Finite-horizon H∞ state estimation for artificial neural networks with component-based distributed delays and stochastic protocol

Communicated by Steven Hoi

Accepted Manuscript

Finite-Horizon H∞ State Estimation for Artificial Neural Networks with Component-Based Distributed Delays and Stochastic Protocol Zhongyi Zhao, Zidong Wang, Lei Zou, Hongjian Liu PII: DOI: Reference:

S0925-2312(18)30971-8 https://doi.org/10.1016/j.neucom.2018.08.031 NEUCOM 19879

To appear in:

Neurocomputing

Received date: Revised date: Accepted date:

8 May 2018 9 August 2018 18 August 2018

Please cite this article as: Zhongyi Zhao, Zidong Wang, Lei Zou, Hongjian Liu, Finite-Horizon H∞ State Estimation for Artificial Neural Networks with Component-Based Distributed Delays and Stochastic Protocol, Neurocomputing (2018), doi: https://doi.org/10.1016/j.neucom.2018.08.031

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT 1

REVISED

Finite-Horizon H∞ State Estimation for Artificial Neural Networks with Component-Based Distributed Delays and Stochastic Protocol

reasons including the limited switching rate of the amplifier. Time-delay could contribute largely to the degradation of the system performance and even the loss of the stability. As such, considerable effort has been devoted to the dynamical analysis of ANNs with time delays. For example, in [32], the finite-time synchronization issue has been studied for a class of CohenGrossberg neural networks with mixed time-delays, where a discontinuous state feedback controller has been designed to ensure the finite-time synchronization of the considered neural networks. In [35], the exponential synchronization problem has been investigated for a class of nonidentically coupled neural networks with time-varying delays, where some criteria achieving the quasi-synchronization of neural networks have been derived based on the extended comparison principle of impulsive systems. For practical applications of ANNs, the state information of primary neurons is often of great importance to certain tasks (e.g. optimization and approximation). Unfortunately, due to a lot of reasons such as the restriction of measuring methods, the full state information might not be fully available/accessible, especially for the ANNs consisting of electrical circuit with a large number of component units and complex structure. In this case, the state estimation technique plays an important role to acquire the state information of ANNs by generating an estimate of the network state with a satisfactory/prescribed accuracy. So far, the state estimation problem for ANNs has stirred a great deal of research interest, see e.g. [1], [23], [27], [31], [36]–[38], [42]. For instance, in [38], the finite-time state estimation problem has been investigated for recurrent delayed neural networks with the component-based event-triggering protocol. Generally speaking, most existing research results concerning the state estimation issue of ANNs have focused on the H∞ estimation, minimum variance estimation and bounded estimation schemes, and these state estimation schemes would lead to different performance requirements regarding the accuracy indices of the estimation results. In fact, appropriate state estimation technique would be selected to accomplish the corresponding state estimation task according to the types of the external noises acting on the ANNs. More specifically, the minimum variance estimation could be employed to deal with the ANNs with externally additive stochastic noises. The setmembership estimation and ultimately bounded estimation are well suited for the ANNs with bounded noises. Considering an ANN with energy-bounded noises, the well-known H∞ state estimation might be the best choice to provide satisfactory

AN US

Abstract—This paper is concerned the H∞ state estimation problem for time-varying artificial neural networks with component-based distributed delays and stochastic protocol scheduling. A shared communication channel is adopted for data transmissions between the sensors and the estimator. For the purpose of avoiding data collisions, the stochastic protocol is used to schedule the transmission opportunities of sensors. A finite-horizon H∞ index is introduced to reflect the performance specification of the estimation. The aim of this paper is to design a time-varying H∞ estimator over a given finite-horizon such that the dynamics of the estimation error satisfy the given H∞ performance requirement. Sufficient conditions are established for the existence of the desired estimator and the explicit expressions of the desired estimator parameters are then given in terms of the solutions to a set of recursive linear matrix inequalities. Finally, a numerical example is given to demonstrate the effectiveness of the developed state estimation scheme.

CR IP T

Zhongyi Zhao, Zidong Wang, Lei Zou and Hongjian Liu

I. I NTRODUCTION

M

Index Terms—Artificial neural networks; component-based distributed delays; finite-horizon H∞ state estimation; stochastic protocol; recursive linear matrix inequalities.

AC

CE

PT

ED

In the past decades, artificial neural networks (ANNs) have received an ever-growing research interest due to their wide application in artificial intelligence and machine learning technologies. Typical practical applications of ANNs include, but are not limited to, pattern recognition, associative memory, optimization calculation and automatic control [3], [15], [16], [19], [21], [22], [33]. Among various research results concerning ANNs, the dynamical analysis issue of ANNs has attracted a great deal of research attention, and a large number of results have been reported in the literature in recent years, see e.g. [24], [26], [32], [35]. On the other hand, timedelay is a common phenomenon in ANNs due to various This work was supported in part by the National Natural Science Foundation of China under Grant 61703245, the Taishan Scholar Project of Shandong Province of China, the China Postdoctoral Science Foundation under the Grant number 2016M600547, the Qingdao Postdoctoral Applied Research Project under the Grant number 2016117, the Postdoctoral Special Innovation Foundation of Shandong under the Grant number 201701015, the Royal Society of the U.K., and the Alexander von Humboldt Foundation of Germany. Z. Zhao and L. Zou are with the College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China. Z. Wang is with the College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China. He is also with the Department of Computer Science, Brunel University London, Uxbridge, Middlesex, UB8 3PH, United Kingdom. (Email: [email protected]) H. Liu is with the School of Mathematics and Physics, Anhui Polytechnic University, Wuhu 241000, China.

ACCEPTED MANUSCRIPT 2

REVISED

CR IP T

that the corresponding estimation error dynamics satisfies the given finite-horizon H∞ performance requirement; and 4) the estimator parameters are derived through the solutions to certain recursive matrix inequalities. The rest of this paper is organized as follows. In Section II, the model of the delayed ANNs to be discussed is provided and the scheduling behavior of the stochastic protocol is detailed. Section III shows the main results of this paper concerning the analysis issue about the estimation error dynamics and the design issue about the estimator parameters. Section IV addresses a simulation example to illustrate the effectiveness of the proposed theory. Section V concludes the paper. Notations: The notation used here is fairly standard except where otherwise stated. R, Rn and Rn×m denote, respectively, the set of all real numbers, the n dimensional Euclidean space and set of all n × m real matrices. The notation X ≥ Y (X > Y ), where X and Y are real symmetric matrices, means that X − Y is positive semi-definite (positive definite). Prob{·} means the occurrence probability of the event “·”. E{x} and E{x|y} will, respectively, denote the expectation of the stochastic variable x and expectation of x conditional on y. 0 represents the zero matrix of compatible dimension. The n-dimensional identity matrix is denoted as In or simply I, if no confusion is caused. The shorthand diag{· · · } stands for a block-diagonal matrix. kxk refers to the Euclidean norm of a vector x. M T ∈ Rn×m represent the transpose of the matrix M ∈ Rm×n . In symmetric block matrices, “ ∗ ” is used as an ellipsis for terms induced by symmetry. Matrices, if they are not explicitly specified, are assumed to have compatible dimensions. The Kronecker delta function δ(a) is a binary function that equals 1 if a = 0 and equals 0 otherwise.

AC

CE

PT

ED

M

AN US

state estimates. Another matter that is worth considering is that, in hardware-implemented ANNs, the strong real-time dynamical ability and fast processing speed would always give rise to certain time-varying parameters of ANNs. Furthermore, the temperature drift of electronic circuits can also lead to the changeable parameters of ANNs. As such, the finite-horizon H∞ state estimation problem has begun to stir some initial research interests for time-varying ANNs with energy-bounded noises, see [13], [14], [18], [20], [30] and the references therein. Thanks to the rapid development of the network-based communication technologies, more and more estimation tasks in engineering applications have adopted the communication networks for data transmissions (e.g. the communication-based train control systems [39]). Compared with the conventional point-to-point communication, the network-based communication is more suitable for data transmission of ANNs with largescale artificial neurons (also called connected units or nodes) due to its advantages such as low cost, simple wiring and high reliability. Though networked systems have already become one of the research fronts, for a communication network with limited bandwidth, it is quite difficult for all network nodes to get access to the network simultaneously without any data collisions. In this case, communication protocols have been introduced to schedule signal transmissions for the sake of preventing data collisions. Till now, the communication protocols extensively studied in the literature include, but are not limited to, Round-Robin protocol [25], [48], Try-oncediscard protocol [17], [46] and stochastic protocol [34], [43], [44], [47]. Among these protocols, the stochastic protocol is a widely utilized protocol in a variety of networked systems including wireless sensor networks. There are two different models characterizing the scheduling behavior of the stochastic protocol in the existing literature. The first one is to describe stochastic protocol scheduling behavior by a Markov chain and the other one is to model it by a sequence of independent and identically distributed (i.i.d.) random variables. Summarizing the discussions made so far, it is of great significance in engineering practice to study the finite-horizon H∞ state estimation problem for delayed ANNs with stochastic protocol. Accordingly, the main questions we need to answer are: 1) how to establish a finite-horizon H∞ state estimation scheme for ANNs with component-based distributed delays and stochastic protocol scheduling? and 2) how to design appropriate estimator gain matrices such that finitehorizon H∞ performance requirement is guaranteed for estimation error dynamics? It is, therefore, the main intention of this paper to launch an initial study in order to answer the above questions. Motivated by the above discussions, in this paper, we aim to study the finite-horizon H∞ state estimation problem for time-varying ANNs with componentbased distributed delays and stochastic protocol. The contributions of this paper are highlighted as follows: 1) the finite-horizon H∞ state estimation problem is, for the first time, investigated for time-varying ANNs with stochastic protocol scheduling; 2) component-based distributed delays are included in the network model to describe the delays in a realistic way; 3) sufficient conditions are obtained to ensure

II. P ROBLEM S TATEMENT AND P RELIMINARIES A. The model of ANNs Consider the following ANNs with nx neurons:  nω X    x (k + 1) = a (k)x (k) + bω  i i i ij (k)ωj (k)    j=1    nx  X    + bfij (k)fj (xj (k − τj (k))) + Ji (k),     j=1    nx nv   X X yi (k) = cij (k)xj (k) + dvij (k)vj (k), (1)   j=1 j=1     i = 1, 2, · · · , ny     nx  X    z (k) = mij (k)xj (k), i = 1, 2, · · · , nz i     j=1     xi (s) = φi (s), 0 ≥ s ≥ − max {τj (k)},  1≤j≤nx

where xi (k) ∈ R, yi (k) ∈ R, zi (k) ∈ R denote, respectively, the i-th state variable, the i-th output and the i-th signal to be estimated of the overall neural networks at time k. ωj (k) and vj (k) denote, respectively, the external process noise and measurement noise belonging to l2 ([0, N ]; R), where l2 ([0, N ]; R) is the space of square-summable real number functions over the interval [0, N ]. fj (.) is the activation

ACCEPTED MANUSCRIPT 3

REVISED

 T x(k) = x1 (k) x2 (k) · · · xnx (k) ,  T f (x(k)) = f1 (x1 (k)) f2 (x2 (k)) · · · fnx (xnx (k)) ,  T ω(k) = ω1 (k) ω2 (k) · · · ωnω (k) ,  T v(k) = v1 (k) v2 (k) · · · vnv (k) ,  T J(k) = J1 (k) J2 (k) · · · Jnx (k) ,  T φ(s) = φ1 (s) φ2 (s) · · · φnx (s) ,

ED

where

(2)

M

1≤j≤nx

nz ×nx

 0, · · · , 0 1 , θj = | {z } j−1

CE

  M (k) = mij (k)

PT

A(k) = diag{a1 (k), a2 (k), · · · , anx (k)},     Bf (k) = bfij (k) , Bω (k) = bω ij (k)  nx ×nω  nx ×nx v , Dv (k) = dij (k) C(k) = cij (k) ny ×nx

AC

C. The structure of the estimator  T Letting y¯(k) = y¯1 (k) y¯2 (k) · · · y¯ny (k) denote the signal received by the estimator, the updating rule for y¯i (k) subject to the stochastic protocol (4) is set to be  yi (k) + dν,i (k)ν(k), i = σ(k) y¯i (k) = (5) y¯i (k − 1), i 6= σ(k) where ν(k) is the transmission noise l2 ([0, N ]; Rnν ), dν,i (k) ∈ R1×nν . Using Kronecker delta function ( 1, x = 0 δ(x) = 0, x 6= 0

belonging

to

and defining

ny ×nv

T 0, · · · , 0 | {z } . nx −j

Assumption 2: The nonlinear vector valued function f (·) in (2) satisfies the following Lipschitz condition kf (x) − f (y)k ≤ kG(x − y)k

i=1

Remark 1: It is worth mentioning that, in [1], the filtering/state estimation problem has been investigated, for the first time, for ANNs subject to the stochastic protocol scheduling, where the scheduling behavior has been modeled by a Markov chain. In this paper, we focus our attention on the finite-horizon H∞ state estimation problem for time-varying ANNs under the effects of the stochastic protocol, for which the scheduling behavior is described by a sequence of i.i.d. variables with the probability distribution shown in (6). In fact, the main results of this paper could be easily extended to the state estimation problem with the stochastic protocol modeled by a Markov chain.

AN US

the ANNs (1) can be rewritten as in the following compact form:  nx X    θj θjT f (x(k − τj (k))) x(k + 1) = A(k)x(k) + B (k)  f    j=1     + Bω (k)ω(k) + J(k)  y(k) = C(k)x(k) + Dv (k)v(k)      z(k) = M (k)x(k)      x(s) = φ(s), s = 0, −1, · · · , − max {τj (k)},

which is independent and identically distributed at every time instant. The probability distribution of σ(k) is given by  Prob{σ(k) = i} = λi , i = 1, 2, · · · , ny   ny X (4)  λi = 1 

CR IP T

function regarding the j-th neuron. φi (s) represents a given initial condition. Ji (k) denotes the exogenous input on the v i-th neuron. ai (k), bfij (k), bω ij (k), cij (k), dij (k), mij (k) are all known real scalars. Assumption 1: For 1 ≤ j ≤ nx , the distributed delay τj (k) in the system (1) satisfies the constraint τjm ≤ τj (k) ≤ τjM where τjm and τjM are all nonnegative integers. Denoting

(3)

where G is a known diagonal matrix. B. The transmission model over the network In this subsection, we shall introduce the mechanism of the stochastic protocol scheduling. It is assumed that, at each transmission instant, only one sensor node is allowed to transmit data via the communication network. In this paper, we use σ(k) to determine which sensor node obtains the access to the communication network at the time instant k. As shown in [34], under the scheduling of the stochastic protocol, σ(k) ∈ {1, 2, · · · , ny } can be modeled by a random variable

Φσ(k) = diag{δ(1 − σ(k)), δ(2 − σ(k)), · · · , δ(ny − σ(k))},  T Dν (k) = dTν,1 (k) dTν,2 (k) · · · dTν,ny (k) ,

we have

y¯(k) = Φσ(k) (y(k) + Dν (k)ν(k)) + (I − Φσ(k) )¯ y (k − 1) = Φσ(k) C(k)x(k) + Φσ(k) Dv (k)v(k)

+ Φσ(k) Dν (k)ν(k) + (I − Φσ(k) )¯ y (k − 1) (6)  T  T Denoting x ¯(k) = x (k) y¯T (k − 1) , f¯(¯ x(k)) =  T T  T T T f (x(k)) 01×ny , v¯(k) = v (k) ν (k) , ω ¯ (k) =  T T  T ¯ ω (k) v¯T (k) and J(k) = J T (k) 0Tny ×1 , the ANNs (1) with the constraints induced by the stochastic protocol can be reformulated as follows:  nx X   ¯σ(k) (k)¯ ¯f (k)f¯(¯ x ¯ (k + 1) = A x (k) + B x(k − τj (k)))  j    j=1  ¯σ(k) (k)¯ ¯ +B ω (k) + J(k)    ¯ ¯  y¯(k) = Cσ(k) (k)¯ x(k) + Dσ(k) (k)¯ ω (k)    ¯ z(k) = M (k)¯ x(k) (7)

ACCEPTED MANUSCRIPT 4

REVISED

 A(k) 0 , Θj = θj θjT , Φσ(k) C(k) I − Φσ(k)     T T ¯f (k) = Bf (k)Θj 0nx ×ny , M ¯ (k) = M (k) B , j 0n ×n 0ny 0ny ×nz  y x  0 0 ¯σ(k) (k) = Bω (k) B , 0 Φσ(k) Dv (k) Φσ(k) Dν (k)   C¯σ(k) (k) = Φσ(k) C(k) I − Φσ(k) ,   ¯ σ(k) (k) = 0 Φσ(k) Dv (k) Φσ(k) Dν (k) . D A¯σ(k) (k) =



Obviously, it is easy to conclude from Assumption 1 that the nonlinear function f¯(·) satisfies the following condition:

The intention of this paper is to design the estimator (9) with parameters Lσ(k) (k) and Kj,σ(k) (k) (j = 1, 2, · · · , nx ) such that the following finite-horizon H∞ performance requirement is satisfied: N X

k=0

≤γ

2

E{k¯ z (k)k2 } N X

k=0

2

E{k¯ ω (k)k } +

0  X

t=−¯ τ

T ˆ ¯ y¯(t) − Cσ(t) (t)φ(t)

 ! ˆ ¯ × R(t) y¯(t) − Cσ(t) (t)φ(t)

(11)

where R(t) (t = 0, −1, · · · , −¯ τ ) are the given positive definite matrices. γ is a given positive scalar representing the given disturbance attenuation level and τ¯ = maxj {τjM }. Remark 3: For conventional H∞ state estimation problems, if the estimation error dynamics is stable, the estimation error is equal to the equilibrium (usually 0) in regardless of initial estimation error when the time tends to infinity. In contrast, in the finite-horizon case, the initial value of estimation error would have an enormous influence on the transient dynamics of the H∞ estimation error. Therefore, when defining the performance index of the state estimation, the influence from the generalized energy of the initial innovation on the estimation error over a finite horizon is taken into account.

M

AN US

¯ x − y¯)k kf¯(¯ x) − f¯(¯ y )k ≤ kG(¯ (8)   ¯ = G 0nx ×ny . where G For the sake of convenience, in the rest of this paper, we let k¯j be k¯j = k − τj (k) for j = 1, 2, · · · , nx . Then, construct the following estimator for ANNs (1):   nx X  ¯ ¯ ¯  Bfj (k)f x x ˆ(k + 1) = Aσ(k) (k)ˆ x(k) + ˆ(k¯j )     j=1         ¯ ¯ ¯  + K (k) y ¯ ( k ) − C (k)ˆ x ( k ) ¯j ) j j j,σ(k)  σ(k     + Lσ(k) (k) y¯(k) − C¯σ(k) (k)ˆ x(k)      ¯  + J(k),     ¯ (k)ˆ  zˆ(k) = M x(k),    ˆ x ˆ(k) = φ(k), k ≤ 0. (9)

  + Kj,σ(k) (k) y¯(k¯j ) − C¯σ(k¯j ) (k¯j )ˆ x(k¯j ) .

CR IP T

where

AC

CE

PT

ED

where x ˆ(k) and zˆ(k) are, respectively, the estimates of x(k) ˆ and z(k). φ(k) is the initial condition of estimator (9). Kj,i (k) and Li (k) (i = 1, 2, · · · , ny , j = 1, 2, · · · , nx ) are gain matrices of estimator (9). Remark 2: It is obvious that, when Kj,i (k) = 0 holds for any j and i, the estimator (9) would degrade into the traditional Luenberger-estimator. Thus, the estimator structure (9) is more general and flexible than that of the traditional Luenbergerestimator due to the introduction of the extra parameters Kj,σ(k) (k). Such kind of estimators has been firstly introduced in [2] where the globally convergent observers have been studied for a class of systems with monotonic nonlinearities. By denoting e(k) = x ¯(k) − x ˆ(k) and z¯(k) = z(k) − zˆ(k), we can have the following estimation error dynamics:  e(k + 1) = (A¯σ(k) (k) − Lσ(k) C¯σ(k) (k))e(k)     ¯τ (k)f˜τ (e(k)) + B ¯σ(k) (k) +B
(10) ¯ σ(k) (k) ω  − L D ¯ (k)  σ(k)   ¯ (k)e(k) z¯(k) = M where

  ¯f (k) B ¯f (k) · · · B ¯f (k) , ¯τ (k) = B B 1 2 nx  T f˜τ (e(k)) = f¯T (e(k¯1 )) f¯T (e(k¯2 )) · · · f¯T (e(k¯nx )) ,  f¯(e(k¯j )) = f¯(¯ x(k¯j )) − f¯ x ˆ(k¯j )

III. M AIN R ESULTS Before proceeding further, the following lemma is introduced which is necessary for the derivation in sequel. Lemma 1: [4] For given constant matrices W1 , W2 , W3 with W1 = W1T and W2 = W2T > 0, then W1 + W3T W2−1 W3 < 0 is equivalent to   W1 W3T < 0. W3 −W2 A. The analysis of finite-horizon H∞ performance

Theorem 1: Consider ANNs (2) subject to the stochastic protocol (4). Let the disturbance attenuation level γ > 0, the weight matrices {R(t)}−¯τ ≤t≤0 , and the estimator parameters {Li (k)}0≤k≤N , {Kj,i (k)}0≤k≤N , (i = 1, 2, · · · , ny , j = 1, 2, · · · , nx ) be given. Assume that there exist  families

of positive definite matrices {Pj (k + 1)}0≤k≤N , Qs (k) ,   Qs,11 (k) Qs,12 (k) (j = 1, 2, · · · , ny , s = Qs,21 (k) Qs,22 (k) 0≤k≤N 1, 2, · · · , nx ) and a set of positive scalars {α(k) > 0}0≤k≤N such that the following matrix inequalities are satisfied: ¯ i (k) , Λ1,i (k) + Λ2,i (k) + Λ3,i (k) < 0 Λ 2

(12)

Pσ(0) (0) ≤ γ R11 (0) (13) nx X ¯ (τsM − τsm + 1)Qs (t) ≤ γ 2 R(t), t = −1, · · · , −¯ τ (14) s=1

ACCEPTED MANUSCRIPT 5

REVISED

 where R11 (0) = I

  0 R(0) I

T 0 and

" " # #T T T C¯σ(t) C¯σ(t) (t) (t) ¯ R(t) = ¯ T R(t) ¯ T , Dσ(t) (t) Dσ(t) (t)  Λ (k) ∗ ∗ ∗ i,11



T   e(k) e(k) E{∆V1 (k)|σ(k) = i} = f˜τ (e(k)) Ωi (k) f˜τ (e(k)) ω ¯ (k) ω ¯ (k)



∗ ∗ ∗ ∗

Λi,22 (k) ∗ ∗  0 −α(k)I ∗ , Λi,41 (k) 0 0 Λi,44 (k) 0 Λi,52 (k) 0 0 Λi,55 (k) ¯ T (k)M ¯ (k) = −Pi (k) + M nx X + (τsM − τsm + 1)Qs,11 (k), s=1

where   Ωi,11 (k) Ωi,12 (k) Ωi,13 (k) Ωi,22 (k) Ωi,23 (k) Ωi (k) =  ∗ ∗ ∗ Ωi,33 (k) Ωi,11 (k) = −Pi (k) + (A¯i (k) − Li (k)C¯i (k))T

CR IP T

0 0

 Λ1,i (k) = 

Λi,11 (k)

From (10), we have

Λi,22 (k) = −diag{Q1,11 (k¯1 ), · · · , Qnx ,11 (k¯nx )}, nx X Λi,41 (k) = (τsM − τsm + 1)Qs,21 (k),

× P¯j (k + 1)(A¯i (k) − Li (k)C¯i (k)) ¯τ (k) Ωi,12 (k) = (A¯i (k) − Li (k)C¯i (k))T P¯j (k + 1)B

s=1

Λi,44 (k) = −γ 2 I +

nx X (τsM − τsm + 1)Qs,22 (k),

Ωi,13 (k) = (A¯i (k) − Li (k)C¯i (k))T P¯j (k + 1) ¯ω (k) − Li (k)D ¯ i (k)) × (B

s=1

Λi,52 (k) = −diag{Q1,21 (k¯1 ), · · · , Qnx ,21 (k¯nx )}, Λi,55 (k) = −diag{Q1,22 (k¯1 ), · · · , Qnx ,22 (k¯nx )}, Λ2,i (k) = ζiT (k)P¯j (k + 1)ζi (k),  ¯τ (k) ζi (k) = A¯i (k) − Li (k)C¯i (k) 0 B  ¯ ¯ Bω (k) − Li (k)Di (k) 0 ,

Λ3,i (k) =

s=1 nx X

AN US

nx X

¯ω (k) − Li (k)D ¯ i (k))T P¯j (k + 1) Ωi,33 (k) = (B ¯ω (k) − Li (k)D ¯ i (k)) × (B

Furthermore, it is easy to see

λs Ps (k + 1),

¯ T Gι ¯ i,j (k), α(k)ιTi,j (k)G

j=1

E{∆V2 (k)|σ(k) = i}  k X − =

M

P¯j (k + 1) =

¯ T (k)P¯j (k + 1)B ¯τ (k) Ωi,22 (k) = B τ T ¯τ (k)P¯j (k + 1)(B ¯ω (k) − Li (k)D ¯ i (k)) Ωi,23 (k) = B

  ιi,j (k) = 0 $i,j1 (k) 0 0 $i,j2 (k) ,   0, · · · , 0 I − Kj,i (k)C¯σ(k¯j ) (k¯j ) 0, · · · , 0 | {z } , $i,j1 (k) = | {z }

ED

j−1

¯ σ(k¯ ) (k¯j ) 0, · · · , 0 −Kj,i (k)D j | {z } .

CE 3 X

Vi (k)

i=1 T

AC

V1 (k) = e (k)Pσ(k) (k)e(k)

V2 (k) =

nx X

k−1 X

η T (t)Qs (t)η(t)

s=1 t=k−τs (k)

V3 (k) =

M −1 k−1 nx τsX X X

E{∆V (k)|σ(k) = i} =

i=1

η (t)Qs (t)η(t)



and E{∆V3 (k)|σ(k) = i} m nx τX k s −1  X X = s=1

p=τsm

q=k+1−p

M −1  nx τsX X

s=1

η T (q)Qs (q)η(q)

E{∆Vi (k)|σ(k) = i}

T

t=k+1−τsM

−

k−1 X

q=k−p



η T (q)Qs (q)η(q)

η T (k)Qs (k)η(k)

p=τsm

T

− η (k − p)Qs (k − p)η(k − p)

s=1 p=τsm q=k−p

3 X

 η T (t)Qs (t)η(t)

η T (k)Qs (k)η(k) − η T (k¯s )Qs (k¯s )η(k¯s )

X

+

=

 T ¯ T (k) . where η(k) = eT (k) ω We first calculate the following conditional expectation:

s=1

t=k−τs (k)

k−τsm

nx −j

Then, the pre-defined H∞ performance requirement (11) is achieved. Proof: Construct the following Lyapunov-like functional: V (k) =

≤

nx −j

PT

j−1

 0, · · · , 0 $i,j2 (k) = | {z }

t=k+1−τs (k+1)

nx  X

k−1 X

=



nx X (τsM − τsm )η T (k)Qs (k)η(k) s=1

−

nx X

k−τsm

X

s=1 t=k+1−τsM

η T (t)Qs (t)η(t)

ACCEPTED MANUSCRIPT 6

REVISED

M

Then, one can see that

+

E{∆V2 (k) + ∆V3 (k)|σ(k) = i} nx X (τsM − τsm + 1)η T (k)Qs (k)η(k) ≤ s=1 nx X

−

s=1

p=τsm

η T (q)Qs (q)η(q)

q=−p

≤ eT (0)Pσ(0) e(0) +

η T (k¯s )Qs (k¯s )η(k¯s )

nx −1 X X (τsM − τsm + 1) η T (q)Qs (q)η(q) s=1

q=−τsM

T

≤ e (0)Pσ(0) e(0)

s=1

Noticing (8), we have nx X

+

−1 X

η T (q)

q=−¯ τ

f¯T (e(k¯j ))f¯(e(k¯j ))

j=1

# T " nx  X (I − Kj,i (k)C¯σ(k¯j ) (k¯j ))T e(k¯j ) ≤ ¯ T ¯ (k¯j )K T (k) −D ω ¯ (k¯j ) j,i σ(kj ) j=1 #T  "  (I − Kj,i (k)C¯σ(k¯j ) (k¯j ))T e(k¯j ) T ¯ G ¯ ×G ¯ T ¯ (k¯j )K T (k) −D ω ¯ (k¯j ) j,i σ(kj )

(15)

nx X (τsM − τsm + 1)Qs (q)η(q) s=1

CR IP T

f˜τT (e(k))f˜τ (e(k)) =

−1 −1 nx τsX X X

and the initial conditions (13)-(14), the finite-horizon H∞ performance requirement (11) can be guaranteed. The proof is now complete.

B. The design of estimator parameters

Theorem 2: Consider ANNs (2) subject to the stochastic protocol (4). Let the disturbance attenuation level γ > 0 and the Denoting weight matrices {R(t)}−¯τ ≤t≤0 be given. Assume that there  T T exist families {Pj (k + 1)}0≤k≤N , T T   of positive definite matrices  eτ (k) = e (k¯1 ) e (k¯2 ) · · · e (k¯nx ) , Q (k) Q (k) s,11 s,12  T T (j = 1, 2, · · · , ny , Qs (k) , Qs,21 (k) Qs,22 (k) 0≤k≤N ¯ (k¯1 ) ω ¯ T (k¯2 ) · · · ω ¯ T (k¯nx ) , ω ¯ τ (k) = ω  T ¯ i (k)}0≤k≤N , {K ¯ j,i (k)}0≤k≤N s = 1, 2, · · · , nx ), matrices {L ξ(k) = eT (k) eTτ (k) f˜τT (e(k)) ω ¯ T (k) ω ¯ τT (k) , (i = 1, 2, · · · , ny , j = 1, 2, · · · , nx ) and a set of positive scalars {α(k) > 0}0≤k≤N such that the following recursive and taking into account all analyses above, we have linear matrix inequalities are satisfied: E{∆V (k)|σ(k) = i}   Λ1,i (k) ∗ ∗ ≤ E{∆V (k)|σ(k) = i} + α(k)κ(k) − α(k)f˜τT (e(k))f˜τ (e(k)) ∗  < 0, (17) Πi (k) = Πi,21 (k) Λi,66 (k) = ξ T (k)Λi (k)ξ(k) Πi,31 (k) 0 Λi,77 (k) = ξ T (k)Λi (k)ξ(k) + z¯T (k)¯ z (k) − γ 2 ω ¯ T (k)¯ ω (k) T 2 T with the initial condition − z¯ (k)¯ z (k) + γ ω ¯ (k)¯ ω (k)  T T 2 T 2 ¯ i (k)ξ(k) − z¯ (k)¯ = ξ (k)Λ z (k) + γ ω ¯ (k)¯ ω (k)   Pσ(0) (0) ≤ γ R11 (0) (16) n x X ¯  (τsM − τsm + 1)Qs (t) ≤ γ 2 R(t), t = −1, −2, · · · , −¯ τ  Then, according to (12) and summing up (16) on both sides

PT

ED

M

AN US

, κ(k)

s=1

from 0 to N with respect to k, we have T

z¯ (k)¯ z (k)

γ2ω ¯ T (k)¯ ω (k) + E{V (0)} − E{V (N + 1)}

AC

≤

k=0 N X

where

CE

N X

≤

k=0 N X

2

T

γ ω ¯ (k)¯ ω (k) + E{V (0)}

Noticing that nx X

−1 X

Λi,72 (k) = diag{Υ1 (k), Υ2 (k), · · · , Υnx (k)}, ¯ C¯σ(k¯ ) (k¯s ), Υs (k) = α(k)G η T (t)Qs (t)η(t)

s=1 t=−τs (0)

+

M nx τs −1

−1 X X X s=1

T

p=τsm

q=−p

≤ e (0)Pσ(0) e(0)

¯τ (k), Λi,63 (k) = P¯j (k + 1)B ¯i (k) − L ¯ i (k)D ¯ i (k), Λi,64 (k) = P¯j (k + 1)B Λi,66 (k) = −P¯j (k + 1),

k=0

E{V (0)} = eT (0)Pσ(0) e(0) +

  Πi,21 (k) = Λi,61 (k) 0 Λi,63 (k) Λi,64 (k) 0 ,   Πi,31 (k) = 0 Λi,72 (k) 0 0 Λi,75 (k) , ¯ i (k)C¯i (k), Λi,61 (k) = P¯j (k + 1)A¯i (k) − L

η T (q)Qs (q)η(q)

s

¯K ¯ s,i (k)C¯σ(k¯ ) (k¯s ), s = 1, 2, · · · , nx , −G s

Λi,75 (k) = diag{Ξ1 , Ξ2 , · · · , Ξnx }, ¯K ¯ s,i (k)D ¯ σ(k¯ ) (k¯s ), s = 1, 2, · · · , nx , Ξs (k) = −G s

Λi,77 (k) = −diag{α(k)I, α(k)I, · · · , α(k)I }. | {z } nx

ACCEPTED MANUSCRIPT 7

REVISED

j=1

¯ s,t (k), Ks,t (k) = α−1 (k)K

s = 1, 2, · · · , nx , t = 1, 2, · · · , ny .   Pny ¯ Proof: Replace j=1 λj Pj (k + 1) Li (k)

and



1.1 cos(k), 0,

k ≥ 2 or k < 5 otherwise

ν(k) = 0.3 cos(0.6k). In this simulation, the probability distribution of σ(k) is ( Prob{σ(k) = 1} = 0.9 Prob{σ(k) = 2} = 0.1

The H∞ disturbance attenuation level γ is selected as γ = 0.98. The initial value of x ¯ and x ˆ are supposed to be  T x ¯(0) = 0.3 0.45 0.3 0 0 ,  T x ˆ(0) = 0 0 0 0 0

and x ¯(t) = x ˆ(t) = 0, t = −1, −2, · · · , −¯ τ . The matrix R11 (0) is given by   2.5797 −0.0042 −0.0100 0 0 −0.0042 2.5841 −0.0012 0 0    −0.0100 −0.0012 2.5730 0 0     0 0 0 2.5239 0  0 0 0 0 2.5239   ¯ R ∗ ¯ The matrix R(−1) is given by ¯ 11 ¯ , where R21 R22   2.4193 −0.0010 −0.0005 0 0 −0.0010 2.4724 −0.0026 0 0    ¯ 11 = −0.0005 −0.0026 2.4313 0 0  R    0 0 0 2.4084 0  0 0 0 0 2.4084   0 0 0 0 0 ¯ 21 = 0 0 0 0 0 R 0 0 0 0 0   2.4084 0 0 ¯ 22 =  0 2.4084 0  R 0 0 2.4084 ¯ and R(−2) is given by 010×10 . The initial value of P1 (0) and P2 (0) are assumed to be   1.2394 −0.0017 −0.0048 0 0 −0.0017 1.2409 −0.0008 0 0    −0.0048 −0.0008 1.2351 0 0     0 0 0 1.2120 0  0 0 0 0 1.2120

IV. N UMERICAL E XAMPLE

M

AN US

¯ i (k) and K ¯ s,t (k). According to Lemma 1, α(k)Ks,t (k) by L it is easy to find that, if the inequality (17) with the initial condition (13)-(14) is satisfied, the condition (12) in Theorem 1 could always be achieved. In other words, the finite-horizon H∞ performance requirement (11) can be guaranteed. The proof is now complete. Remark 4: In this paper, the finite-horizon H∞ estimator has been designed for ANNs with component-based distributed delays and stochastic protocol. Note that a parameter design approach for the estimator (9) is developed in Theorem 2, which could ensure that the prescribed finite-horizon H∞ performance requirement (11) for the estimation error system (10) is achieved. It can be found from the design of the estimator parameters that all the important factors contributing to the system complexity have been reflected which include 1) the time-varying parameters of ANNs; 2) the probability distribution of stochastic protocol; and 3) the prescribed disturbance attenuation level.

v(k) =

CR IP T

Then, the pre-specified H∞ performance requirement (11) is achieved. Furthermore, the desired estimator parameters can be calculated by X −1 ny ¯ ¯ i (k), i = 1, 2, · · · , ny Li (k) = λj Pj (k + 1) L

ED

In this section, a simulation example is provided in order to illustrate the effectiveness of our finite-horizon H∞ estimator. Consider ANNs (2) with the following parameters:

AC

CE

PT

A(k) = diag{0.9, 0.2 + 0.1 sin(0.15k), 0.35e−0.4(k+1) },   1.9 0.6 0.5 Bf (k) = 0.04 0.15 0.03 , 0.12 0.11 0.12  T Bω = 0.01 0.015 0.03 ,     0.1 0.15 0.2 0.01 C= , Dv = , 0.2 0.16 0.1 0.012     0.01 M = 0.02 0.05 0.015 , Dν = , 0.08

G = diag{0.1, 0.24, 0.15}.

Let the nonlinear activation function f (·) be   0.1 tanh(x1 (k))) f (x(k)) = 0.24 tanh(0.7x2 (k)) . 0.13 tanh(x3 (k))

The delay parameters of ANNs (2) are assumed to be τ1 (k) = mod( k2 ), τ2 (k) = 0 and τ3 (k) = mod( k3 ). The process noise ω(k), output noise v(k) and channel noise ν(k) of ANNs (2) are, respectively, set to be  1.2 sin(0.7k), k ≥ 5 or k ≤ 15 ω(k) = 0, otherwise

and



 −0.0048 0 0 −0.0004 0 0   1.2360 0 0   0 1.2120 0  0 0 1.2120   Q1,11 ∗ The matrix Q1 (−1) is given by , where Q1,21 Q1,22   0.4647 −0.0002 −0.0001 0 0 −0.0002 0.4749 −0.0005 0 0    −0.0001 −0.0005 0.4670 0 0  Q1,11 =     0 0 0 0.4626 0  0 0 0 0 0.4626 1.2381 −0.0023  −0.0048   0 0

−0.0023 1.2409 −0.0004 0 0

ACCEPTED MANUSCRIPT 8

CR IP T

REVISED

Fig. 3: The state evolution x3 (k) and its estimate x ˆ3 (k).

Fig. 4: The state evolution z(k) and its estimate zˆ(k).

ED

M

AN US

Fig. 1: The state evolution x1 (k) and its estimate x ˆ1 (k).

 0 = 0 0

0 0 0

CE

Q1,21

PT

Fig. 2: The state evolution x2 (k) and its estimate x ˆ2 (k).

AC

Q1,22

 0.4626 = 0 0

0 0 0

0 0 0

0 0.4626 0

 0 0 0

 0 0  0.4626

and the matrix Q3 (−1) is supposed to equal Q1 (−1). The value of other initial matrices are Q2 (−1) = 0 and Qs (−2) = 0, s = 1, 2, 3. Using the Matlab LMI Toolbox, the recursive linear matrix inequalities (17) can be solved. Corresponding simulation results are shown in Figs. 1-5. The states of the ANNs (2) and their estimations are depicted in Figs. 1-3. The signal to be estimated z(k) and its estimation zˆ(k) are shown in Fig. 4. The evolution of estimation error z¯(k) is depicted in Fig. 5. The simulation results have verified that the proposed state estimation algorithm performs well, which show that our developed state estimation method in this paper is valid.

Fig. 5: The estimation error evolution z¯(k). V. C ONCLUSION This paper has investigated the finite-horizon H∞ state estimation for a class of ANNs with component-based distributed delays as well as the stochastic protocol. The effects of the stochastic protocol scheduling have been modeled by a series of independent and identically distributed (i.i.d.) random variables. A finite-horizon H∞ performance requirement has been proposed to appraise the estimation performance for the con-

ACCEPTED MANUSCRIPT 9

REVISED

R EFERENCES

CE

PT

ED

M

[1] F. E. Alsaadi, Y. Luo, Y. Liu and Z. Wang, “State estimation for delayed neural networks with stochastic communication protocol: The finite-time case,” Neurocomputing, vol. 281, pp. 86-95, Mar. 2018 [2] M. Areak and P. Kokotovi´c, “Nonlinear observers: a circle criterion design and robustness analysis,” Automatica, vol. 37, no. 12, pp. 19231930, Dec. 2001 [3] P. T. Bharathi and P. Subashini, “Optimization of image processing techniques using neural networks: a review,” WSEAS Transactions on Information Science and Applications, vol. 8, no. 8, pp. 300-328, Aug. 2011. [4] S. Boyd, L. E. Ghaoul, E. Feron and V. Balakrishnan, “Linear matrix inequalities in systems and control theory,” Society for Industrial and Applied Mathematics, 1994. [5] Z. Bu, G. Gao, H.-J. Li, and J. Cao, CAMAS: A cluster-aware multiagent system for attributed graph clustering, Information Fusion, vol. 37, pp. 10-21, 2017. [6] J. Cao, Z. Wu, J. Wu, and H. Xiong, SAIL: Summation-based incremental learning information-theoretic text clustering, IEEE Transactions on Cybernetics, vol. 43, no. 2, pp. 570-584, 2013. [7] J. Cao, Z. Wu, and J. Wu, Scaling up cosine interesting pattern discovery: a depth-first method, Information Sciences, vol. 266, pp. 31-46, 2014. [8] J. Cao, B. Wang, and B. Douglas, Similarity based leaf image retrieval using multiscale R-angle description, Information Sciences, vol. 374, pp. 51-64, 2016. [9] J. Cao, Z. Wu, J. Wu, and W. Liu, Towards information-theoretic Kmeans clustering for image indexing, Signal Processing, vol. 93, no. 7, pp. 2026-2037, 2013. [10] J. Cao, Z. Wu, Y. Wang, and Y. Zhuang, Hybrid collaborative filtering algorithm for bidirectional web service recommendation, Knowledge and Information Systems, vol. 36, no. 3, pp. 607-627, 2013. [11] J. Cao, Z. Wu, B. Mao, and Y. Zhang, Shilling attack detection utilizing semi-supervised learning method for collaborative recommender system, World Wide Web Journal: Internet and Web Information Systems, vol. 16, no. 5-6, pp. 729-748, 2013. [12] J. Cao, Z. Bu, G. Gao, and H. Tao, Weighted modularity optimization for crisp and fuzzy community detection in large-scale networks, Physica A: Statistical Mechanics and its Applications, vol. 462, pp. 386-395, 2016. [13] H. Chen, M. Liu and S. Zhang, “Event-triggered distributed dynamic state estimation with imperfect measurements over a finite horizon,” IET Control Theory and Applications , vol. 11, no. 15, pp. 2607-2622, Oct. 2017. [14] Y. Chen, Z. Wang, W. Qian and F.E. Alsaadi, “Finite-horizon H∞ filtering for switched time-varying stochastic systems with random sensor nonlinearities and packet dropouts,” Signal Processing, vol. 138, pp. 138-145, Sep. 2017.

AC

CR IP T

[15] L. O. Chua and L. Yang, “Cellular neural networks: applications,” IEEE Transactions on Circuits and Systems, vol. 35, no. 10, pp. 1273-1290, Oct. 1998. [16] M. A. Cohen and S. Grossberg, “Absolute stability of global pattern formation and parallel memory storage by competitive neural networks,” IEEE Transactions on Systems, Man, and Cybernetics, vol. SMC-13, no. 5, pp. 815-826, Sep./Oct. 1983. [17] M. C. F. Donkers, W. P. M. H. Heemels, N. van de Wouw and L. Hetel, “Stability analysis of networked control systems using a switched linear systems approach,” IEEE Transactions on Automatic Control, vol. 56, no. 9, pp. 2101-2115, Sep. 2011. [18] X. Feng, F. Zhou and C. Wen, “Robust fusion filtering for multisensor time-varying uncertain systems: the finite horizon case,” Discrete Dynamics in Nature and Society , Article ID 2720549, 13 pages, 2016. [19] W. Gao and R. R. Selmic, “Neural network control of a class of nonlinear systems with actuator saturation,” IEEE Transactions on Neural Networks, vol. 17, no. 1, pp. 147-156, Jan. 2006. [20] F. Han, G. Wei, D. Ding and Y. Song, “Finite-horizon bounded H∞ synchronisation and state estimation for discrete-time complex networks: local performance analysis,” IET Control Theory and Applications, vol. 11, no. 6, pp. 827-837, Apr. 2017. [21] J. J. Hopfield and D. W. Tank, “Computing with neural circuits: a model,” Science, vol. 233, no. 4764, pp. 625-633, Aug. 1986. [22] G. Joya, M. Atencia and F. Sandoval, “Hopfield neural networks for optimization: study of the different dynamics,” Neurocomputing, vol. 43, pp. 219-237, Mar. 2002. [23] X. Kan, H. Shu and Z. Li, “Robust state estimation for discrete-time neural networks with time-delays, linear fractional uncertainties and successive packet dropouts,” Neurocomputing, vol. 135, pp. 130-138, Jul. 2016. [24] D. Lin, J. Wu, J. Cai and J. Li, “ Mean-square asymptotic synchronization control of discrete-time neural networks with restricted disturbances and missing data,” IEEE Access, vol. 6, pp. 10240-10248, Mar. 2018. [25] J. Li, G. Wei, D. Ding and Y. Li, “Set-membership filtering for discrete time-varying nonlinear systems with censored measurements under Round-Robin protocol,” Neurocomputing, vol. 281, pp. 20-26, Mar. 2018. [26] N. Li, H. Sun and Q. Zhang, “Adaptive semi-periodically intermittent and lag synchronization control of neural networks with mixed delays,” IEEE Access, vol. 6, pp. 4742-4749, 2018. [27] Q. Li, B. Shen and Y. Liu, “Event-triggered H∞ state estimation for discrete-time neural networks with mixed time delays and sensor saturations,” Neurocomputing, vol. 28, no. 12, pp. 3815-3825, Dec. 2017. [28] J. Liu, Q. Liu, J. Cao, and Y. Zhang, Adaptive event-triggered H∞ filtering for T-S fuzzy system with time delay, Neurocomputing, vol. 189, pp. 86-94, 2016. [29] J. Liu, J. Cao, Z. Wu, and Q. Qi, State estimation for complex systems with randomly occurring nonlinearities and randomly missing measurements, International Journal of Systems Science, vol. 45, no. 7, pp. 1364-1374, 2014. [30] Y. Liu, Z. Wang, X. He and D. Zhou, “Finite-horizon quantized H∞ filter design for a class of time-varying systems under eventtriggered transmissions,” Systems and Control Letters, vol. 103, pp. 3844, May 2017. [31] Y. Luo, Z. Wang, G. Wei, F.E. Alsaadi and H. Tasawar, “State estimation for a class of artificial neural networks with stochastically corrupted measurements under Round-Robin protocol,” Neural Networks, vol. 77, pp. 70-79, Mar. 2016. [32] D. Peng, X. Li, C. Aouiti and F. Miaadi, “Finite-time synchronization for Cohen-Grossberg neural networks with mixed time-delays,” Neurocomputing, vol. 294, pp. 39-47, Jun. 2018. [33] M. Seera, C. Lim, K. Tan and W. Liew, “Classification of transcranial Doppler signals using individual and ensemble recurrent neural networks,” Neurocomputing, vol. 249, pp. 337-344, Aug. 2017. [34] M. Tabbara and D. Neˇsi´c, “Input-output stability of networked control systems with stochastic protocols and channels,” IEEE Transactions on Automatic Control, vol. 53, no. 5, pp. 1160-1175, Jun. 2008. [35] Z. Tang, J. H. Park and J. Reng, “Impulsive effects on quasisynchronization of neural networks with parameter mismatches and time-varying delay,” IEEE Transactions on Neural Networks and Learning Systems, vol. 29, no. 4, pp. 908-919, Apr. 2018. [36] H. Wang and G. Zhou, “State of charge prediction of supercapacitors via combination of Kalman filtering and backpropagation neural network,” IET Electric Power Applications, vol. 12, no. 4, pp. 588-594, Apr. 2018. [37] J. Wang, X. Zhang and Q. Han, “Event-triggered generalized dissipativity filtering for neural networks with time-varying delays,” IEEE

AN US

sidered ANNs. By constructing a Lyapunov-like functional and using the stochastic analysis method, sufficient conditions have been established to guarantee the estimation error satisfying the prescribed finite-horizon H∞ performance requirement. Next, the expression of estimator parameters has been obtained in terms of a set of recursive linear matrix inequalities. In the end, a simulation example has been provided to illustrate the effectiveness of our state estimation method. Further research topics based on this work include: 1) the extended Kalman estimation problem for time-varying ANNs subject to mixed time-delays and the stochastic protocol scheduling effects; 2) the ultimate boundedness estimation problem for ANNs subject to other communication protocols (e.g. the Round-Robin protocol and Try-Once-Discard protocol); 3) the distributed state estimation problem for time-varying ANNs subject to Component-Based Distributed Delays and Stochastic Protocol; and 4) the utilization of deep learning techniques for parameter identification and applications of the ANNs, see [5], [6], [7], [8], [9], [10], [11], [12], [28], [29], [40], [45].

ACCEPTED MANUSCRIPT 10

REVISED

[45] [46] [47]

[48]

CR IP T

[44]

Zidong Wang (SM’03-F’14) was born in Jiangsu, China, in 1966. He received the B.Sc. degree in mathematics in 1986 from Suzhou University, Suzhou, China, and the M.Sc. degree in applied mathematics in 1990 and the Ph.D. degree in electrical engineering in 1994, both from Nanjing University of Science and Technology, Nanjing, China. He is currently a Professor of Dynamical Systems and Computing in the Department of Computer Science, Brunel University London, U.K. From 1990 to 2002, he held teaching and research appointments in universities in China, Germany and the UK. Prof. Wang’s research interests include dynamical systems, signal processing, bioinformatics, control theory and applications. He has published more than 400 papers in refereed international journals. He is a holder of the Alexander von Humboldt Research Fellowship of Germany, the JSPS Research Fellowship of Japan, William Mong Visiting Research Fellowship of Hong Kong. Prof. Wang serves (or has served) as the Editor-in-Chief for Neurocomputing, Deputy Editor-in-Chief for International Journal of Systems Science, and an Associate Editor for 12 international journals including IEEE Transactions on Automatic Control, IEEE Transactions on Control Systems Technology, IEEE Transactions on Neural Networks, IEEE Transactions on Signal Processing, and IEEE Transactions on Systems, Man, and Cybernetics-Part C. He is a Fellow of the IEEE, a Fellow of the Royal Statistical Society and a member of program committee for many international conferences.

AN US

[43]

M

[42]

ED

[41]

PT

[40]

CE

[39]

AC

[38]

Transactions on Neural Networks and Learning Systems, vol. 27, no. 1, pp. 77-88, Jan. 2016. L. Wang, Z. Wang, G. Wei and F. E. Alsaadi, “Finite-time state estimation for recurrent delayed neural networks with component-based event-triggering protocol,” IEEE Transactions on Neural Networks and Learning Systems, vol. 29, no. 4, pp. 1046-1057, Apr. 2018. T. Wen, L. Zou, J. Liang and C. Roberts, “Recursive filtering for communication-based train control systems with packet dropouts,” Neurocomputing, vol. 275, pp. 948-957, Jan. 2018. Z. Wu, J. Cao, H. Tao, and Y. Zhuang, A novel noise filter based on interesting pattern mining for bag-of-features images, Expert Systems with Applications, vol. 40, no. 18, pp. 7555-7561, 2013. S. W. Yun, Y. J. Choi and P. Park, “Dynamic output-feedback guaranteed cost control for linear systems with uniform input quantization,” Nonlinear Dynamics, vol. 62, no. 1, pp. 95-104, Oct. 2010. L. Zha, J. Fang, J. Liu and E. Tian, “Event-triggering non-fragile state estimation for delayed neural networks with randomly occurring sensor nonlinearity,” Neurocomputing, vol. 273, pp. 1-8, Jan. 2018. J. Zhang, C. Peng, M. Fei and Y. Tian, “Output feedback control of networked systems with a stochastic communication protocol,” Journal of the Franklin Institute-Engineering and Applied Mathematics, vol. 354, no. 9, pp. 3838-3853, Jun. 2017. Y. Zhang and M. van der Schaar, “Robust reputation protocol design for online communities: a stochastic stability analysis,” IEEE Journal of Selected Topics in Signal Processing, vol. 1, no. 5, pp. 907-920, Oct. 2013. G. Zhu, J. Cao, C. Li, and Z. Wu, A recommendation engine for travel products based on topic sequential patterns, Multimedia Tools and Application, vol. 76, no. 16, pp. 17594-17612, 2017. L. Zou, Z. Wang and H. Gao, “Set-membership filtering for time-varying systems with mixed time-delays under round-robin and weighted tryonce-discard protocols,” Automatica, vol. 74, pp. 341-348, Jan. 2016. L. Zou, Z. Wang, J. Hun and H. Gao, “On H∞ finite-horizon filtering under stochastic protocol: dealing with high-rate communication networks,” IEEE Transactions on Automatic Control , vol. 62, no. 9, pp. 4884-4890, Sep. 2017. L. Zou, Z. Wang, H. Gao and X. Liu, “State estimation for discretetime dynamical networks with time-varying delays and stochastic disturbances under the Round-Robin protocol,” IEEE Transactions on Neural Networks and Learning Systems, vol. 28, no. 5, pp. 1139-1151, May 2017.

Zhongyi Zhao received the B.Eng. degree in electrical engineering and automation from Shandong University of Science and Technology, Qingdao, China, in 2016. He is currently pursuing the Ph.D. degree from the College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao, China. His current research interests include the control and filtering for networked system.

Lei Zou received the B.Sc. degree in automation from Beijing Institute of Petrochemical Technology, Beijing, China, in 2008, the M.Sc. degree in control science and engineering from China University of Petroleum (Beijing Campus), Beijing, China, in 2011 and the Ph.D degree in control science and engineering in 2016 from Harbin Institute of Technology, Harbin, China. From October 2013 to October 2015, he was a visiting Ph.D. student with the Department of Computer Science, Brunel University London, Uxbridge, U.K. He is currently a postdoctoral researcher with the college of electrical engineering and automation, Shandong University of Science and Technology, Qingdao, China. His research interests include nonlinear stochastic control and filtering, as well as networked control under various communication protocols. He is a very active reviewer for many international journals.

Hongjian Liu received his B.Sc. degree in applied mathematics in 2003 from Anhui University, Hefei, China and the M.Sc. degree in detection technology and automation equipments in 2009 from Anhui Polytechnic University, Wuhu, China, and the Ph.D. degree in control theory and control engineering in 2018 from Donghua University, Shanghai, China. He is currently an Associate Professor in the School of Mathematics and Physics, Anhui Polytechnic University, Wuhu, China. Dr. Liu’s current research interests include filtering theory, memristive neural networks and network communication systems. He is a very active reviewer for many international journals.