtering error covariance and the filter gain is then designed to minimize such an upper bound. Theorem 1: Consider the one-step prediction error covariance Ph+1|h and the filtering error covariance Ph+1|h+1 given in (12) and (13). Let γh and εi (i = 0, 1, . . . , 10) be positive scalars. If the following two Riccati-like difference equations: −1 Πh+1|h = Fh (Πh|h − γh LTh Lh )−1 FhT + γh−1 Sh ShT r n X m Ψh tr (1 + ε0 )Πh|h + (1 + ε−1 xh|h + 0 )ˆ m=1

 o × xˆTh|h Γm + Bh Qh BhT , h

(14)

Πh+1|h+1 = ε¯1 (Πh+1|h − p0 Gh+1 Ch+1 Πh+1|h − p0 Πh+1|h T × Ch+1 GTh+1 ) + Gh+1 Φh+1 GTh+1

+ σλmax {Φh+1 }I

(15)

with initial condition Π0|0 = P0|0 > 0 have positive-definite solutions Πh+1|h and Πh+1|h+1 such that the following constraint γh−1 I − Lh Πh|h LTh > 0 is satisfied, then with the filter gain Gh+1 given by T Gh+1 = ε¯1 p0 Πh+1|h Ch+1 Φ−1 h+1

where ε¯1 , 1 + ε1 + ε2 + ε3 + ε4 , ε¯2 , 1 + ε−1 1 + ε5 + ε6 + ε7 ,

(16)

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−1 ε¯3 , 1 + ε−1 2 + ε5 + ε8 + ε9 , −1 −1 ε¯4 , 1 + ε−1 3 + ε6 + ε8 + ε10 , −1 −1 −1 ε¯5 , 1 + ε−1 4 + ε7 + ε9 + ε10 , q X T pj Ch+1−j Φh+1 , ε¯1 p0 Ch+1 Πh+1|h Ch+1 + ε¯2 j=1 q

T × Πh+1−j|h−j Ch+1−j + ε¯3

X

pj Ch+1−j

j=0

×

T xˆh+1−j|h−j xˆTh+1−j|h−j Ch+1−j

×

T Dh+1−j Rh+1−j Dh+1−j

+ ε¯4

q X

pj

j=0

+ ε¯5 Ch+1−d

T , × xˆh+1−d|h−d xˆTh+1−d|h−d Ch+1−d

the matrix Πh+1|h+1 is an upper bound for Ph+1|h+1 , that is, Ph+1|h+1 ≤ Πh+1|h+1 . Moreover, the upper bound Πh+1|h+1 is minimized by the filter gain Gh+1 given by (16). Proof: The mathematical induction is utilized for the proof. It is known from the initial value that P0|0 ≤ Π0|0 is ensured. By assuming that Ph|h ≤ Πh|h , what we need to do is to prove that Ph+1|h+1 ≤ Πh+1|h+1 . To begin with, let us prove Ph+1|h ≤ Πh+1|h . Applying Lemma 2 to the first term on the right-hand-side of (12), one has (Fh + Sh Θh Lh )Ph|h (Fh + Sh Θh Lh )T −1 ≤Fh (Ph|h − γh LTh Lh )−1 FhT + γh−1 Sh ShT .

With Lemma 1 and the fact that xh = eh|h + xˆh|h , we have  E{xh xTh } = E (eh|h + xˆh|h )(eh|h + xˆh|h )T

≤ (1 + ε0 )Ph|h + (1 + ε−1 xh|h xˆTh|h . 0 )ˆ

(17)

(18)

The inequalities (17) and (18), together with the assumption Ph|h ≤ Πh|h , imply that −1 Ph+1|h ≤ Fh (Πh|h − γh LTh Lh )−1 FhT + γh−1 Sh ShT r n X + Ψm tr (1 + ε0 )Πh|h + (1 + ε−1 xh|h h 0 )ˆ m=1

×

xˆTh|h

 mo Γh + Bh Qh BhT

= Πh+1|h .

Next, we are about to show that Ph+1|h+1 ≤ Πh+1|h+1 . In view of Lemma 1, we have −C1,h+1 − CT1,h+1 n  ≤ε1 E I − δ(τh+1 )(Gh+1 + Λh+1 )Ch+1

(19)

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 × Ph+1|h I − δ(τh+1 )(Gh+1 + Λh+1 ) n T o × Ch+1 + ε−1 E (Gh+1 + Λh+1 ) 1 o × A1,h+1 (Gh+1 + Λh+1 )T ,

−C2,h+1 − CT2,h+1 n  ≤ε2 E I − δ(τh+1 )(Gh+1 + Λh+1 )Ch+1  × Ph+1|h I − δ(τh+1 )(Gh+1 + Λh+1 ) n T o × Ch+1 + ε−1 E (Gh+1 + Λh+1 ) 2 o × A2,h+1 (Gh+1 + Λh+1 )T ,

−C3,h+1 − CT3,h+1 n  ≤ε3 E I − δ(τh+1 )(Gh+1 + Λh+1 )Ch+1  × Ph+1|h I − δ(τh+1 )(Gh+1 + Λh+1 ) n T o × Ch+1 + ε−1 E (Gh+1 + Λh+1 ) 3 o × A3,h+1 (Gh+1 + Λh+1 )T ,

C4,h+1 + CT4,h+1 n  ≤ε4 E I − δ(τh+1 )(Gh+1 + Λh+1 )Ch+1  × Ph+1|h I − δ(τh+1 )(Gh+1 + Λh+1 ) n T o × Ch+1 + ε−1 E (Gh+1 + Λh+1 ) 4 o × A4,h+1 (Gh+1 + Λh+1 )T , C5,h+1 + CT5,h+1 ≤ ε5 A1,h+1 + ε−1 5 A2,h+1 ,

C6,h+1 + CT6,h+1 ≤ ε6 A1,h+1 + ε−1 6 A3,h+1 ,

−C7,h+1 − CT7,h+1 ≤ ε7 A1,h+1 + ε−1 7 A4,h+1 , C8,h+1 + CT8,h+1 ≤ ε8 A2,h+1 + ε−1 8 A3,h+1 ,

−C9,h+1 − CT9,h+1 ≤ ε9 A2,h+1 + ε−1 9 A4,h+1 ,

−C10,h+1 − CT10,h+1 ≤ ε10 A3,h+1 + ε−1 10 A4,h+1 . Noting that Λh is independent of Gh and E{Λh } = 0, the first term on the right-hand-side of (13) can be handled by n  E I − δ(τh+1 )(Gh+1 + Λh+1 )Ch+1 Ph+1|h  T o × I − δ(τh+1 )(Gh+1 + Λh+1 )Ch+1 T =Ph+1|h − p0 Gh+1 Ch+1 Ph+1|h − p0 Ph+1|h Ch+1 GTh+1 T + p0 Gh+1 Ch+1 Ph+1|h Ch+1 GTh+1

T + p0 E{Λh+1 Ch+1 Ph+1|h Ch+1 ΛTh+1 }.

12

FINAL

Similarly, it can be derived that o n E (Gh+1 + Λh+1 )A1,h+1 (Gh+1 + Λh+1 )T =Gh+1

q X

T GTh+1 pj Ch+1−j Ph+1−j|h−j Ch+1−j

j=1

q o n X T pj Ch+1−j Ph+1−j|h−j Ch+1−j ΛTh+1 , + E Λh+1 j=1

n o T E (Gh+1 + Λh+1 )A2,h+1 (Gh+1 + Λh+1 ) =Gh+1

q X

T pj Ch+1−j xˆh+1−j|h−j xˆTh+1−j|h−j Ch+1−j

j=0

×

GTh+1

q n X pj Ch+1−j xˆh+1−j|h−j + E Λh+1 j=0

o T × xˆTh+1−j|h−j Ch+1−j ΛTh+1 ,

n o E (Gh+1 + Λh+1 )A3,h+1 (Gh+1 + Λh+1 )T =Gh+1

q X

T pj Dh+1−j Rh+1−j Dh+1−j GTh+1

j=0

q n o X T T + E Λh+1 pj Dh+1−j Rh+1−j Dh+1−j Λh+1 , j=0

n o E (Gh+1 + Λh+1 )A4,h+1 (Gh+1 + Λh+1 )T

T =Gh+1 Ch+1−d xˆh+1−d|h−d xˆTh+1−d|h−d Ch+1−d n × GTh+1 + E Λh+1 Ch+1−d xˆh+1−d|h−d o T × xˆTh+1−d|h−d Ch+1−d ΛTh+1 .

Summarizing the above derivations, we have P h+1|h+1

≤ ε¯1 (Ph+1|h − p0 Gh+1 Ch+1 Ph+1|h − p0 Ph+1|h T × Ch+1 GTh+1 ) + Gh+1 Ξh+1 GTh+1 + Σh+1 ,

where Ξh+1 ,

T ε¯1 p0 Ch+1 Ph+1|h Ch+1

+ ε¯2

q X

pj Ch+1−j

j=1 q

T × Ph+1−j|h−j Ch+1−j + ε¯3

×

X

pj Ch+1−j

j=0

T xˆh+1−j|h−j xˆTh+1−j|h−j Ch+1−j

+ ε¯4

q X j=0

pj

(20)

13

FINAL

T × Dh+1−j Rh+1−j Dh+1−j + ε¯5 Ch+1−d T × xˆh+1−d|h−d xˆTh+1−d|h−d Ch+1−d ,

T Σh+1 , ε¯1 p0 E{Λh+1 Ch+1 Ph+1|h Ch+1 ΛTh+1 } q n X pj Ch+1−j Ph+1−j|h−j + ε¯2 E Λh+1 j=1

q o n X T × Ch+1−j ΛTh+1 + ε¯3 E Λh+1 pj Ch+1−j j=0

×

T xˆh+1−j|h−j xˆTh+1−j|h−j Ch+1−j ΛTh+1

×

ΛTh+1

o

q n X T pj Dh+1−j Rh+1−j Dh+1−j + ε¯4 E Λh+1

o

j=0

+ ε¯5 E{Λh+1 Ch+1−d xˆh+1−d|h−d

T × xˆTh+1−d|h−d Ch+1−d ΛTh+1 }.

Together with the fact of E{Λh ΛTh } ≤ σI, the inequality (19) implies that Ξh+1 ≤ Φh+1 ,

Σh+1 ≤ σλmax {Φh+1 }I.

(21) (22)

From (19)-(22), it follows that P h+1|h+1 T ≤ ε¯1 (Πh+1|h − p0 Gh+1 Ch+1 Πh+1|h − p0 Πh+1|h Ch+1

× GTh+1 ) + Gh+1 Φh+1 GTh+1 + σλmax {Φh+1 }I.

Then, according to the mathematical induction, Ph+1|h+1 ≤ Πh+1|h+1 always holds. Now, we are ready to determine the filter gain by minimizing Πh+1|h+1 . Taking partial derivative of the trace of Πh+1|h+1 , we have ∂tr(Πh+1|h+1 ) T = −2¯ ε1 p0 Πh+1|h Ch+1 + 2Gh+1 Φh+1 . ∂Gh+1 Letting

∂tr(Πh+1|h+1 ) = 0, ∂Gh+1

the filter gain can be derived as shown in (16). Moreover, it is easily known that such a filter gain minimizes the upper bound on the filtering error covariance. The proof is complete. Remark 5: In this paper, the resilient filtering problem is addressed for a class of nonlinear stochastic systems with packet disorders. The RTDs, which occur unavoidably during data transmission, are characterized by a random variable with known probability distribution. The actual input to the filter is properly written in a novel expression by introducing a certain stochastic Kronecker delta function. By utilizing an integer-valued function, a novel innovation structure is proposed, which is dependent on the RTD-induced packet disorders. Subsequently, a recursive filter is developed where the random fluctuations on the filter gain and the effects of packet disorders are simultaneously considered. The nonlinear function is linearized

14

FINAL

by employing the Taylor expansion and the high-order terms are adequately tackled. Moreover, an upper bound for the filtering error covariance is guaranteed and the filter gain is then obtained by minimizing the upper bound in Theorem 1. It is worth mentioning that, the proposed filtering scheme is still suitable for the case without considering the phenomena of stochastic nonlinearities, gain variations and packet disorders. IV. A S IMULATION E XAMPLE In this section, a numerical simulation example is given to illustrate the effectiveness of the proposed recursive filtering scheme. Let us consider the nonlinear stochastic system (1) with # " # " 0.01 −0.46x1,h + 0.98x1,h x2,h , Bh = , f (xh ) = 0.03 0.9x2,h − x1,h x2,h h i Ch = 8 10 sin(h) , Dh = 0.2,

where xi,h (i = 1, 2) is the ith element of the state xh . The stochastic nonlinearity g(xh , ξh ) is chosen as " # 0.2 g(xh , ξh ) = [0.3sign(x1,h )x1,h ξ1,h 0.3

+ 0.4sign(x2,h )x2,h ξ2,h ], where ξ1,h and ξ2,h are zero-mean uncorrelated Gaussian white noises with unity covariance. It is easy to know that g(xh , ξh ) satisfies  E g(xh , ξh )|xh = 0,  E g(xh , ξh )g(xh , ξh )T |xh " # " # 0.04 0.06 T 0.09 0 = x xh . 0.06 0.09 h 0 0.16 h iT In this simulation, the initial conditions are set as xˆ0|0 = x¯0 = 0.35 0.04 , xˆ−2|−3 = xˆ−1|−2 = " # h iT 0 0 xˆ0|−1 = 0 0 , Π0|0 = 0.01I2 , and Π−2|−3 = Π−1|−2 = Π0|−1 = . The other parameters are 0 0 chosen as q = 3, φ−3 = φ−2 = φ−1 = 0, Qh = Rh = 0.05, Sh = Lh = 0.01I2 , σ = 0.0001, γh = 0.1, ε0 = 0.5, and εi = 0.1 (i = 1, 2, . . . , 10). Let RMSE represent the root mean squared error (RMSE) for the estimate of the state with v u M u1 X  (j) (j) (j) (j) (x1,h − xˆ1,h|h )2 + (x2,h − xˆ2,h|h )2 RMSE = t M j=1

where M is the number of the samples (M = 100 in this example). According to Theorem 1, the filter gain Gh and the minimum upper bound Πh|h can be recursively calculated at each time step. The simulation results are given in Figs. 2-5, where Figs. 2-3 show the trajectories of states xi,h and their estimation xˆi,h (i = 1, 2), which illustrate that the filter can estimate the actual state well. In other words, the proposed filtering scheme performs well.

15

FINAL

To quantify the accuracy of the estimation, the upper bound Πh|h of the filtering error covariance and RMSE of the state xh are plotted in Fig. 4, which indicates the RMSE stays below its upper bound. In Fig. 5, RMSE0 , RMSE1 , RMSE2 and RMSE3 represent the cases of d = 0, d = 1, d = 2 and d = 3, respectively. It can be seen from Fig. 5 that RMSE increases when d becomes large, which shows the influences on the filter performance brought by the RTD-induced packet disorders. From these simulation results, we further illustrate the feasibility and usefulness of the developed filtering scheme.

0.35

Actual state x1,h Estimate state x ˆ1,h

0.3 0.25

x1,h and its estimate

0.2 0.15 0.1 0.05 0 −0.05 −0.1 −0.15

0

10

20

30

40

50

Time (h) Fig. 2: The state x1,h and its estimate xˆ1,h .

V. C ONCLUSION In this paper, we have considered the resilient filtering problem for a class of time-varying systems with stochastic nonlinearity and packet disorders. The RTDs existing in the transmission process have been characterized by a random variable with known probability distribution. A recursive filter has been designed, for the first time, to take into account both the random perturbations on the filter gain and the impacts brought by the packet disorders. With the proposed filtering scheme, an upper bound has been derived for the filtering error covariance and the filter gain has been computed by minimizing such an upper bound. Finally, a simulation example has been exploited to show the validity of developed filtering

16

FINAL

0.035

Actual state x2,h Estimate state x ˆ2,h

0.03

x2,h and its estimate

0.025 0.02 0.015 0.01 0.005 0 0

10

20

30

40

50

Time (h) Fig. 3: The state x2,h and its estimate xˆ2,h . scheme. The possible topics for the future research would be the extension of the main results in this paper to the filtering problems for more complex systems [4], [9]. R EFERENCES [1] M. Basin, P. Shi and D. Calderon-Alvarez, Joint state filtering and parameter estimation for linear stochastic time-delay systems, Signal Processing, vol. 91, no. 4, pp. 782–792, Apr. 2011. [2] G. Calafiore, Reliable localization using set-valued nonlinear filters, IEEE Transactions on Systems, Man and Cybernetics-Part A: Systems and Humans, vol. 35, no. 2, pp. 189–197, Mar. 2005. [3] Y. Chen, Z. Wang, Y. Yuan and P. Date, Distributed H∞ filtering for switched stochastic delayed systems over sensor networks with fading measurements, IEEE Transactions on Cybernetics, vol. 50, no. 1, pp. 2-14, Jan. 2020. [4] D. Ding, Z. Wang, H. Dong and H. Shu, Distributed H∞ state estimation with stochastic parameters and nonlinearities through sensor networks: the finite-horizon case, Automatica , vol. 48, no. 8, pp. 1575–1585, Aug. 2012. [5] Y. Duan, B. Du, Y. Li, C. Shen, X. Zhu, X. Li and J. Chen, Improved sufficient LMI conditions for the robust stability of time-delayed neutral-type Lur’e systems, International Journal of Control, Automation and Systems, vol. 16, no. 5, pp. 2343–2353, Sept. 2018. [6] Y. Duan, Y. Li and J. Chen, Further stability analysis for rime-delayed neural networks based on an augmented Lyapunov functional, IEEE Access, vol. 7, pp. 104655–104666, Jul. 2019. [7] G. A. Einicke and L. B. White, Robust extended Kalman filtering, IEEE Transactions on Signal Processing, vol. 47, no. 9, pp. 2596– 2599, Sept. 1999.

17

FINAL

0.5 RMSE Upper bound

0.45

RMSE of xh and upper bound

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

0

10

20

30

40

50

Time (h) Fig. 4: RMSE and the upper bound. [8] J. Hu, Z. Wang, H. Gao and L. K. Stergioulas, Extended Kalman filtering with stochastic nonlinearities and multiple missing measurements, Automatica, vol. 48, no. 9, pp. 2007–2015, Sept. 2012. [9] J. Hu, Z. Wang, G.-P. Liu and H. Zhang, Variance-constrained recursive state estimation for time-varying complex networks with quantized measurements and uncertain inner coupling, IEEE Transactions on Neural Networks and Learning Systems, to be published. doi: 10.1109/TNNLS.2019.2927554. [10] J. Hu, Z. Wang, B. Shen and H. Gao, Quantised recursive filtering for a class of nonlinear systems with multiplicative noises and missing measurements, International Journal of Control, vol. 86, no. 4, pp. 650–663, Jan. 2013. [11] J. Hu, Z. Wang, B. Shen and H. Gao, Gain-constrained recursive filtering with stochastic nonlinearities and probabilistic sensor delays, IEEE Transactions on Signal Processing, vol. 61, no. 5, pp. 1230–1238, Mar. 2013. [12] A. G. Kallapur, I. R. Petersen and S. G. Anavatti, A discrete-time robust extended Kalman filter for uncertain systems with sum quadratic constraints, IEEE Transactions on Automatic Control, vol. 54, no. 4, pp. 850–854, Apr. 2009. [13] C. Keliris, M. M. Polycarpou and T. Parisini, A distributed fault detection filtering approach for a class of interconnected continuous-time nonlinear systems, IEEE Transactions on Automatic Control, vol. 58, no. 8, pp. 2032–2047, Aug. 2013. [14] P. P. Khargonekar, M. A. Rotea and E. Baeyens, Mixed H2 /H∞ filtering, International Journal of Robust and Nonlinear Control, vol. 6, no. 4, pp. 313–330, May 1996. [15] S. Kluge, K. Reif and M. Brokate, Stochastic stability of the extended Kalman filter with intermittent observations, IEEE Transactions on Automatic Control, vol. 55, no. 2, pp. 514–518, Feb. 2010. [16] W. Li, C. Meng, Y. Jia and J. Du, Recursive filtering for complex networks using non-linearly coupled UKF, IET Control Theory & Applications, vol. 12, no. 4, pp. 549–555, Mar. 2018. [17] X. Li, W. Zhang, J. Fang and H. Li, Finite-time synchronization of memristive neural networks with discontinuous activation functions

18

FINAL

RMSE 0 RMSE 1 RMSE 2 RMSE 3

RMSE in the cases of d = 0, 1, 2, 3

0.12

0.1

0.08

0.06

0.04

0.02 0

10

20

30

40

50

Time (h) Fig. 5: RMSE in the cases of d = 0, d = 1, d = 2 and d = 3. and mixed time-varying delays, Neurocomputing, vol. 340, pp. 99–109, May 2019. [18] A. Liu, W. A. Zhang, B. Chen and L. Yu, Networked filtering with Markov transmission delays and packet disordering, IET Control Theory & Applications, vol. 12, no. 5, pp. 687–693, Mar. 2018. [19] A. Liu, W. A. Zhang, L. Yu, S. Liu and M. Z. Q. Chen, New results on stabilization of networked control systems with packet disordering, Automatica, vol. 52, pp. 255–259, Feb. 2015. [20] D. Liu, Y. Liu and F. Alsaadi, Recursive state estimation based-on the outputs of partial nodes for discrete-time stochastic complex networks with switched topology, Journal of the Franklin Institute, vol. 355, no. 11, pp. 4686–4707, Jul. 2018. [21] G.-P. Liu, Y. Xia, D. Rees and W. Hu, Design and stability criteria of networked predictive control systems with random network delay in the feedback channel, IEEE Transactions on Systems, Man, and Cybernetics-Part C: Applications and Reviews, vol. 37, no. 2, pp. 173–184, Mar. 2007. [22] Q. Liu, Z. Wang, X. He, G. Ghinea and F. E. Alsaadi, A resilient approach to distributed filter design for time-varying systems under stochastic nonlinearities and sensor degradation, IEEE Transactions on Signal Processing, vol. 65, no. 5, pp. 1300–1309, Mar. 2017. [23] S. Liu, G. Wei, Y. Song and Y. Liu, Extended Kalman filtering for stochastic nonlinear systems with randomly occurring cyber attacks, Neurocomputing, vol. 207, pp. 708–716, Sept. 2016. [24] S. Liu, Z. Wang, Y. Chen and G. Wei, Protocol-based unscented Kalman filtering in the presence of stochastic uncertainties, IEEE Transactions on Automatic Control, in press, DOI:10.1109/TAC.2019.2929817. [25] Y. Liu, B. Shen and H. Shu, Finite-time resilient H∞ state estimation for discrete-time delayed neural networks under dynamic event-triggered mechanism, Neural Networks, vol. 121, pp. 356–365, 2020. [26] Y. Liu, Z. Wang, L. Ma and F. Alsaadi, A partial-nodes-based information fusion approach to state estimation for discrete-time delayed stochastic complex networks, Information Fusion, vol. 49, pp. 240–248, Sept. 2019.

FINAL

19

[27] Y. Liu, Z. Wang, Y. Yuan and W. Liu, Event-triggered partial-nodes-based state estimation for delayed complex networks with bounded distributed delays, IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 49, no. 6, pp. 1088–1098, Jun. 2019. [28] W. M. McEneaney, Robust/H∞ filtering for nonlinear systems, Systems & control letters, vol. 33, no. 5, pp. 315–325, Apr. 1998. [29] R. Mehra, On the identification of variances and adaptive Kalman filtering, IEEE Transactions on Automatic Control, vol. 15, no. 2, pp. 175–184, Apr. 1970. [30] A. Ray, L. W. Liou and J. H. Shen, State estimation using randomly delayed measurements, Journal of Dynamic Systems, Measurement, and Control, vol. 115, no. 1, pp. 19–26, Mar. 1993. [31] K. Reif, and R. Unbehauen, The extended Kalman filter as an exponential observer for nonlinear systems, IEEE Transactions on Signal processing, vol. 47, no. 8, pp. 2324–2328, Aug. 1999. [32] S. Sarkka, On unscented Kalman filtering for state estimation of continuous-time nonlinear systems, IEEE Transactions on Automatic Control, vol. 52, no. 9, pp. 1631–1641, Sept. 2007. [33] Y. Shen, Z. Wang, B. Shen, F. E. Alsaadi and F. E. Alsaadi, Fusion estimation for multi-rate linear repetitive processes under weighted Try-Once-Discard protocol, Information Fusion, vol. 55, pp. 281–291, 2020. [34] B. Sinopoli, L. Schenato, M. Franceschetti, K. Poolla, M. I. Jordan and S. S. Sastry, Kalman filtering with intermittent observations, IEEE Transactions on Automatic Control, vol. 49, no. 9, pp. 1453–1464, Sept. 2004. [35] S. Sun, Optimal linear filters for discrete-time systems with randomly delayed and lost measurements with/without time stamps, IEEE Transactions on Automatic Control, vol. 58, no. 6, pp. 1551–1556, Jun. 2013. [36] L. Wang, Z. Wang, G. Wei and F. E. Alsaadi, Variance-constrained H∞ state estimation for time-varying multi-rate systems with redundant channels: The finite-horizon case, Information Sciences, vol. 501, pp. 222–235, Oct. 2019. [37] T. Wang, Y. Zhang, J. Qiu and H. Gao, Adaptive fuzzy backstepping control for a class of nonlinear systems with sampled and delayed measurements, IEEE Transactions on Fuzzy Systems, vol. 23, no. 2, pp. 302–312, Apr. 2015. [38] Y. Wang, L. Xie and C. E. de Souza, Robust control of a class of uncertain nonlinear systems, Systems & Control Letters, vol. 19, no. 2, pp. 139–149, Aug. 1992. [39] Z. Wang, X. Liu, Y. Liu, J. Liang and V. Vinciotti, An extended Kalman filtering approach to modeling nonlinear dynamic gene regulatory networks via short gene expression time series, IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol. 6, no. 3, pp. 410–419, Jul.-Sept. 2009. [40] Z. Wang, F. Yang, D. W. C. Ho and X. Liu, Robust H∞ filtering for stochastic time-delay systems with missing measurements, IEEE Transactions on Signal Processing, vol. 54, no. 7, pp. 2579–2587, Jul. 2006. [41] L. Xie, L. Lu, D. Zhang and H. Zhang, Improved robust H2 and H∞ filtering for uncertain discrete-time systems, Automatica, vol. 40, no. 5, pp. 873–880, May 2004. [42] K. Xiong, C. Wei and L. Liu, Robust extended Kalman filtering for nonlinear systems with stochastic uncertainties, IEEE Transactions on Systems, Man and Cybernetics-Part A: Systems and Humans, vol. 40, no. 2, pp. 399–405, Mar. 2010. [43] F. Yang and Y. Li, Set-Membership fuzzy filtering for nonlinear discrete-Time systems, IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics), vol. 40, no. 1, pp. 116–124, Feb. 2010. [44] G.-H. Yang and J. L. Wang, Robust nonfragile Kalman filtering for uncertain linear systems with estimator gain uncertainty, IEEE Transactions on Automatic Control, vol. 46, no. 2, pp. 343–348, Feb. 2001. [45] L. Zhang, Y. Zhu, P. Shi and Y. Zhao, Resilient asynchronous H∞ filtering for Markov jump neural networks with unideal measurements and multiplicative noises, IEEE Transactions on Cybernetics, vol. 45, no. 12, pp. 2840–2852, Dec. 2015. [46] S. Zhang, Z. Wang, D. Ding, G. Wei and F. E. Alsaadi, Non-fragile H∞ state estimation for discrete-time complex networks with randomly occurring time-varying delays and channel fadings, IMA Journal of Mathematical Control and Information, vol. 36, no. 1, pp. 247-269, Mar. 2019. [47] X.-M. Zhang and Q.-L. Han, Robust H∞ filtering for a class of uncertain linear systems with time-varying delay, Automatica, vol. 44, no. 1, pp. 157–166, Jan. 2008. [48] X.-M. Zhang and Q.-L. Han, State estimation for static neural networks with time-varying delays based on an improved reciprocally convex inequality, IEEE Transactions on Neural Networks and Learning Systems, vol. 29, no. 4, pp. 1376–1381, Apr. 2018.

Declaration of interests ☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: