H∞ filtering for networked systems with hybrid-triggered communication mechanism and stochastic cyber attacks

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Hybrid-driven-based H filtering design for networked systems under stochastic cyber attacks Jinliang Liu, Lili Wei, Engang Tian, Shumin Fei, Jie Cao PII: DOI: Reference:

S0016-0032(17)30524-0 10.1016/j.jfranklin.2017.10.007 FI 3176

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Journal of the Franklin Institute

Received date: Revised date: Accepted date:

4 April 2017 10 September 2017 14 October 2017

Please cite this article as: Jinliang Liu, Lili Wei, Engang Tian, Shumin Fei, Jie Cao, Hybrid-driven-based H filtering design for networked systems under stochastic cyber attacks, Journal of the Franklin Institute (2017), doi: 10.1016/j.jfranklin.2017.10.007

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Hybrid-driven-based H∞ filtering design for networked systems under stochastic cyber attacks Jinliang Liu*a , Lili Weia , Engang Tianb , Shumin Feic , Jie Caoa a College

of Information Engineering, Nanjing University of Finance and Economics, Nanjing, Jiangsu 210023, PR China of Information and Control Engineering Technology, Nanjing Normal University, Nanjing, Jiangsu 210042, PR China c School of Automation, Southeast University, Nanjing, Jiangsu 210096, PR China

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b Institute

Abstract

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This paper concentrates on investigating hybrid-driven-based H∞ filtering design for networked systems under stochastic cyber attacks. Random variables satisfying Bernoulli distribution are introduced to describe the hybrid triggered scheme and stochastic cyber attacks respectively. Firstly, a mathematical H∞ filtering error model with hybrid triggered scheme is constructed under the stochastic cyber attacks. Secondly, by using Lyapunov stability theory and linear matrix inequality (LMI) techniques, the sufficient conditions which can guarantee the stability of augmented filtering system are obtained and the parameters of the designed filter can be presented in an explicit form. Finally, numerical examples are given to demonstrate the feasibility of the designed filter. Key words: Event-triggered scheme, H∞ filtering design, stochastic cyber attacks, hybrid triggered scheme

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1. Introduction

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Networked control systems (NCSs) are a kind of control systems wherein feedback signals and control signals are exchanged through a network in a form of information package. It’s characterized by enabling the execution of several tasks far away when it connects cyber-space to physical space [1]. Due to the advantages such as high flexibility, low cost and simple installation [2, 3], NCSs are applied in different kinds of fields referring to aircrafts, automobiles, vehicles and so on [4–6]. Therefore, more and more scholars have been interested in NCSs owing to the development and the advantages of the Internet [7]. In [8], the robust H∞ controller is designed for networked control systems with uncertainties such like network-induced delay and data dropouts. The synchronization problem of feedback control is investigated in [9] with time-varying delay for complex dynamic networks. The authors in [10] study the robust fault tolerant control problem for distributed network control systems. In the past few years, time-driven method (periodic sampling) is widely adopted for system modeling and analysis in the NCSs, however, periodic sampling will generate lots of redundant signals if all the sampled data is transmitted through the network. To make full use of of the limited network resource, lots of researchers propose the event-triggered schemes to overcome the problem caused by periodic sampling, for example, a novel event-triggered scheme is proposed in [11], in which a H∞ controller is designed for NCSs. The core idea of the novel event-triggered scheme in [11] is that whether the newest sampled data is released or not is dependent on a threshold, and the adoption of event-triggered scheme can largely help alleviate the burden of the network [12]. Consequently, there are large numbers of researchers interested in the investigations about the novel eventtriggered scheme proposed in [11]. An event-triggered non-parallel distribution compensation control problem in [13] is addressed for networked Takagi-Sugeno fuzzy systems. The authors of [14] consider the event-triggered filtering problem for discrete-time linear system with package dropouts satisfying Bernoulli distribution. In [15], a discrete event-triggered scheme is proposed for fuzzy filtering design in a class of nonlinear NCSs. Inspired by the aforementioned event-triggered scheme in [11], the hybrid triggered scheme which consists of time-triggered scheme and event-triggered scheme is firstly proposed in [16], which investigates the problem of control stabilization for networked control systems under the hybrid triggered scheme. Based on the hybrid triggered scheme ∗ Corresponding Author. Tel:+02586718440; Email address : [email protected]

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above, the authors in [17] are concerned with the hybrid-driven-based reliable control design for a class of T-S fuzzy systems with probabilistic actuator faults and nonlinear perturbations. Motivated by the proposed hybrid triggered scheme in [16], this paper is devoted to the hybrid-driven-based H∞ filtering design subjecting to stochastic cyber attacks on the measurement outputs. Due to the insertion of the network in the control systems, challenges including packet dropouts, networkinduced delay and randomly occurring nonlinearities [18–21] are inevitable. It is nonnegligible that another phenomena named cyber attacks can be more destroyable. Cyber attacks are offensive maneuvers which target networked information systems, infrastructures and networked devices by various means of malicious acts. By hacking into a susceptible system, cyber attacks can be the biggest threat to the security of network. As the description in [22], there are three kinds of common attacks containing denial of service attacks [23, 24], relay attacks [25, 26] and deception attacks [27, 28]. With the rapid development of the network, the influence of cyber attacks can not be neglected any more. Based on the cyber attacks mentioned above, lots of researches are investigated and impressive results are yielded. The authors are concerned with extended Kalman filtering design for stochastic nonlinear systems under cyber attacks in [29]. The distributed recursive filtering problem is studied in [30] with quantization and deception attacks for a class of discrete time-delayed systems. In [31], a novel state filtering approach and sensor scheduling co-design with random deception attacks are presented. This paper addresses the issue about a hybrid-driven-based H∞ filtering design for networked systems under stochastic cyber attacks. The main contributions of this paper are as follows. (1) In order to make full use of networked bandwidth and guarantee the desired system performance, the hybrid triggered scheme which consists of time-triggered scheme and event-triggered scheme is introduced. (2) Due to the insertion of network, the stochastic cyber attacks are considered, and the launching probability of cyber attacks is governed by Bernoulli random variable. (3) By taking the hybrid triggered scheme and stochastic cyber attacks into consideration, an H∞ filter is designed for networked systems. Although there are several researches concerned with filtering design, to the best of our knowledge, there is no research investigating the H∞ filtering design by considering both hybrid triggered scheme and cyber attacks for networked systems. The rest of this paper is organized as follows. In the Section 2, a filtering error system is constructed by introducing the hybrid triggered scheme and taking the stochastic cyber attacks into account. Section 3 gives the sufficient conditions which can guarantee the augmented filtering system stable by using Lyapunov functional approach and LMI techniques. Moreover, the design algorithm of H∞ filter is presented and the filtering parameters are obtained in an explicit form. Section 4 gives an illustrative example to demonstrate the usefulness of desired H∞ filter. Notation: Rn and Rn×m denote the n-dimensional Euclidean space, and the set of n × m real matrices; the su-perscript T stands for matrix transposition; I is the identity matrix of appropriate dimension; the notation X > 0(respectively, X ≥ 0), for X ∈ Rn×n means that the matrix X is real symmetric positive definite (respectively, " # A ∗ positive semi-definite). For a matrix B and two symmetric matrices A, C and denotes a symmetric matrix, B C where ∗ denotes the entries implied by symmetry.

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2. Problem description and preliminaries

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In this paper, a hybrid-driven-based H∞ filter for networked system under stochastic cyber attacks is investigated. The framework of hybrid-driven-based H∞ filtering for NCSs under stochastic cyber attacks is shown as Fig.1. From Fig.1, one can see that the framework consists of the sensor, the hybrid triggered scheme, the filter, a zero-order-hold (ZOH), a network channel. Consider the following continuous-time linear system.    x˙(t) = Ax(t) + Bw(t)     (1) y(t) = Cx(t)      z(t) = Lx(t)

where x(t)∈Rn is the state vector, y(t)∈Rm is the ideal measurement, z(t)∈R p is the signal to be estimated, w(t)∈ L2 [0, +∞) represents the disturbance input vector, A, B, C and L are known real matrices with appropriate dimensions. 2

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Hybrid triggered scheme

Time-triggered scheme

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Sensor

Filter

ZOH

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Event-triggered scheme

Network

Cyber attacks

Figure 1: The framework of hybrid-driven-based H∞ filter for NCSs under stochastic cyber attacks

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The purpose of this paper is to design a hybrid-driven-based H∞ filter under stochastic cyber attacks for networked system. Consider the following filter system.     x˙ f (t) = A f x f (t) + B f yˆ (t) (2)   z f (t) = C f x f (t)

where x f (t) is the filter state vector, yˆ (t) is the real input of the filter, z f (t) is the estimation of the z(t), A f , B f and C f are filter parameters to be determined.

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Remark 1. Different from the classic filtering design, this paper investigates the H∞ filtering problem with hybrid triggered scheme and stochastic cyber attacks. As is shown in Fig.1, the data transmission is assumed to work in a non-ideal network condition, hence some uncertainties such as network-induced delay and cyber attacks are taken into consideration.

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Considering the limitation of the networked resources, a hybrid triggered scheme in Fig.1 is used for data transmission to alleviate the burden of the networked communication, which consists of time-triggered scheme and event-triggered scheme. Both of the two schemes are discussed in detail respectively in the following. Time-triggered scheme: Suppose that the sensor is time-driven and each process signal is sampled at periodic intervals. The ideal measurement y1 (t) can be expressed as follows. y1 (t) = Cx(tk h), t ∈ [tk h + τtk , tk+1 h + τtk+1 )

(3)

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where h represents the sampling period, tk (k = {0, 1, 2, · · · }) are integers and {t1 , t2 , t3 , . . .} ⊂ {1, 2, 3, . . .}, the corresponding network-induced delay is represented by τtk . Similar to [8], define τ(t) = t − tk h, eqnarray (3) can be written as follows. y1 (t) = Cx(t − τ(t))

(4)

where τ(t) ∈ [0, τ M ], τ M is the upper bound of the networked delay. Event-triggered scheme: To further enhance the bandwidth utilization, by taking [11] as a reference, an event-triggered scheme is applied to determine whether the current measurements should be transmitted or not. We use kh and tk h to represent the sampling instants and the triggering instants. Once y(tk h) is transmitted, whether the next triggered instant y(tk+1 h) should be transmitted or not is determined by comparing the latest transmitted sampled-data with the error between the latest transmitted data which is shown as follows . tk+1 h = tk h + inf { jh|eTk (tk h)Ωek (tk h) > σyT (tk h)Ωy(tk h)} j>1

where Ω > 0, σ ∈ [0, 1) and j = 1, 2,· · · . The threshold error ek (tk h) = y(tk h) − y(tk h + jh). 3

(5)

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For analyzing more easily, the interval [tk h+τtk , tk+1 h+τtk+1 ) can be divided into several subintervals. Suppose S that there exists a constant g which satisfying [tk h + τtk , tk+1 h + τtk+1 ) = gj=1 Λ j , where Λ j = [tk h + jh + ηt+ j , tk h + jh + h + ηt+ j+1 ], j = {1, 2, · · · , g}, g = tk+1 − tk − 1. Define η(t) = t − tk h − jh, 0 ≤ τtk ≤ η(t) ≤ h + ηtk+ j+1 , η M . Let ek (t) = x(tk h) − x(tk h + lh), the measurement y(t) can be written as y2 (t) = Cx(t − η(t)) + Cek (t)

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Remark 2. From the event-triggered judgement algorithm above, for a given period h, the sampler samples the data can be found at time kh, and the next sensor measurement is at time (k + 1)h. Suppose that t0 h, t1 h, t2 h, · · · are the release time, then it is easily obtained that si h = ti+1 h − ti h denote the release period of event generator in (5), where si h mean that the sampling instants between the two conjoint releasing instants. Remark 3. According to the event-triggered algorithm (5), the set of the releasing instants {t1 , t2 , t3 , . . .} ⊆ {0, 1, 2, · · · }. Learn from the research [8], it is not required that tk+1 > tk . The packet dropout will not occur only when {t1 , t2 , t3 , . . .} = {1, 2, 3, . . .}. If tk+1 = tk + 1, then h + τtk+1 > τtk , two special cases about τtk = τ¯ and τtk < h imply the packet dropout and the network-induced delay, where τ¯ is a constant. Thus, the frequency of releasing instants depends on the value of σ and the variation of the sensor measurements.

y¯ (t) = =

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Combine (4) in time-triggered scheme and (6) in event-triggered scheme, similar to [16], the measurement y¯ (t) via hybrid triggered scheme can be expressed as follows. α(t)y1 (t) + (1 − α(t))y2 (t)

  α(t)Cx(t − τ(t)) + (1 − α(t)) Cx(t − η(t)) + Cek (t)

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where α(t) ∈ [0, 1], α¯ is utilized to represent the expectation of α(t), and ρ21 represents the mathematical variance of α(t).

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Remark 4. To describe the stochastic switching rule between the time-triggered scheme and the event-triggered scheme, the random variable α(t) which satisfies the Bernoulli distribution is introduced. In (7), when α(t) = 1, y¯ (t) = Cx(t − τ(t)), it is observed that the data is transmitted via time-triggered scheme; When α(t) = 0, y¯ (t) = Cx(t − η(t)) + Cek (t), then, the event-triggered scheme is activated in data transmission.

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The cyber attacks in this paper belong to deception attacks which aim to destroy the stability and performance of networked system. A nonlinear function f (x(t)) is utilized to describe the deception attacks which is assumed to satisfy the following condition.

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Assumption 1. Suppose that deception attacks f (x(t)) satisfy the following condition. || f (x(t))||2 ≤ ||Gx(t)||2

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where G is a constant matrix representing the upper bound of the nonlinearity.

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Remark 5. In order to describe the restrictive condition of nonlinear perturbation, the information of upper bound is introduced in [35, 36]. Similarly, we use matrix G to represent the upper bound of stochastic cyber attacks in Assumption 1, and its value depends on the actual situation of networked attacks. In the transmitting process, normal signals are subject to deception attacks randomly in the networked channel, then we use variable d(t) to represent the time-varying delay of the aggressive signals which are delivered to the filter. By using the similar methods in [22, 32], the Bernoulli variable θ(t) is introduced to govern the stochastic cyber-attacks, then the real input yˆ (t) of filter can be written as yˆ (t) = =

θ(t)C f (x(t − d(t))) + (1 − θ(t))¯y(t)    θ(t)C f (x(t − d(t))) + (1 − θ(t)) α(t)Cx(t − τ(t)) + (1 − α(t)) Cx(t − η(t)) + Cek (t)

(9)

where d(t)∈[0, d M ], θ(t) ∈ [0, 1]. θ¯ is utilized to represent the expectation of θ(t), ρ22 is utilized to represent the mathematical variance of θ(t). 4

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Remark 6. Bernoulli variables are used to describe the stochastic characteristic in the control systems. In [33], the stochastic delay is described by random Bernoulli variable for NCSs. In [34], the occurring probabilities of the two different sampling periods are described by Bernoulli variable. In this paper, the random variables α(t) and θ(t) which satisfy Bernoulli distribution are used to describe the stochastic switching changes between the two different schemes and the stochastic cyber attacks, respectively. It is noted that the Bernoulli variables α(t) and θ(t) are mutually independent.

Define

e(t) =

"

# x(t) , z˜(t) = z(t) − z f (t) x f (t)

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According to (9), we can obtain the real input yˆ (t) of the filter. Substitute yˆ (t) into (2), then the filter can be written as follows.    x˙ f (t) = A f x f (t) + B f {θ(t)C f (x(t − d(t))) + (1 − θ(t)){α(t)Cx(t − τ(t))       (10) +(1 − α(t)) Cx(t − η(t)) + Cek (t) }}      z f (t) = C f x f (t)

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Based on (1) and (10), the filtering error system can be described as   ¯  e˙ (t) = A¯ f e(t) + (1 − θ(t))α(t) B¯ f He(t − τ(t)) + Bw(t) + (1 − θ(t))(1 − α(t)) B¯ f He(t − η(t))     +(1 − θ(t))(1 − α(t)) B¯ f ek (t) + θ(t) B¯ f f (x(t − d(t)))      z˜(t) = C¯ f e(t)

where

# " # " # h 0 ¯ 0 B ¯ , Bf = ,B = ,H = I Af Bf C 0

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" A ¯ Af = 0

Some important lemmas are introduced in the following.

i h 0 , C¯ f = L

−C f

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Lemma 1. [37] For any vectors x, y ∈ Rn , and positive definite matrix Q ∈ Rn×n , the following inequality holds. 2xT y ≤ xT Qx + yT Q−1 y

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Lemma 2. [38] Suppose τ(t) ∈ [0, τ M ], d(t) ∈ [0, d M ], η(t) ∈ [0, η M ], Ξ1 , Ξ2 , Ξ3 , Ξ4 , Ξ5 , Ξ6 and Ω are matrices with appropriate dimensions, then τ(t)Ξ1 + (τ M − τ(t))Ξ2 + d(t)Ξ3 + (d M − d(t))Ξ4 + η(t)Ξ5 + (η M − η(t))Ξ6 + Ω < 0

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if and only if

3. Main results

   τ M Ξ1 + d M Ξ3 + η M Ξ5 + Ω < 0,       τ M Ξ1 + d M Ξ4 + η M Ξ5 + Ω < 0,    τ M Ξ1 + d M Ξ3 + η M Ξ6 + Ω < 0,      τ M Ξ1 + d M Ξ4 + η M Ξ6 + Ω < 0,

τ M Ξ2 + d M Ξ3 + η M Ξ5 + Ω < 0 τ M Ξ2 + d M Ξ4 + η M Ξ5 + Ω < 0 τ M Ξ2 + d M Ξ3 + η M Ξ6 + Ω < 0 τ M Ξ2 + d M Ξ4 + η M Ξ6 + Ω < 0

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In this section, by using Lyapunov functional approach, the main results will be summarized and the sufficient conditions which can guarantee the stability of networked system will be obtained. ¯ α, Theorem 1. For given positive parameters θ, ¯ τ M , η M , d M , σ, ρi (i = 1, 2) and k , matrix G, with hybrid triggered scheme and deception attacks, system (11) is asymptotically stable with an H∞ disturbance attenuation level γ, if there exist matrixes P > 0, Z > 0, Qk > 0, Rk > 0 (k = 1, 2, 3), Ω > 0 and M, N, U, S , W, V with appropriate dimensions satisfying 5

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Ω21

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Ω41 (1)

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Ω41 (5)

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∗ ∗ ∗ ∗ −Q2 0 0 0 0 0

∗ ∗ ∗ ∗ ∗ ∗ Ω77 0 0

∗ ∗ ∗ ∗ ∗ 0 0 0 0 0

∗ ∗ ∗ ∗ ∗ ∗ −Q3 0 0 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ −C T ΩC 0 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ Ω88 0

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Ω61

Ω71

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Ω81

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Ω91

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θ¯1

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Ω55

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 ∗   ∗   ∗   ∗   ∗  < 0(s = 1, · · · , 8)  ∗   ∗   ∗  Ω99

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∗ ∗ ∗ Φ33 0 0 0 0 0 0

∗ ∗ ∗ ∗ ∗ Ω66 0 0 0

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −γ2 I 0

 ∗   ∗   ∗  ∗   ∗   ∗   ∗  ∗   ∗  ¯  −θZ

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∗ ∗ ∗ ∗ Ω55 0 0 0 0

PA¯ f + A¯ Tf P + Q1 + Q2 + Q3 , Φ21 = θ¯1 αH ¯ T B¯ Tf P, Φ31 = θ¯1 α¯ 1 H T B¯ Tf P σC T H T ΩHC, Φ41 = θ¯1 α¯ 1 B¯ Tf P, Υ1 = θ¯1 αP ¯ B¯ f H, Υ2 = θ¯1 α¯ 1 P B¯ f H h i M + U + W N − M −N −U + S −S −W + V −V 0 0 0 h i h i C¯ f 0 0 0 0 0 0 0 0 0 , Ω31 = 0 0 0 0 0 θ¯2 ZGH 0 0 0 0 √  √  √  √   τ M M T   τ M M T   τ M M T   τ M M T  √ √ √  √       T T T T  √ η M U  , Ω41 (2) =  √η M U  , Ω41 (3) =  √ η M S  , Ω41 (4) =  √η M S  dM W T dM V T dM W T dM V T        √ √ √ √  τ M N T   τ M N T   τ M N T   τ M N T  √ √ √        √ T T T T  √ η M U  , Ω41 (6) =  √η M U  , Ω41 (7) =  √ η M S  , Ω41 (8) =  √η M S  dM W T dM V T dM W T dM V T √  √ √ √ √ √ τ M Υ1 0 τ M Υ2 0 0 0 θ¯1 α¯ 1 τ M P B¯ f τ M P B¯ θ¯ τ M P B¯ f   τ M PA¯ f √ √ √ √ √  √   √η M PA¯ f √η M Υ1 0 √η M Υ2 0 0 0 θ¯1 α¯ 1 √η M P B¯ f √η M P B¯ θ¯ √η M P B¯ f  d M PA¯ f d M Υ1 0 d M Υ2 0 0 0 θ¯1 α¯ 1 d M P B¯ f d M P B¯ θ¯ d M P B¯ f   √ √ √ 0 θ¯1 ρ1 τ M P B¯ f H 0 −θ¯1 ρ1 τ M P B¯ f H 0 0 0 −θ¯1 ρ1 τ M P B¯ f 0 0 √ √ √   0 θ¯1 ρ1 √η M P B¯ f H 0 −θ¯1 ρ1 √η M P B¯ f H 0 0 0 −θ¯1 ρ1 √η M P B¯ f 0 0 0 θ¯1 ρ1 d M P B¯ f H 0 −θ¯1 ρ1 d M P B¯ f H 0 0 0 −θ¯1 ρ1 d M P B¯ f 0 0   √ √ √ ¯ 2 τ M P B¯ f H 0 α¯ 1 ρ2 τ M P B¯ f H 0 0 0 α¯ 1 ρ2 τ M P B¯ f 0 0 0 αρ √ √ √   ¯ 2 √η M P B¯ f H 0 α¯ 1 ρ2 √η M P B¯ f H 0 0 0 α¯ 1 ρ2 √η M P B¯ f 0 0 0 αρ  0 αρ ¯ 2 d M P B¯ f H 0 α¯ 1 ρ2 d M P B¯ f H 0 0 0 α¯ 1 ρ2 d M P B¯ f 0 0   √ √ √ 0 ρ1 ρ2 τ M P B¯ f H 0 −ρ1 ρ2 τ M P B¯ f H 0 0 0 −ρ1 ρ2 τ M P B¯ f 0 0 √ √ √   0 ρ1 ρ2 √η M P B¯ f H 0 −ρ1 ρ2 √η M P B¯ f H 0 0 0 −ρ1 ρ2 √η M P B¯ f 0 0 ¯ ¯ ¯ 0 ρ1 ρ2 d M P B f H 0 −ρ1 ρ2 d M P B f H 0 0 0 −ρ1 ρ2 d M P B f 0 0   √ 0 0 0 0 0 0 0 0 0 ρ2 τ M P B¯ f  √   0 0 0 0 0 0 0 0 0 ρ2 √η M P B¯ f  ¯ 0 0 0 0 0 0 0 0 0 ρ2 d M P B f p ¯ Ω44 = diag{−R1 , −R2 , −R3 } ¯ α¯ 1 = 1 − α, 1 − θ, ¯ ρ21 = α¯ α¯ 1 , ρ22 = θ¯ θ¯1 , θ¯2 = θ,

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∗ 0 0 0 0 0 0 0 0 0

∗ ∗ ∗ Ω44 0 0 0 0 0

CE

Ω11

  Φ11  Φ  21  0   Φ31   0  0   0   Φ41  ¯ T  B P θ¯ B¯ Tf P

∗ ∗ −Z 0 0 0 0 0 0

ED

where

=

∗ −I 0 0 0 0 0 0 0

PT

Ω(s)

 T Ω11 + Γ + Γ  Ω21  Ω31   Ω (s) 41   Ω 51   Ω61  Ω71   Ω 81  Ω91

−1 −1 Ω66 = Ω77 = Ω88 = Ω99 = {−PR−1 1 P, −PR2 P, −PR3 P} 6

(15)

ACCEPTED MANUSCRIPT

MT

=

UT

=

WT

=

h

M1T

h U1T h W1T

M2T

0

0

0

0

U4T

0

0

0

0

0

0

0

0

W6T

i h i 0 , N T = 0 N2T N3T 0 0 0 0 0 0 0 i h i 0 0 0 0 , S T = 0 0 0 S 4T S 5T 0 0 0 0 0 i h i 0 0 0 0 , V T = 0 0 0 0 0 V6T V7T 0 0 0 0

0

0

Proof. Choose the following Lyapunov functional candidate as

V(t) = V1 (t) + V2 (t) + V3 (t)

(16)

where eT (t)Pe(t) Zt Zt Zt T T e (s)Q1 e(s)ds + e (s)Q2 e(s)ds + eT (s)Q3 e(s)ds

=

V2 (t) =

t−τ M

t−η M

Zt Zt

V3 (t) =

t−d M

e˙ T (v)R1 e˙ (v)dvds +

t−τ M s

Zt Zt

CR IP T

V1 (t)

e˙ T (v)R2 e˙ (v)dvds +

t−η M s

Zt Zt

e˙ T (v)R3 e˙ (v)dvds

t−d M s

E{LV2 (t)}

=

E{LV3 (t)}

=

2eT (t)P[A¯ f e(t) + θ¯1 α¯ B¯ f He(t − τ(t)) + θ¯1 α¯ 1 B¯ f He(t − η(t)) + θ¯1 α¯ 1 B¯ f ek (t) ¯ + Bw(t) + θ¯ B¯ f f (x(t − d(t)))]

(17)

−eT (t − d M )Q3 e(t − d M )

(18)

eT (t)(Q1 + Q2 + Q3 )e(t) − eT (t − τ M )Q1 e(t − τ M ) − eT (t − η M )Q2 e(t − η M ) n o E e˙ T (t)(τ M R1 + η M R2 + d M R3 )˙e(t) − −

Zt

M

=

Zt

t−τ M

e˙ T (s)R1 e˙ (s)ds −

Zt

e˙ T (s)R2 e˙ (s)ds

t−η M

e˙ T (s)R3 e˙ (s)ds

ED

E{LV1 (t)}

AN US

and P>0, Qk >0, Rk >0 (k = 1, 2, 3). By applying the infinitesimal operator (15) for Vk (t) (k = 1, 2, 3) and taking expectation on it, we can obtain

(19)

t−d M

PT

Notice that

where

CE

n o E e˙ T (t)(τ M R1 + η M R2 + d M R3 )˙e(t)

A =

˜ + θ¯12 ρ21 BT1 RB ˜ 1 + ρ22 BT2 RB ˜ 2 + ρ21 ρ22 BT1 RB ˜ 1 AT RA +ρ22 f T (x(t − d(t))) B¯ Tf R˜ B¯ f f (x(t − d(t)))

(20)

¯ A¯ f e(t) + θ¯1 α¯ B¯ f He(t − τ(t)) + θ¯1 α¯ 1 B¯ f He(t − η(t)) + θ¯1 α¯ 1 B¯ f ek (t) + Bw(t) + θ¯ B¯ f f (x(t − d(t))) B¯ f [He(t − τ(t)) − He(t − η(t)) − ek (t)]

AC

B1

=

B2 R˜

=

=

=

B¯ f [αHe(t ¯ − τ(t)) + α¯ 1 He(t − η(t)) + α¯ 1 ek (t)]

τ M R1 + η M R2 + d M R3

From the assumption in (8), we can obtain ¯ T (t − d(t))H T GT ZGHe(t − d(t)) − θ¯ f T (x(t − d(t))Z f (x(t − d(t))) ≥ 0 θe

Applying the free-weighting matrices method [8, 39], it can be obtained that   Zt     2ξT (t)M e(t) − e(t − τ(t)) − e˙ (s)d s  = 0   7

t−τ(t)

(21)

(22)

ACCEPTED MANUSCRIPT

  t−τ(t) Z     e˙ (s)d s  = 0 2ξ (t)N e(t − τ(t)) − e(t − τ M ) −   T

(23)

t−τ M

  Zt      T  e˙ (s)d s  = 0 2ξ (t)U e(t) − e(t − η(t)) −  

(24)

t−η(t)

CR IP T

  t−η(t) Z     T e˙ (s)d s  = 0 2ξ (t)S e(t − η(t)) − e(t − η M ) −   t−η M   Zt     e˙ (s)d s  = 0 2ξT (t)W e(t) − e(t − d(t)) −   t−d(t)

  t−d(t) Z      T  e˙ (s)d s  = 0 2ξ (t)V e(t − d(t)) − e(t − d M ) −   t−d M

(25)

(26)

(27)

By Lemma 1, we have

−2ξ (t)M

Zt

T

t−τ(t) Z

e˙ (s)ds ≤ τ(t)ξ

t−τ(t)

T (t)MR−1 1 M ξ(t)

e˙ (s)ds ≤ (τ M − τ(t))ξ

t−τ M

t−η(t)

t−η(t) Z

T

CE

−2ξ (t)S

t−η M

AC

−2ξT (t)W T

−2ξ (t)V

e˙ (s)ds ≤ (η M − η(t))ξ

Zt

T

e˙ T (s)R1 e˙ (s)ds

T (t)NR−1 1 N ξ(t)

(28)

+

t−τ(t) Z

e˙ T (s)R1 e˙ (s)ds

(29)

t−τ M

Zt

e˙ T (s)R2 e˙ (s)ds

(30)

t−η(t)

T

T (t)S R−1 2 S ξ(t)

t−η(t) Z

e˙ T (s)R2 e˙ (s)ds

+

(31)

t−η M

T e˙ (s)ds ≤ d(t)ξT (t)WR−1 3 W ξ(t) +

t−d(t)

Zt

e˙ T (s)R3 e˙ (s)ds

(32)

t−d(t)

t−d(t) Z

t−d M

+

T e˙ (s)ds ≤ η(t)ξT (t)UR−1 2 U ξ(t) +

PT

−2ξT (t)U

Zt

Zt

t−τ(t)

ED

−2ξ (t)N

T

M

T

AN US

where N, M, T, S , W, V are matrices with appropriate dimensions, and h i ξT (t) = ξ1T (t) ξ2T (t) h i ξ1T (t) = eT (t) eT (t − τ(t)) eT (t − τ M ) eT (t − η(t) eT (t − η M ) h i ξ2T (t) = eT (t − d(t) eT (t − d M ) eTk (t) wT (t) f T (x(t − d(t)))

e˙ (s)ds ≤ (d M − d(t))ξ

T

T (t)VR−1 3 V ξ(t)

+

t−d(t) Z

e˙ T (s)R3 e˙ (s)ds

(33)

t−d M

Considering the condition of event-triggered scheme (5), we can obtain that σeT (t − η(t))C T H T ΩHCe(t − η(t)) − eTk (t)C T ΩCek (t) ≥ 0

(34)

By applying free-weighting matrixes method and reciprocally convex approach, combining (16) − (34), we can obtain that

=

E{LV(t) + z˜T (t)˜z(t) − γ2 wT (t)w(t)}

2eT (t)PA + eT (t)(Q1 + Q2 + Q3 )e(t) − eT (t − τ M )Q1 e(t − τ M ) − eT (t − η M )Q2 e(t − η M ) 8

ACCEPTED MANUSCRIPT

−e (t − d M )Q3 x(t − d M ) + τ(t)ξ

T

T (t)MR−1 1 M ξ(t)

T

+2ξ (t)U

e˙ (s)ds + (η M − η(t))ξ

t−η(t)

T T +d(t)ξT (t)WR−1 3 W ξ(t) + 2ξ (t)W

T

Zt

t−d(t) Z

t−d M

e˙ (s)ds

t−τ(t) Z

T e˙ (s)ds + η(t)ξT (t)UR−1 2 U ξ(t)

t−τ M

T (t)S R−1 2 S ξ(t)

t−d(t)

+2ξT (t)V

+ 2ξ (t)M

Zt

t−τ(t)

T T +(τ M − τ(t))ξT (t)NR−1 1 N ξ(t) + 2ξ (t)N

Zt

T

T

+ 2ξ (t)S

t−η(t) Z

e˙ (s)ds

t−η M

CR IP T

T

T e˙ (s)ds + (d M − d(t))ξT (t)VR−1 3 V ξ(t)

˜ + θ¯1 ρ21 BT1 RB ˜ 1 + ρ22 BT2 RB ˜ 2 + ρ21 ρ22 BT1 RB ˜ 1 e˙ (s)ds + AT RA

−θ f (x(t − d(t))Z f (x(t − d(t)))

AN US

+ρ22 f T (x(t − d(t))) B¯ Tf R˜ B¯ f f (x(t − d(t))) + σeT (t − η(t))C T H T ΩHCe(t − η(t)) ¯ T (t − d(t))H T GT ZGHe(t −eTk (t)C T ΩCek (t) − γ2 wT (t)w(t) + eT (t)C¯ Tf C¯ f e(t) + θe ¯ T

− d(t))

ξT (t)[Ω11 + Γ + ΓT − ΩT21 Ω21 − ΩT31 PΩ31 + ΩT51 Ω55 Ω51 + ΩT61 Ω66 Ω61 + ΩT71 Ω77 Ω71

≤

T −1 T −1 T +ΩT81 Ω88 Ω81 + ΩT91 Ω99 Ω91 + τ(t)MR−1 1 M + (τ M − τ(t))NR1 N + η(t)UR2 U

T −1 T −1 T +(η M − η(t))S R−1 2 S + d(t)WR3 W + (d M − d(t))VR3 V ]ξ(t)

(35)

M

Similar to [12, 40], by using Schur complement and Lemma 2, we can obtain that (15) can guarantee E{L(V(t)) + z˜T (t)˜z(t) − γ2 wT (t)w(t)} < 0. The proof is completed. 

ED

Based on Theorem 1, the sufficient conditions have been obtained which can guarantee the stability of system. In order to solve the nonlinear terms in Theorem 1, LMI techniques are used to compute the parameters of the desired filter in the following.

AC

CE

PT

¯ α, Theorem 2. For given positive parameters γ, θ, ¯ τ M , η M , d M , σ, ρi (i = 1, 2) and k , matrix G, system (11) is asymptotically stable with hybrid triggered scheme and stochastic cyber attacks, if there exist matrices P1 > 0, ¯ N, ¯ U, ¯ S¯ , W, ¯ V¯ are matrixes with appropriate P¯ 3 > 0, Z > 0, Q¯ k > 0, R¯ k > 0 (k = 1, 2, 3), Ω > 0, Aˆ f , Bˆ f , Cˆ f , M, dimensions, such that the following LMIs hold. ˜  ∗ ∗ ∗ ∗ ∗ ∗ ∗  Ω11 + Γ˜ + Γ˜ T ∗   ˜ 21 Ω −I ∗ ∗ ∗ ∗ ∗ ∗ ∗    ˜ 31  Ω 0 −Z ∗ ∗ ∗ ∗ ∗ ∗    ˜ 41 (s) ˜ 44 0 0 Ω ∗ ∗ ∗ ∗ ∗   Ω  ˜ 51 ˜ 55 ˜ Ω 0 0 0 Ω ∗ ∗ ∗ ∗  < 0(s = 1, · · · , 8) Ω(s) =  (36)  ˜ 61 ˜ 66 Ω 0 0 0 0 Ω ∗ ∗ ∗     ˜ 71 ˜ 77 Ω 0 0 0 0 0 Ω ∗ ∗    ˜ 81 ˜ 88 Ω 0 0 0 0 0 0 Ω ∗   ˜ 91 ˜ 99 Ω 0 0 0 0 0 0 0 Ω P1 − P¯ 3 > 0 (37)

where ˜ 11 Φ

=

ψ11

=

Γ˜

=

˜ 21 Ω

=

h i h i ψ11 + ψT11 + Q˜ 1 + Q˜ 2 + Q˜ 3 , ψ31 = C T Bˆ Tf C T Bˆ Tf , ψ41 = BT P1 BT P¯ 3 " # " T T # " T # " # C Bˆ f C T Bˆ Tf P1 A Aˆ f σC ΩC 0 ˜ P1 p¯ 3 , ψ21 = , ψ44 = ,P = ¯ 0 0 P3 p¯ 3 P¯ 3 A Aˆ f 0 0 h i ˜ + U˜ + W ˜ N˜ − M ˜ −N˜ −U˜ + S˜ −S˜ −W ˜ + V˜ −V˜ 0 0 0 M h i h i ψ51 0 0 0 0 0 0 0 0 0 , ψ51 = L −Cˆ f 9

ACCEPTED MANUSCRIPT

˜ 41 (5) Ω

=

˜ 55 Ω

=

˜ 51 Ω

=

˜ 61 Ω

=

˜ 71 Ω

=

˜ 81 Ω

=

˜ 91 Ω

=

˜T M

=

U˜ T

=

˜T W

CR IP T

=

AN US

˜ 41 (1) Ω

M

=

ED

˜ 31 Ω

PT

=

CE

˜ 11 Ω

  ˜ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗   Φ11   θ¯ αψ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗   1 ¯ 21 0   0 0 −Q˜ 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗    0 ψ44 ∗ ∗ ∗ ∗ ∗ ∗  θ¯1 α¯ 1 ψ21 0   0 0 0 0 −Q˜ 2 ∗ ∗ ∗ ∗ ∗     0 0 0 0 0 0 ∗ ∗ ∗ ∗    0 0 0 0 0 0 −Q˜ 3 ∗ ∗ ∗    ¯ T 0 0 0 0 0 −C ΩC ∗ ∗  θ1 α¯ 1 ψ31 0   ψ 2 0 0 0 0 0 0 0 −γ I ∗  41  ¯ 31 ¯  θψ 0 0 0 0 0 0 0 0 −θZ h i ˜ 44 = diag{−R˜ 1 , −R˜ 2 , −R˜ 3 } 0 0 0 0 0 θ¯2 ZG 0 0 0 0 , Ω √    √ √ √ ˜ T ˜ T ˜ T  ˜ T   τ M M  τ M M  τ M M  τ M M  √ ˜ T  ˜  √ ˜ T  ˜  √ ˜ T  ˜  √ ˜ T   √ η M U  , Ω41 (2) =  √η M U  , Ω41 (3) =  √ η M S  , Ω41 (4) =  √η M S  ˜T ˜T dM W d M V˜ T dM W d M V˜ T √ ˜T  √ ˜T  √ ˜T  √ ˜T   τ M N   τ M N   τ M N   τ M N  √  √ ˜ T  ˜  √ ˜ T   √ ˜ T  ˜ T ˜ η S , Ω (8) = M  √η M S    √ η M U  , Ω41 (6) =  √η M U˜  , Ω 41 (7) =  41   √   ˜T ˜T dM W d M V˜ T dM W d M V˜ T ˜ 66 = Ω ˜ 77 = Ω ˜ 88 = Ω ˜ 99 = {−21 P˜ +  2 R˜ 1 , −22 P˜ +  2 R˜ 2 , −23 P˜ +  2 R˜ 3 } Ω 1 2 3  √ √ √ √ √ √ τ M ψT31 θ¯ τ M ψT41   τ M ψ11 θ¯1 α¯ τ M ψT21 0 θ¯1 α¯ 1 τ M ψT21 0 0 0 θ¯1 α¯ 1 τ M ψT41 √ √ √ √ √  √ T T T T T   √η M ψ11 θ¯1 α¯ √η M ψ21 0 θ¯1 α¯ 1 √η M ψ21 0 0 0 θ¯1 α¯ 1 √η M ψ41 √η M ψ31 θ¯ √η M ψ41  d M ψ11 θ¯1 α¯ d M ψT21 0 θ¯1 α¯ 1 d M ψT21 0 0 0 θ¯1 α¯ 1 d M ψT41 d M ψT31 θ¯ d M ψT41   √ √ √ 0 θ¯1 ρ1 τ M ψT21 0 −θ¯1 ρ1 τ M ψT21 0 0 0 −θ¯1 ρ1 τ M ψT41 0 0 √ √ √   T T T 0 θ¯1 ρ1 √η M ψ21 0 −θ¯1 ρ1 √η M ψ21 0 0 0 −θ¯1 ρ1 √η M ψ41 0 0 T T T ¯ ¯ ¯ 0 θ1 ρ1 d M ψ21 0 −θ1 ρ1 d M ψ21 0 0 0 −θ1 ρ1 d M ψ41 0 0   √ √ √ ¯ 2 τ M ψT21 0 α¯ 1 ρ2 τ M ψT21 0 0 0 α¯ 1 ρ2 τ M ψT41 0 0 0 αρ √ √ √  0 αρ ¯ 2 √η M ψT21 0 α¯ 1 ρ2 √η M ψT21 0 0 0 α¯ 1 ρ2 √η M PψT41 0 0   T T T 0 αρ ¯ 2 d M ψ21 0 α¯ 1 ρ2 d M ψ21 0 0 0 α¯ 1 ρ2 d M ψ41 0 0   √ √ √ 0 ρ1 ρ2 τ M ψT21 0 −ρ1 ρ2 τ M ψT21 0 0 0 −ρ1 ρ2 τ M ψT41 0 0 √ √ √ 0 ρ1 ρ2 η M ψT 0 −ρ1 ρ2 η M ψT 0 0 0 −ρ1 ρ2 η M ψT 0 0 21 21 41   √ √ √ 0 ρ1 ρ2 d M ψT21 0 −ρ1 ρ2 d M ψT21 0 0 0 −ρ1 ρ2 d M ψT41 0 0   √ 0 0 0 0 0 0 0 0 0 ρ2 τ M ψT41  √ 0 0 0 0 0 0 0 0 0 ρ2 η M ψT  41   √  0 0 0 0 0 0 0 0 0 ρ2 d M ψT41 i h i h ˜T M ˜ T 0 0 0 0 0 0 0 0 , N˜ T = 0 N˜ T N˜ T 0 0 0 0 0 0 0 M 1 2 2 3 i h i h U˜ 1T 0 0 U˜ 4T 0 0 0 0 0 0 , S˜ T = 0 0 0 S˜ 4T S˜ 5T 0 0 0 0 0 h i h i ˜T 0 0 0 0 W ˜ T 0 0 0 0 , V˜ T = 0 0 0 0 0 V˜ T V˜ T 0 0 0 W 1 7 6 6

=

AC

Moreover, if the above conditions are feasible, the parameter matrices of the filter are given by    A f = Aˆ f P¯ −1  3    B f = Bˆ f      C f = Cˆ f P¯ −1 3

Proof. Due to

(Rk − k−1 P)Rk−1 (Rk − k−1 P) ≥ 0, (k = 1, 2, 3) we have 2 −PR−1 k P ≤ −2k P + k Rk 10

(38)

ACCEPTED MANUSCRIPT

1

2

(39)

CR IP T

2 Substitute −PR−1 k P with −2k P + k Rk into (15), we obtain (39) above.  ˜ ∗ ∗ ∗ ∗ ∗ ∗ ∗  Ω11 + Γ˜ + Γ˜ T ∗  ˜ 21  Ω −I ∗ ∗ ∗ ∗ ∗ ∗ ∗    ˜ 31  Ω 0 −Z ∗ ∗ ∗ ∗ ∗ ∗    ˜ 41 (s) ˜ 44 0 0 Ω ∗ ∗ ∗ ∗ ∗   Ω   ˜ 51 ˆ 55 ˆ Ω 0 0 0 Ω ∗ ∗ ∗ ∗  < 0 (s = 1, · · · , 8) Ω(s) =    ˜ 61 ˆ 66 Ω 0 0 0 0 Ω ∗ ∗ ∗     ˜ 71 ˆ 77 Ω 0 0 0 0 0 Ω ∗ ∗    ˜ 81 ˆ 88  Ω 0 0 0 0 0 0 Ω ∗   ˜ 91 ˆ 99  Ω 0 0 0 0 0 0 0 Ω ˆ 55 = Ω ˆ 66 = Ω ˆ 77 = Ω ˆ 88 = Ω ˆ 99 = {−21 P +  2 R1 , −22 P +  2 R2 , −23 P +  2 R3 } Ω 3

Since P¯ 3 > 0, there exist P2 and P3 > 0 satisfying P¯ 3 = Define " # " # P1 PT2 I 0 P= ,J = , χ = diag{J, · · · , J , I, · · · , I , J, · · · , J } | {z } | {z } | {z } P2 P3 0 PT2 P−1 3 PT2 P−1 3 P2 .

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By Schur complement, P>0 is equivalent to P1 −P¯ 3 >0." Multiplying (39) by χ from the left side and its # ¯3 P P 1 T transpose from the right side, and defining P˜ = JPJ = ¯ , Q˜ k = JQk J T , R˜ k = JRk J T (k = 1, 2, 3), P3 P¯ 3 ˜ v1 = JMv1 J T , N¯ v2 = JNv2 J T , U¯ v3 = JUv3 J T , S¯ v4 = JS v4 J T , W ¯ v5 = JWv5 J T ,V¯ v6 = JVv6 J T , (v1 = 1, 2; v2 = M 2, 3; v3 = 1, 4; v4 = 4, 5; v5 = 1, 6; v6 = 6, 7), then, we can derive (36) and (37). Define variables    Aˆ f = A˜ f P¯ 3 , A˜ f = PT2 A f P−T  2   ˆ (40) B f = PT2 B f      −T Cˆ f = C˜ f P¯ 3 , C˜ f = C f P 2

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Based on the descriptions above, similar to the analysis of [41], the filter parameters (A f , B f , C f ) can be −T ˆ ˜ ˜ expressed by (P−T 2 A f P2 , P2 B f , C f P2 ), the filter model (2) can be written as  −T ˆ  ˜  ˆ (t)  x˙ f (t) = P−T 2 A f P2 x f (t) + P2 B f y (41)   z f (t) = C˜ f P2 x f (t)

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Define xˆ(t) = PT2 x f (t), (41) can be written as follows.     xˆ f (t) = A˜ f xˆ(t) + Bˆ f yˆ (t)   z f (t) = C˜ f xˆ(t)

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That is why (A˜ f , Bˆ f , C˜ f ) can be chosen as the filter parameters. This completes the proof.  According to Theorem 2, the H∞ filtering parameters are obtained in an explicit form when the designed filter is under the hybrid triggered scheme. To make the main results more abundant and more substantial, two corollaries are given to describe the time-triggered filtering system and event-triggered filtering system, respectively. Thus, the filtering parameters are expressed precisely.

¯ τ M , d M , σ, ρ2 and k , matrix G, system is asymptotically stable Corollary 1. For given positive parameters γ, θ, with time-triggered scheme and stochastic cyber attacks, if there exist matrices P1 > 0, P¯ 3 > 0, Z > 0, Q¯ k > 0, ¯ N, ¯ W, ¯ V¯ are matrixes with appropriate dimensions, such that the R¯ k > 0 (k = 1, 2), Ω > 0, Aˆ f , Bˆ f , Cˆ f , M, following LMIs hold.   T ∗ ∗ ∗ ∗  Ξ11 + Θ + Θ   Ξ21 −I ∗ ∗ ∗    Ξ31 0 −Z ∗ ∗  < 0, (s = 1, 2, 3, 4) (43)    Ξ41 (s) 0 0 Ξ44 ∗   Ξ51 0 0 0 Ξ55 P1 − P¯ 3 > 0 (44) 11

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  ∗ ∗ ∗ ∗ ∗  ψ11 + ψT11 + Q˜ 1 + Q˜ 2 ∗   θ¯1 ψ21 0 ∗ ∗ ∗ ∗    0 0 −Q˜1 ∗ ∗ ∗ ∗    0 0 0 0 ∗ ∗ ∗     0 0 0 0 −Q˜2 ∗ ∗     ψ41 0 0 0 0 −γ2 I ∗   ¯ 41 ¯  θψ 0 0 0 0 0 −θZ h i ˜ +W ˜ N˜ − M ˜ −N˜ −W ˜ + V˜ −V˜ 0 0 M h i h i ψ51 0 0 0 0 0 0 , Ξ31 = 0 0 0 θ¯2 ZG 0 0 0 "√ # "√ # "√ # "√ # ˜T ˜T ˜T ˜T √τ M M T , Ξ41 (2) = √τ M MT , Ξ41 (3) = √ τ M N T , Ξ41 (4) = √τ M N T ˜ ˜ dM W d M V˜ dM W d M V˜ √ √ √ √   √τ M ψ11 θ¯1 √τ M ψT21 0 0 0 √τ M ψT41 θ¯ √τ M ψT31   d ψ T d M ψT41 θ¯ d M ψT31   M 11 θ¯1 √d M ψ21 0 0 0 √  T  0 ρ2 √τ M ψ21 0 0 0 0 ρ2 √τ M ψT31   T T  0 ρ2 d M ψ21 0 0 0 0 ρ2 d M ψ31 2 diag{−R˜ 1 , −R˜ 2 }, Ξ55 = {−21 P˜ + 1 R˜ 1 , −22 P˜ + 22 R˜ 2 , −21 P˜ + 12 R˜ 1 , −22 P˜ + 22 R˜ 2 }

Besides, the parameters matrices of the filter under the time-triggered scheme are obtained if the conditions above are hold.    A f = Aˆ f P¯ −1  3    (45) B f = Bˆ f      C f = Cˆ f P¯ −1 3

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Corollary 1 has discussed the filtering system which is under the time-triggered scheme. In Corollary 2, the case of event-triggered scheme is described meticulously and the filtering parameters are expressed accurately.

  ∗ ∗ ∗ ∗ ∗ ∗  ψ11 + ψT11 + Q˜ 2 + Q˜ 3 ∗   θ¯1 ψ21 ψ44 ∗ ∗ ∗ ∗ ∗ ∗    0 0 −Q˜2 ∗ ∗ ∗ ∗ ∗    0 0 0 0 ∗ ∗ ∗ ∗  Ξ¯ 11 =   0 0 0 0 −Q˜3 ∗ ∗ ∗     ∗ ∗  θ¯1 ψ31 0 0 0 0 −C T ΩC   ψ41 0 0 0 0 0 −γ2 I ∗   ¯ 31 ¯ θψ 0 0 0 0 0 0 −θZ h i ˜ −U˜ + S˜ −S˜ −W ˜ + V˜ −V˜ 0 0 0 ¯ = U˜ + W Θ h i h i Ξ¯ 21 = ψ51 0 0 0 0 0 0 0 , Ξ¯ 31 = 0 0 0 θ¯2 ZG 0 0 0 0 # "√ # "√ # "√ # "√ ˜T ˜T ˜T ˜T √ η M U T , Ξ¯ 41 (2) = √η M UT , Ξ¯ 41 (3) = √ η M S T , Ξ¯ 41 (4) = √η M S T Ξ¯ 41 (1) = ˜ ˜ dM W d M V˜ dM W d M V˜ 12

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¯ η M , d M , σ, ρ2 and k , matrix G, system is asymptotically stable Corollary 2. For given positive parameters γ, θ, with event-triggered scheme and stochastic cyber attacks, if there exist matrices P1 > 0, P¯ 3 > 0, Z > 0, Q¯ k > 0, ¯ S¯ , W, ¯ V¯ are matrixes with appropriate dimensions, such that the R¯ k > 0 (k = 2, 3), Ω > 0, Aˆ f , Bˆ f , Cˆ f , U, following LMIs hold. ¯  ¯ +Θ ¯T ∗ ∗ ∗ ∗  Ξ11 + Θ   Ξ¯ 21 −I ∗ ∗ ∗    ¯ Ξ31 0 −Z ∗ ∗  < 0, (s = 1, 2, 3, 4) (46)    ¯ ¯  Ξ (s) 0 0 Ξ ∗ 41 44   Ξ¯ 51 0 0 0 Ξ¯ 55 P1 − P¯ 3 > 0 (47)

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√  √ √ √ √  √η M ψ11 θ¯1 √η M ψT21 0 0 0 θ¯1 √η M ψT31 √η M ψT41 θ¯ √η M ψT31   d ψ T T T T d M ψ41 θ¯ d M ψ31   M 11 θ¯1 √d M ψ21 0 0 0 θ¯1 √d M ψ31 √  T T  0 ρ2 √τ M ψ21 0 0 0 ρ2 √τ M ψ41 0 ρ2 √η M ψT31   T T T  0 ρ2 d M ψ21 0 0 0 ρ2 d M ψ41 0 ρ2 d M ψ31 2 2 diag{−R˜ 2 , −R˜ 3 }, Ξ¯ 55 = {−22 P˜ + 2 R˜ 2 , −23 P˜ + 3 R˜ 3 , −22 P˜ + 22 R˜ 2 , −23 P˜ + 32 R˜ 3 }

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The parameter matrices of the filter under the event-triggered scheme are given as follows.    A f = Aˆ f P¯ −1  3    ˆ B = B  f f     C f = Cˆ f P¯ −1 3

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Remark 7. In this section, two corollaries are added to make a comparison with Theorem 2 which presents the accurate expression of filtering parameters under the case of hybrid triggered scheme. By comparing with the reference [33] which investigates the problem of H∞ filtering for systems with time-varying delay, Corollary 1 describes that the H∞ filtering design with stochastic cyber attacks is feasible when the system is under the time-triggered scheme. In [42], the event-based H∞ filtering for networked systems with communication delay is investigated. Compared with the reference [42], an extension is made in Corollary 2 which shows the algorithm of designed event-triggered H∞ filter with stochastic cyber-attacks. Moreover, in the simulation section, three examples are given to demonstrate the effectiveness of the designed H∞ filter with stochastic cyber attacks under hybrid triggered scheme, time-triggered scheme and event-triggered scheme, respectively. 4. Simulation examples

In this section, three examples are given to demonstrate the effectiveness of the designed filter.

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" −2.1 A= 1

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Example 1. Consider the system (1) with the following matrix parameters.

i h 0.6 , L = −0.2

   1, 5 ≤ t ≤ 10   i   0.3 , ω(t) =  −1, 15 ≤ t ≤ 20     0, else

# −tanh(0.1x2 (t)) . Set the −tanh(0.01x1 (t)) upper bound G = diag{0.01, 0.1}, it can be verified easily that the upper bound confirm to the restrictive condition of nonlinear stochastic cyber attacks (8). h iT h iT Choose the initial condition x(0) = 1 −1 , x f (0) = 0.8 −0.8 , sampling period h = 0.01, networkinduced delays τ M = 0.6, d M = 0.6, η M = 0.3 and γ = 20. Set α¯ = 0.2 and σ = 0.8, then the designed filter is under the hybrid triggered scheme. Set θ¯ = 0.1, by using equation (38) in Theorem 2, we can derive the filtering parameters as follows. " # " # h i −1.3180 0.0084 100.1870 Af = , Bf = , C f = 0.0020 −0.0028 0.2887 −1.6150 64.2797

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The function of cyber attacks f (x(t)) which is shown in Fig.2 is supposed as f (x(t)) =

Fig.3 shows the response of z˜(t) when the designed filter is under the hybrid triggered scheme (9). The diagram of release instants and intervals is shown in Fig.4, and Fig.5 presents the Bernoulli distribution diagram of variable α(t). From Fig.3, it is easy to know that the hybrid triggered filter for networked systems under stochastic cyber attacks is feasible.

Example 2. Consider a mechanical example borrowed from [33], x1 and x2 are the positions of masses m1 and m2 , respectively, and k1 and k2 are the spring constants. c denotes viscous friction coefficient between the massed 13

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and the horizontal surface. The parameters of this system (1) are given as follows.     0 1 0   0  0    0  0  h i h 0 0 1    , B =  1  , C = 1 0 0 0 , L = 0 A = − k1 +k2 k2 c  m  − m1 0   m1 m1   1  k2 k2 c  0 − 0 − m2 m2 m2

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where m1 = 1, m2 = 0.5, k1 = k2 = 1, hc = 0.5. i h i Set the initial condition x(0) = 1; 1; −1; −1 , x f (0) = 1; 1; −1; −1 , sampling period h = 0.05, ω(t) is assumed to be the same in Example 1. The system is under the time-triggered scheme (4), let the probability of cyber attacks θ¯ = 0.1, disturbance attenuation level γ = 1, time delay t M = 0.1, d M = 0.1. The cyber attacks are denoted by f T (x(t)) = [−tanhT (x2 (t)) − tanhT (0.05x1 (t)) − tanhT (x4 (t)) − tanhT (0.1x2 (t) + 0.1x3 (t) +   1 0 0   0  0.05 0 0 0   can be obtained. Based 0.1x4 (t))]T . According to the inequation (8), the upper bound G =  0 0 1   0 0 0.1 0.1 0.1 on the equation (45) in Corollary 1, we can derive the filtering parameters as follows.

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    0.2235 4.1480 −1.1117 −2.2813  0.2053   0.2087 −0.6597 −2.2168 2.1096  −0.5329     −0.8093 −0.0256 −1.9716 0.0647  , B f =  0.1968      −1.3808 0.0774 −0.9883 −1.0724 0.3317 h i −0.0758 −0.3208 0.0407 0.0853

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Fig.6 indicates the diagram of cyber attacks. Fig.7 represents the response of z˜(t) when the designed filter is under the time-triggered scheme, which illustrates the effectiveness of the designed filter with stochastic cyber attacks. Example 3. Consider the system with the following parameters. " # " # h −2.1 0 1 A = ,B = , C = 0.8 0 −2 −0.2

i h 0.6 , L = −0.2

i 0.3

The disturbance ω(t) and cyber attacks f (x(t)) are set as the same in Example 1, the sampling period h = 0.01. Set θ¯ = 0.1, γ = 1 and the triggered factor σ = 0.2, time delay η M = 0.1, d M = 0.1, the system is under the event-triggered scheme (6). By applying equation (48) in Corollary 2, the parameters of the designed filter are obtained as follows. " # " # h i −12.7860 2.1508 38.8105 Af = , Bf = , C f = 0.0875 −0.0177 −43.1202 6.5667 260.1745 15

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Under the event-triggered scheme, Fig.8 depicts the response of z˜(t) in (11) and Fig.9 represents the released instants and event-triggered intervals. According to Fig.8, the designed H∞ filter with event-triggered scheme and stochastic cyber attacks is useful. 5. Conclusion

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6. Acknowledgements

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This paper is devoted to the investigation of the hybrid-driven-based H∞ filtering problem for networked systems under stochastic cyber attacks. In order to alleviate the burden of the network, a hybrid triggered scheme is introduced, in which the switching rule between the time-triggered scheme and the event-triggered scheme is described by Bernoulli variable. By taking the hybrid triggered scheme and the effects of stochastic cyber attacks into consideration, a mathematical H∞ filtering model has been constructed for networked systems. By applying Lyapunov stability theory and the LMI techniques, sufficient conditions for the stability of filtering error system have been developed. In addition, the parameters of designed H∞ filter are obtained in an explicit form. Illustrative examples are given to demonstrate the usefulness of desired filter under hybrid triggered scheme and cyber attacks.

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This work is partly supported by the National Natural Science Foundation of China (no. 61403185), the Natural Science Foundation of Jiangsu Province of China(no. BK20171481), Six Talent Peaks Project in Jiangsu Province (no. 2015-DZXX-21), major project supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (no. 15KJA120001), a Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), Collaborative Innovation Center for Modern Grain Circulation and Safety, Jiangsu Key Laboratory of Modern Logistics (Nanjing University of Finance & Economics), Jiangsu Agricutural Science and Technology Independent Innovation Fund Project (no. CX(15)1051), and Jiangsu Overseas Research & Training Program for University Prominent Young & Middle-aged Teachers and Presidents. References

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