具有混合触发通信机制和随机网络攻击的网络系统的H∞滤波.pdf-资料库

Available online at www.sciencedirect.com Journal of the Franklin Institute 354 (2017) 8490–8512 www.elsevier.com/locate/jfranklin H ∞ ﬁltering for networked systems with hybrid-triggered communication mechanism and stochastic cyber attacks Jinliang Liu ∗ a , , Lili Wei a , Engang Tian b , Shumin Fei a c , Jie Cao a College of Information Engineering, Nanjing University of Finance and Economics, Nanjing, Jiangsu 210023, PR China b Institute of Information and Control Engineering Technology, Nanjing Normal University, Nanjing, Jiangsu c School of Automation, Southeast University, Nanjing, Jiangsu 210096, PR China 210042, PR China Received 4 April 2017; received in revised form 10 September 2017; accepted 14 October 2017 Available online 24 October 2017 Abstract This paper concentrates on investigating H ∞ ﬁltering for networked systems with hybrid-triggered communication mechanism and stochastic cyber attacks. Random variables satisfying Bernoulli distri- bution are introduced to describe the hybrid-triggered scheme and stochastic cyber attacks, respectively. Firstly, a mathematical H ∞ ﬁltering error model with hybrid-triggered communication mechanism 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 aug- mented ﬁltering error 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. © 2017 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. 1. Introduction 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 ∗ Corresponding author. E-mail address: liujinliang@vip.163.com (J. Liu). https://doi.org/10.1016/j.jfranklin.2017.10.007 0016-0032/© 2017 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

J. Liu et al. / Journal of the Franklin Institute 354 (2017) 8490–8512 8491 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, NCSs are attracting more and more interest owing to the development and the advantages of the Internet [7] . In [8] , the robust H ∞ con- troller 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-triggered method (periodic sampling) is widely adopted for system modeling and analysis in the NCSs [11] , however, periodic sampling will generate lots of redundant signals if all the sampled data is transmitted through the network. To make full use 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 [12] , in which a H ∞ controller is designed for NCSs. The core idea of the novel event-triggered scheme in [12] 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 [13] . Consequently, there are large numbers of researchers interested in the investigations about the novel event-triggered scheme proposed in [12] . An event-triggered non-parallel distribution compensation control problem in [14] is addressed for networked Takagi–Sugeno (T–S) fuzzy systems. The authors of [15] consider the event- triggered ﬁltering problem for discrete-time linear system with package dropouts satisfying Bernoulli distribution. In [16] , a discrete event-triggered scheme is proposed for fuzzy ﬁlter design in a class of nonlinear NCSs. Inspired by the aforementioned event-triggered scheme in [12] , the hybrid-triggered scheme which consists of time-triggered scheme and event-triggered scheme is ﬁrstly proposed in [17] , which investigates the problem of control stabilization for networked control systems under the hybrid-triggered scheme. Based on the hybrid-triggered scheme above, the authors in [18] 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 [17] , this paper is devoted to the hybrid-triggered H ∞ ﬁltering subject to stochastic cyber attacks on the measurement outputs. Due to the insertion of the network in the control systems, challenges including packet dropouts, network-induced delay and randomly occurring nonlinearities [19–22] are inevitable. It is nonnegligible that another phenomena named cyber attacks can be more destroyable. Cy- ber attacks are offensive maneuvers which target networked information systems, infrastruc- tures 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 [23] , there are three kinds of common attacks containing denial of service attacks [24,25] , relay attacks [26,27] and deception attacks [28,29] . With the rapid development of the net- work, 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 ﬁlter design for stochastic nonlinear systems under cyber attacks in [30] . The distributed recursive ﬁltering problem is studied in [31] with quantization and deception attacks for a class of discrete time-delayed systems. In [32] , a novel state ﬁltering approach and sensor scheduling co-design with random deception attacks are presented.

8492 J. Liu et al. / Journal of the Franklin Institute 354 (2017) 8490–8512 This paper addresses the issue about H ∞ ﬁltering for networked systems under stochastic cyber attacks with hybrid-triggered communication mechanism. 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 communication mechanism and stochastic cyber attacks into consideration, an H ∞ ﬁlter is designed for net- worked systems. Although there are several researches concerned with ﬁlter design, to the best of our knowledge, there is no research investigating the H ∞ ﬁlter design for networked systems by considering both hybrid-triggered scheme and cyber attacks. The rest of this paper is organized as follows. In the Section 2 , a ﬁltering error system is constructed by introducing the hybrid-triggered communication mechanism and taking the stochastic cyber attacks into account. Section 3 gives the sufﬁcient conditions which can guar- antee 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 param- eters are obtained in an explicit form. Section 4 gives illustrative examples to demonstrate the usefulness of desired H ∞ × m real matrices; the su-perscript T stands for matrix transposition; I is the identity matrix of × n means that the appropriate dimension; the notation X matrix X is real symmetric positive deﬁnite (respectively, positive semi-deﬁnite). For a matrix ∗ ×m denote the n-dimensional Euclidean space, and the set of n ≥ 0), for X ∈ n R B and two symmetric matrices A, C and denotes a symmetric matrix, where 0 (respectively, X Notation: R n n and R ﬁlter. > ∗ A B C denotes the entries implied by symmetry. 2. Problem description and preliminaries In this paper, the problem of H ∞ ﬁltering for networked system with hybrid-triggered communication mechanism and stochastic cyber attacks is investigated. The framework of hybrid-triggered H ∞ ﬁltering for networked systems 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. The purpose of this paper is to design a hybrid-triggered H ∞ ﬁlter under stochastic cyber attacks for networked systems. Consider the following ﬁlter system. = A f x f = C f x f (t ) (t ) + ˆ y B f (t ) (2) (t ) (t ) ˙ x f z f + Bw(t ) ⎧ ⎨ Consider the following continuous-time linear system. ˙ ⎩ x (t ) y(t ) z(t ) = Ax(t ) = Cx(t ) = L x (t ) ∈ where x ( t ) R be estimated, w(t ) known real matrices with appropriate dimensions. ∈ n is the state vector, y ( t ) R 2 [0, ∞ + L ∈ ) ∈ m is the ideal measurement, z ( t ) R p is the signal to represents the disturbance input vector, A , B , C and L are (1)

J. Liu et al. / Journal of the Franklin Institute 354 (2017) 8490–8512 8493 Fig. 1. The framework of H ∞ attacks. ﬁltering for networked systems with hybrid-triggered scheme and stochastic cyber ˆ y (t ) is the real input of the ﬁlter, z f ( t ) is the estimation where x f ( t ) is the ﬁlter state vector, of the z ( t ), A f , B f and C f are ﬁlter parameters to be determined. Remark 1. Different from the classic ﬁlter 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 packet dropouts along with cyber attacks are taken into consideration. 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 communica- tion, which consists of time-triggered scheme and event-triggered scheme. Both of the two schemes are discussed in detail in the following, respectively. Time-triggered scheme: Suppose that the sensor is time-triggered and each process signal t , (t ) ∈ + [ t k h = Cx(t k h) + t k+1 h , τt k is sampled at periodic intervals. The ideal measurement y 1 ( t ) can be expressed as follows: y 1 τt k+1 ) where h represents the sampling period, t k { } the corresponding network-induced delay is represented by 1 3 2, , , , . . = − t k h, deﬁne t τ (t ) − τ (t )) = Cx(t ∈ τ M ], [0, τ ( t ) (3) } ⊂ { are integers and t 3 t 2 t 1 . , , , . Similar to [8] , τt k τ M is the upper bound of the networked delay. Eq. (3) can be written as follows. y 1 where = (k 1 2, , (4) (t ) } . ) . { 0, . . . . Event-triggered scheme: To further enhance the bandwidth utilization, by taking [12] as a reference, an event-triggered scheme is applied to determine whether the current measurements should be transmitted or not. We use kh and t k h to represent the sampling instants and the triggering instants. Once y ( t k h ) is transmitted, whether the next triggered instant y(t k+1 h) should be transmitted or not is determined by comparing the latest transmitted sampled-data with the error which is shown as follows: = + { t k+1 h inf t k h l 1 T (t k h)y(t k h) T e (t k h)e k k (t k h) lh| (5) σ y }

8494 J. Liu et al. / Journal of the Franklin Institute 354 (2017) 8490–8512 . . 0, > σ ∈ where = 1 2, , [0, 1) and l . The threshold error e k (t k h) . τt k+1 ) + + t k+1 h For analyzing more easily, the interval [ t k h τt k , + + + ηt+ ηt+ h lh t k h , l ≤ τt k ≤ η(t ) ≤ h + 0 can be written as = y(t k h) + subintervals. Suppose that there exists a constant g which satisﬁes [ t k h g l=1 1 2, where , , l − 1 . Deﬁne t k + x(t k h lh) = Cx(t = + + [ t k h lh l = − t k h − lh, η(t ) t the measurement y 2 (t ) , − η(t )) − y(t k h + . lh) can be divided into several = + t k+1 h τt k+1 τt k , ) = = { g} − l+1 ] t k+1 l g , , . . . , = − ηM . Let e k (t ) x(t k h) ηt k+ l+1 + Ce k (6) (t ) (t ) y 2 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 + ··· are the release time, then it is easily obtained 1) time (k = − t i h denote the release period of event generator in (5) , where s i h mean that t i+1 h that s i h the sampling instants between the two conjoint releasing instants. h. Suppose that t 0 h, t 2 h, t 1 h, . . 0, } . 1 2, , ⊆ { ···} Remark 3. According to the event-triggered algorithm (5) , a set of the releasing instants { . Learning from the research [8] , it is not required that t k+1 t k . t 1 t 3 t 2 > , , , { = + = . If t k+1 The packet dropouts will not occur only when 1 t 1 t 3 t 2 t k . . , , , , ¯τ and = + h imply the packet dropouts and then h τt k τt k ¯τ is a constant. Thus, the frequency of releasing instants the network-induced delay, where depends on the value of σ and the variations of the sensor measurements. two special cases about , τt k τt k+1 } . < 1 2, , } . 3 , > { . . Combine Eq. (4) in time-triggered scheme and Eq. (6) in event-triggered scheme, similar via hybrid-triggered scheme can be expressed as follows: to [17] , the measurement ¯y (t ) = = α(t ) α(t ) ¯y (t ) − α(t + (1 )) + (1 − τ (t )) y 1 (t ) Cx(t ∈ [0, 1], α( t ) where mathematical variance of α( t ). y 2 (t ) − α(t )) − η(t )) [ Cx(t + Ce k (t ) ] ¯α is utilized to represent the expectation of (7) α( t ), and ρ2 1 represents the Remark 4. To describe the stochastic switching rule between the time-triggered scheme and α( t ) which satisﬁes the Bernoulli distribution the event-triggered scheme, the random variable − τ (t )) = it is observed that the data is is introduced. In (7) , when Cx(t , = ¯y = then, 0, transmitted via time-triggered scheme; When Cx(t α(t ) (t ) , the event-triggered scheme is activated in data transmission. − η(t )) + Ce k = 1 , ¯y (t ) α(t ) (t ) 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 assumption: Assumption 1. Suppose that deception attacks f ( x ( t )) satisfy the following condition: || ≤ || || || f (x(t )) 2 Gx(t ) 2 (8) where G is a constant matrix representing the upper bound of the nonlinearity. Remark 5. In order to describe the restrictive condition of nonlinear perturbation, the infor- mation of upper bound is introduced in [36,37] . Similarly, we use matrix G to represent the

J. Liu et al. / Journal of the Franklin Institute 354 (2017) 8490–8512 8495 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 cyber 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 [23,33] , the Bernoulli variable θ( t ) is introduced to govern ˆ y (t ) of ﬁlter can be written as the stochastic cyber attacks, then the real input ˆ y (t ) + − d (t ))) − θ (t (1 { + −d (t ))) −θ (t )) (1 ¯y (t ) Cx(t α(t ) (x(t (x(t θ (t ) θ (t ) C f C f = = )) −τ (t )) + −α(t )) (1 −η(t )) [ Cx(t + Ce k (t ) } ] ∈ [0, d M ], ∈ [0, 1]. ¯θ is utilized to represent the expectation of where d ( t ) utilized to represent the mathematical variance of θ( t ) θ( t ). (9) θ( t ), ρ2 2 is Remark 6. Bernoulli variables are used to describe the stochastic characteristic in the control systems. In [34] , the stochastic delay is described by random Bernoulli variable for NCSs. In [35] , the occurring probability of the two different sampling periods are described by Bernoulli variable. In this paper, the random variables 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. α( t ) and According to (9) , we can obtain the real input of the ﬁlter. Substitute ˆ y (t ) − θ (t )) { α(t Cx(t ) − τ (t )) ˆ y (t ) into Eq. (10) ⎧ ⎨ ˙ ⎩ x f z f (2) , then the ﬁlter can be written as follows: + − d (t ))) (1 + }} ] (t ) Ce k C f (x(t − η(t )) (t ) + { (t ) θ (t ) B f − α(t )) [ Cx(t (t ) (t ) = A f x f + (1 = C f x f x(t ) (t ) x f Deﬁne = e (t ) ˜ z (t ) , = z(t ) − z f (t ) (12) ⎧ ⎨ Based on Eqs. (1) and (10), the ﬁltering error system can be described as ˙ ⎩ e (t ) ˜ z (t ) −τ (t (t )) + θ (t ) (t ) −α(t )) )) (1 ¯A f e = + ¯C f e = + −θ (t (1 (t ) α(t ) − α(t − θ (t ))(1 )) (t ) ¯B f H e ¯B f e k ¯B w(t + + −θ (t ))(1 (1 ) ¯B f f − d (t ))) (x(t 0 A 0 A f ¯B f = , ¯B = , 0 B f C = H , B 0 I 0 , ¯C f = L −C f where = ¯A f Some important lemmas are introduced in the following: ¯B f H e −η(t (t )) (11) ∈ n , and positive deﬁnite matrix Q n R × n , the following ∈ Lemma 1 ( [38] ) . For any vectors x , y R inequality holds: + T Qx y ≤ x −1 y T Q T y 2x

8496 J. Liu et al. / Journal of the Franklin Institute 354 (2017) 8490–8512 ∈ [0, τ ( t ) ∈ [0, d M ], τ M ], d ( t ) Lemma 2 ( [39] ) . Suppose 6 and are matrices with appropriate dimensions, then + τ (t )1 − τ (t ))2 + + d (t )3 − d (t ))4 + (d M (τM ∈ [0, η( t ) + η(t )5 − η(t ))6 + (ηM < 0 ηM ], 1 , 2 , 3 , 4 , 5 , ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ if and only if + d M 1 + d M 1 + d M 1 + d M 1 + 3 + 4 + 3 + 4 τM τM τM τM + 5 + 5 + 6 + 6 ηM ηM ηM ηM < < < < 0, 0, 0, 0, + d M 2 + d M 2 + d M 2 + d M 2 + 3 + 4 + 3 + 4 τM τM τM τM + 5 + 5 + 6 + 6 ηM ηM ηM ηM < < < < 0 0 0 0 (13) (14) 3. Main results 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. ¯θ , ¯α, = 1 2) (i , > ρi σ , τ M , ηM , d M , γ , if there exist matrices P disturbance attenuation level 0, Q k > 0 and M , N , U , S , W , V with appropriate dimensions satisfying Theorem 1. For given positive parameters and matrix G , with hybrid-triggered communication mechanism and cyber attacks, system (11) is asymptoti- cally stable with an H ∞ 0, R k > T ∗ −I 0 0 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ 88 0 ∗ ∗ ∗ ∗ ∗ ∗ 77 0 0 ∗ ∗ ∗ ∗ ∗ 66 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ 99 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ = 1 0(s , ··· , (15) (s) 8) > < where 0 = 41 ∗ ∗ −I 0 0 0 0 0 0 ∗ ∗ ∗ 44 0 0 0 0 0 ∗ ∗ ∗ ∗ 55 0 0 0 0 ⎡ = 1 2, 3) (k , , ⎢ + + ⎢ ⎢ 11 ⎢ ⎢ 21 ⎢ ⎢ 31 ⎢ ⎢ (s) ⎢ ⎢ 51 ⎢ ⎣ 61 71 81 91 ⎡ ⎢ ∗ ∗ ∗ ⎢ ⎢ 11 ∗ ∗ ⎢ ⎢ 0 21 −Q 1 ∗ ⎢ ⎢ 0 0 ⎢ ⎢ 0 0 33 31 ⎢ ⎢ 0 0 0 0 ⎢ ⎢ 0 0 0 0 ⎢ ⎣ 0 0 0 0 0 0 0 41 ¯B T P 0 0 0 ¯θ ¯B T P 0 0 0 f ¯A ¯A f = + + = + + Q 3 Q 2 Q 1 P 11 21 , ¯θ1 ¯B = = ¯α1 = T H C, T H T σC P, ϒ1 41 33 f + − M −N −U + = U W N M ∗ ∗ ∗ ∗ −Q 2 0 0 0 0 0 11 T P f = ∗ ∗ ∗ ∗ ∗ 0 0 0 0 0 ∗ ∗ ∗ ∗ ∗ ∗ −Q 3 0 0 0 ¯θ1 T ¯B ¯αH ¯θ1 ¯B f H, ¯αP + −S S ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ −C ∗ T C −γ 2 I 0 0 0 ¯θ1 T ¯B ¯α1 H = T P, 31 f ¯θ1 ¯B f H = ¯α1 P ϒ2 −W + −V V ⎤ ⎥ ∗ ⎥ ⎥ ∗ ⎥ ⎥ ∗ ⎥ ⎥ ∗ ⎥ ⎥ ∗ ⎥ ⎥ ∗ ⎥ ⎥ ∗ ⎥ ⎦ ∗ ∗ − ¯θI T P f 0 0 0

8497 , = = = = = = , , , , , , = = = (6) 61 21 51 41 (2) 41 (7) 41 (4) 41 (3) 41 (5) 41 (1) = 31 ⎤ ⎦ ⎤ ⎦ J. Liu et al. / Journal of the Franklin Institute 354 (2017) 8490–8512 ⎡ √ ⎣ √ T τM M √ ⎡ T ηM U T d M V √ ⎣ √ T τM N √ T ηM U T d M V √ √ ϒ2 0 0 0 τM √ ϒ2 0 0 0 ηM ϒ2 0 0 0 d M √ − ¯θ1 √ τM P ρ1 − ¯θ1 √ ηM P ρ1 − ¯θ1 d M P ρ1 √ ¯α1 √ τM P ρ2 ¯α1 √ ηM P ρ2 ¯α1 d M P ρ2 √ −ρ1 √ τM P ρ2 −ρ1 √ ηM P ρ2 −ρ1 d M P ρ2 ¯C f 0 0 0 0 0 0 0 0 0 ⎡ ⎤ ⎡ ⎡ ⎤ ⎤ ¯θ2 GH 0 0 0 0 0 0 0 0 0 √ √ √ ⎣ ⎦ ⎣ ⎣ ⎦ ⎦ √ √ √ T T T τM M τM M τM M √ √ √ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ T T T ηM U ηM S ηM S T T T d M W d M W d M V √ √ √ ⎣ ⎦ ⎦ ⎣ ⎦ ⎣ √ √ √ T T T τM N τM N τM N = = √ √ √ ⎡ ⎤ T T T ηM U ηM S ηM S (8) 41 T T T d M W d M W d M V √ √ √ √ τM P ⎣ ⎦ ¯θ ¯θ1 ¯A f ¯B f ¯B f ¯B ¯α1 √ √ √ √ ηM P τM P τM P τM P ¯θ ¯θ1 ¯A f ¯B ¯B f ¯B f ¯α1 √ √ √ √ ⎡ ⎤ ηM P ηM P ηM P ¯θ1 ¯θ ¯A f ¯B f ¯B f ¯B ¯α1 d M P d M P d M P d M P √ √ ⎣ ⎦ ¯θ1 − ¯θ1 ¯B f H 0 0 0 ¯B f 0 0 √ √ 0 τM P τM P ρ1 ρ1 ¯θ1 − ¯θ1 ¯B f 0 0 ¯B f H 0 0 0 √ √ ⎡ 0 ηM P ηM P ρ1 ρ1 ¯θ1 − ¯θ1 ¯B f H 0 0 0 ¯B f 0 0 0 d M P d M P ρ1 ρ1 √ √ ⎣ ¯B f H ¯B f ¯αρ2 ¯α1 √ √ 0 τM P τM P ρ2 ¯B f H 0 0 0 ¯B f ¯αρ2 ¯α1 √ √ ⎡ ⎤ 0 ηM P ηM P ρ2 ¯B f H 0 0 0 ¯B f ¯αρ2 ¯α1 0 d M P d M P ρ2 √ √ ⎣ ⎦ ¯B f 0 0 ¯B f H 0 0 0 −ρ1 √ √ 0 τM P τM P ρ2 ρ1 ρ2 ¯B f 0 0 ¯B f H 0 0 0 −ρ1 √ √ ⎡ ⎤ 0 ηM P ηM P ρ2 ρ1 ρ2 ¯B f H 0 0 0 ¯B f 0 0 −ρ1 0 d M P d M P ρ1 ρ2 ρ2 √ ⎣ ⎦ ¯B f √ 0 0 0 0 0 0 0 0 0 τM P ρ2 ¯B f √ 0 0 0 0 0 0 0 0 0 91 ηM P ρ2 ¯B f 0 0 0 0 0 0 0 0 0 d M P ρ2 ¯θ , ¯θ2 ¯θ ¯θ1 ¯θ1 − ¯θ , = = = = diag{−R 1 ρ2 1 44 2 −P R {−P R = −P R = } −1 −1 −1 P, P P, 66 55 3 2 1 T = T = 0 0 0 0 0 0 0 0 N M , T = T = 0 0 0 0 0 0 U S , T = T = 0 0 0 0 W V , 41 √ √ ϒ1 0 τM √ ϒ1 0 ηM ϒ1 0 d M ¯B f H 0 ¯B f H 0 ¯B f H 0 ¯B f H ¯B f H 0 ¯B f H 0 ¯B f H 0 ¯B f H 0 ¯B f H 0 T M 2 T 0 0 U 4 T 0 0 0 0 W 6 T T 0 N N 3 2 T 0 0 0 S 0 0 0 0 0 4 T 0 0 0 0 0 V 6 0 0 0 0 0 0 0 T S 5 0 0 0 − ¯α, = ¯α ¯α1 ρ2 1 = = 99 88 −R 3 } −R 2 ¯α1 = 1 = 77 0 0 0 0 0 0 T M 1 T U 1 T W 1 0 0 0 71 81 T V 7 = = = , , 0 , , ⎤ ⎦ Proof. Choose the following Lyapunov functional candidate as = V 1 (t ) + V 2 (t ) + V 3 (t ) V (t ) (16) t t T (t ) Pe (t ) V 1 (t ) where = e = (t ) V 2 t t t−τM = V 3 (t ) T (s) e Q 1 e (s) t ˙ T (v) e ˙ e R 1 (v) T (s) e t + ds t−ηM + d vd s t−ηM = 3) 1 2, (k , 0 s . s t−τM 0, Q k > and P > 0, R k > Q 2 e (s) t ˙ T (v) e + ds t−d M ˙ e R 2 (v) T (s) e Q 3 e (s) ds t t + d vd s t−d M s ˙ T (v) e ˙ e R 3 (v) d vd s

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具有混合触发通信机制和随机网络攻击的网络系统的H∞滤波.pdf

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