Quasi-synchronization of heterogeneous nonlinear multi-agent systems subject to DOS attacks with impulsive effects

Neurocomputing 366 (2019) 131–139

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Quasi-synchronization of heterogeneous nonlinear multi-agent systems subject to DOS attacks with impulsive effectsR Dan Ye a,b,∗, Yuyuan Shao a a b

College of Information Science and Engineering, Northeastern University, Shenyang 110819, Liaoning, China State Key Laboratory of Synthetical Automation of Process Industries, Northeastern University, Shenyang 110819, Liaoning, China

a r t i c l e

i n f o

Article history: Received 6 March 2019 Revised 2 July 2019 Accepted 29 July 2019 Available online 6 August 2019 Communicated by Prof. S. Arik Keywords: DOS attacks Impulsive effects Quasi-synchronization Nonlinear multi-agent systems

a b s t r a c t This paper is concerned with the quasi-synchronization of heterogeneous nonlinear multi-agent systems under DOS attacks with impulsive effects. The impulsive effects play either a positive role (impulsive control) or a negative role (impulsive disturbance) in the synchronization problem. The communication topology subjected to DOS attacks is considered, which destroys the communication channel among agents. A connectivity restoration mechanism is assumed. Only considering the positive impulsive effect, the controller does not need to operate constantly, which will reduce the controller resource consumption. Only taking the negative impulsive effect into account, a feedback controller is designed to counteract with the impulsive disturbance and DOS attacks. By using the concept of ‘average impulsive interval’, the criterion of quasi-synchronization with two impulsive effects is unified. Based on the impulsive theorem, the relationship among attack strength, control gain and average impulsive interval is analyzed. The results are illustrated by a simulation example. © 2019 Elsevier B.V. All rights reserved.

1. Introduction The past few decades have witnessed increasing attention on the multi-agent systems due to wide applications of many fields, such as team robots, formation control of vehicles, unmanned aerial and so on [1–8]. The most important issues in multi-agent systems are that the synchronization can be achieved by designing a control strategy. However, due to some inevitable reasons, such as the time-delay, the parameter mismatch and heterogeneous property, the complete synchronization cannot be satisfied. The quasi-synchronization is a kind of approximate synchronization, where the systems error dynamics could converge on a bounded set as time goes to infinity. In [9], the quasi-synchronization in multi-agent systems under time-delay can be achieved by appropriately choosing the coupling strengths. The condition for reaching quasi-synchronization is established. The authors in [10] solve the quasi-synchronization of two delayed dynamic systems with parameter mismatch. In [11], the leader-following quasi-synchronization of heterogeneous nonlinear multi-agent

R This work is supported by National Natural Science Foundation of China (Nos. 61773097, U1813214), Fundamental Research Funds for the Central Universities (No. N160402004). ∗ Corresponding author at: College of Information Science and Engineering, Northeastern University, Shenyang 110819, Liaoning, China. E-mail address: [email protected] (D. Ye).

https://doi.org/10.1016/j.neucom.2019.07.095 0925-2312/© 2019 Elsevier B.V. All rights reserved.

systems via impulsive control is achieved. The quasisynchronization in [11] is caused by heterogeneous characteristic, which widely exists on the multi-agent systems. Meanwhile, the impulsive control, which can achieve the synchronization of systems and save the communication resource of the controller, is designed by the concept of “synchronizing impulsive”. The systems exist either the positive impulsive effect(impulsive control) or negative impulsive effect (impulsive disturbance) owing to the switching phenomena or sudden noise in [12–19]. The positive and negative impulsive effect either facilitates or does harm to the synchronization of systems, respectively. Therefore, it is necessary to research the synchronization of multi-agent systems with two impulsive effects. In [20], the multi-agent systems with fixed topology based on the local information from agents is considered. Based on impulsive control theorem, some sufficient conditions are given to guarantee the consensus of the linear multiagent systems. The author in [21] investigates the consensus of leader-following nonlinear multi-agent under network-induced delays via distributed impulsive control. In [11], the periodic impulsive control interval is used to guarantee the synchronization of systems. However, the impulsive disturbance interval is aperiodic. The obtained results cannot be applied to study the synchronization in multi-agent systems under impulsive disturbance. The problem of cooperative synchronization of nonlinear multi-agent systems with time delays and impulsive disturbances is investigated in [22]. However, the lower bound of impulsive interval is

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applied to guarantee that the impulsive disturbance does not appear too frequently, which would cause very conservative results. With the development of communication and network technologies, the multi-agent systems are valuable to cyber attacks [23–28]. For the multi-agent systems under cyber attacks, the communication topology among agents is destroyed by adversaries. A kind of such cyber attacks is called DOS attacks, which can be executed by jamming the communication topology. In [29], a hybrid control protocol is designed for satisfying the mean-square secure consensus subject to DOS attacks. Assume that there has a connectivity restoration mechanism, which can make the communication topology recover from the destructive structure. By using Lyapunov’s method, some conditions related to attack frequency and attack length rate are derived. In [30], the nonlinear multi-agent systems subject to DOS attacks is considered. Based on the multiple Lyapunov function method, some sufficient conditions are proposed to achieve synchronization. However, the quasisynchronization of nonlinear multi-agent systems under DOS attacks with both impulsive effects and heterogeneous property has not been considered. Motivated by the above discussion, we solve the quasisynchronization problem of heterogeneous nonlinear multi-agent systems under DOS attacks with two impulsive effects. The contribution to this paper is threefold: (1) The DOS attacks in [29,30], which the topology is damaged by adversaries, is first considered in the nonlinear multiagent systems with heterogeneous property and two impulsive effects in this paper. Impulsive controller will further reduce the communication consumption of controller. Based on the relationship between impulsive gain and attacks strength, the multi-agent systems can tolerate bigger DOS attacks strength by choosing appropriate impulsive strength. (2) Considering positive and negative impulsive effects separately, the attacks strength is represented by attacks length rate, maximum attack ratio and minimum communication length, respectively. The average impulsive interval is applied to unify the quasi-synchronization criteria with two impulsive effects. Compared with minimum impulsive interval in [15,22], the bigger attacks strength in multi-agent systems can be tolerated by using average impulsive interval. (3) By designing a new control strategy and using impulsive theorem, the relationship among control gain, average impulsive interval and attacks strength is analyzed to achieve the secure quasi-synchronization of heterogeneous nonlinear multi-agent systems with two impulsive effects. The structure of this paper is divided into four parts. In part II, preliminaries and problem formulation are presented. The control protocols and main results are discussed in part III. In order to show the efficiency of results, a simulation example is given in part VI. Some conclusions are drawn in Section 5. The following notations are used in this paper. Let Rn and Rn × n denotes the n-dimensional Euclidean space and the set of n × n matrix, respectively. Symbol  stands for the Kronecker product. λmax (A ) and λmin (A ) represent the maximal eigenvalues and minimal eigenvalue of matrix A, respectively. 2. Problem formulation and preliminaries 2.1. Communication graphs





Let  = , , A˜ be a graph, where  = {0, 1, . . . , N} is the node set, and  ⊆  ×  is the edgeset. The neighbor set of agent i is Ni = { j ∈ |( j, i ) ∈ , j = i}. A˜ = ai j be the adjacency maN×N trix of followers. If there is a directed path from node j to i(j = i),

then aij > 0; otherwise, ai j = 0. G = diag{g1 , g2 , . . . , gN } be the adjacency matrix of the leader. Denote D˜ = diag{d1 , d2 , . . . , dN } be the  degree matrix, where di = j∈NI ai j . L = D˜ − A˜ is the Laplacian matrix, and H = L + G. If there exists a root node and exists a directed path from this root node to any other nodes, then this kind of topology graph is said to contain a directed spanning tree. 2.2. Problem formulations For a heterogeneous nonlinear multi-agent systems consisting of N followers and a leader. The system dynamics of the ith follower agent and leader agent are described as follows:



x˙ i (t ) = Ai xi (t ) + Bi f (xi (t )) + ui (t ) x˙ 0 (t ) = A0 xi (t ) + B0 f (x0 (t ))

(1)

where xi (t) ∈ Rn , x0 (t) ∈ Rn are the states of the ith following agent T and leader agent, respectively. f (· ) = ( f1 (· ), f1 (· ), . . . , fn (· ) ) is nonlinear function. ui (t) is control input. In order to enlarge application range, we consider the multi-agent systems containing heterogeneous characteristics, which means that the system matrix Ai , Bi can be different. The following assumptions, lemmas and definition are necessary to derive our main results. Assumption 1. For the nonlinear function f(·), there exists a positive constant lij , such that the following Lipschitz condition is satisfied.

| fi (xi ) − f0 (x0 )| ≤

N 

li j |xi − x0 |

(2)

j=1

Assumption 2. The information of neighbor agent will be updated as soon as the communication topology is recovered from the paralyzed topology. Assumption 3. The initial communication topology contains a directed spanning tree with the leader agent as the root node. Lemma 1. If the communication topology G contains a directed spanning tree with the leader agent as the root node, the Laplacian matrix L contains a simple zero eigenvalue and all the other eigenvalues have positive real parts. Lemma 2. For A ∈ R, denote matrix measure by μ(A ) = −1 limh→0+ I+hA , where I is the identity matrix with approprih ate dimension. The matrix measure is calculated by

  1 μ(A ) = λmax A + AT 2

Definition 1. For any initial conditions, when there exists a set M such that the error x˜i (t ) = xi (t ) − x0 (t ) converges into the set, then the multi-agent systems is said to achieve the quasisynchronization. Definition 2. The average impulsive interval sequence  = {t1 , t2 , . . . , } is equal to Tb . Assume that there exist positive interval I0 , such that

T −t T −t − I0  N (t, T )  + I0 Tb Tb

(3)

where N(t, T) is the number of impulsive instances during the interval (t, T). Remark 1. To unify the positive or negative impulsive effects and get a less conservative result, the concept of ”average impulsive interval” is first proposed for delayed complex dynamical networks in [31]. The author in [14] achieves the quasi-synchronization of nonidentical neural networks with two impulsive effects. According to the concept of average impulsive interval, some conditions

D. Ye and Y. Shao / Neurocomputing 366 (2019) 131–139

are derived. However, the average impulsive interval is used to unify two impulsive effects and get a less conservative result for heterogeneous nonlinear multi-agent systems in this paper. 2.3. Connectivity restoration mechanism For achieving secure quasi-synchronization for multi-agent systems subject to DOS attacks, the following Assumption is given. Assumption 4. The initial communication topology is destroyed when the systems suffer from DOS attacks. Due to the existence of connectivity restoration mechanism, after a period of time, the communication topology can recover to the initial topology structure. Remark 2. It is noticed that a connectivity restoration mechanism is assumed in [29,30] to ensure that the secure quasisynchronization for multi-agent systems can be realized. The initial topology structure contains a directed spanning tree. Remark 3. Due to the existence of DOS attacks, the topology structure of multi-agent systems is not fixed. To facilitate analysis, we   assume that there exists an infinite time interval hk , hk+1 and the DOS attacks appear at instants hk + τk , where 0 < τk < hk+1 − hk . Based on Assumption 3, the communication topology will be destroyed at instants hk + τk . Therefore, we define the time inter val [hk , hk + τk ) and hk + τk , hk+1 as communication time interval and attack time interval, respectively. The impact of DOS attacks will be removed at instant hk+1 by using the restoration mechanism. 2.4. Controller design The ith agent obtains its neighbors’ information and controller updates within the communication [hk , hk + τk ). Dur time interval  ing the attack time interval hk + τk , hk+1 , the controller will sleep. Based on the above analysis, when only considering the positive impulsive effect(impulsive control), the controller is designed as:

  ∞

ui (t ) =

c

k=0

pi (t )δ (t − tk ) 0

t ∈ [hk , hk + τk ) t ∈ [hk + τk , hk+1 )

(4)

where k ∈ N, the positive constant c is the impulsive control gain, and δ (·) denotes the Dirac function. The local error signal pi (t) is defined as

pi (t ) = −

N 

li j x j (t ) − di (xi (t ) − x0 (t ))

j=1

where di is pinning gain, di > 0 if and only if there exists a directed path from the leader to the ith node. Remark 4. The controllers in [29,30] require to communicate with neighbor agents constantly, which causes systems subject to continuous DOS attacks. In order to avoid to be attacked by continuous DOS attacks, we design impulsive control method which only need to communicate with neighbor agents at sampling instants. Compared with the continuous controllers in [29,30], impulsive controller will further reduce the communication consumption of controller. Based on (1) and (4), when t ∈ [hk , hk + τk ), the error dynamics is described by



x˜˙ i (t ) = Ai x˜i (t ) + Bi g(xi (t )) + Wi (x0 (t )) x˜i (tk ) =

( ) + cpi (t )

x˜i tk−

t ∈ (tk , tk+1 ] t = tk

133

where g(xi (t ))= f (xi (t )) − f (x0 (t )), Wi (x0 (t )) = (Bi −B0 ) f (x0 (t )) + (Ai − A0 )x0 (t ). g(xi (t)) is nonlinear function satisfying Lipschitz condition. Because the state of the leader is bounded and f(x0 (t)) is a Lipschitz function, the heterogeneous function Wi (x0 (t)) is bounded. Wi (x0 (t)) ≤ ξ , where ξ is a constant. To facilitate the following results analysis, based on the Kronecker product, the error dynamics can be described the compact form as follows:



x˜˙ (t ) = Ax˜(t ) + BG(t ) + W (t ) x˜(tk ) = (IN − c (L + D )  In )x˜(tk− )

t ∈ (tk , tk+1 ] t = tk

(5)

T

where G(t ) = [g(x1 (t )) g (x2 (t )) · · · g (xN (t ))] , W (t ) = [W1 (x1 (t )) · · · WN (xN (t ))]T , A = diag(A1 A2 · · · AN ), B = diag(B1 B2 · · · BN ). When only taking the negative impulsive effect (impulsive disturbance) into account, the controller is designed as:

ui (t ) =

 θ pi (t ) 0

t ∈ [h k , h k + τ k )

(6)

t ∈ [hk + τk , hk+1 )

During the process of information transmission, the multi-agent systems is assumed to be disturbed by an impulsive disturbance, which can be described by ∞ 

cpi (t )δ (t − tk )

(7)

k=1

According to (1) and (6), considering the impulsive disturbance (7), when t ∈ [hk , hk + τk ), the compact form of error systems are described as follows:



x˜˙ (t ) = (A − θ I )x˜(t ) + BG(t ) + W (t )

t ∈ (tk , tk+1 ]

x˜(tk ) = (IN − c (L + D )  In )x˜(tk− )

t = tk

(8)

2.5. Attacks strength For achieving the quasi-synchronization of heterogeneous nonlinear multi-agent systems, some conditions related to attacks strength have to be imposed. The following definitions are proposed to get the attacks strength. Definition 3. When t > 0, we define Ta as the total attack time length in multi-agent systems during [t0 , t). Then, Ta /(t − t0 ) is defined as attack length rate. Definition 4. When t > 0, we define amax as the maximum attack interval in multi-agent systems during [t0 , t). τmin is defined as minimum communication interval in multi-agent systems during [t0 , t). Then, we denote amax /τmin as maximum attack ratio. Remark 5. Although the initial topology provides the probability of quasi-synchronization for multi-agent systems, the existence of DOS attacks may totally destroy the quasi-synchronization. Therefore, it is necessary to get the constraint condition about attacks strength. The attacks strength can be represented by Ta /(t − t0 ), amax /τmin and τmin . In Definition 3, Ta is used to show the whole attack time. The maximum attack interval amax can be satisfied by the connectivity restoration mechanism. 3. Main results Both the positive impulsive effect and the negative impulsive effect on the heterogeneous nonlinear multi-agent systems under DOS attacks are considered. Some sufficient conditions are derived under the two types of impulsive effects, respectively. Theorem 1. Suppose that Assumption 1,2,3 and 4 hold, 0 < μ < 1. Under the proposed impulsive control strategy (4), the heterogeneous

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nonlinear multi-agent systems (1) under DOS attacks can achieve quasi-synchronization if the following conditions are satisfied:

IN − c(L + D )  In  < μ Ta  t amax

τmin

η + (α + ln μ Tb

<

− (α +

α

ln μ Tb

(9)

x˜(t ) 

(10)

+ γ2 e

(11)

D+ x˜(t ) = lim sup m→0+

= lim sup

+

lnμ 1 μ  e(α+ Tb )(t−s)

(13)

μ

t hl

W (t, s )ξ ds

(14)

1 (α + lnTμ )(t−hl ) b e

(α + x˜(hl ) + γ1 (e Tb





μ

lnμ

)(t−hl )

ξ x˜(hl−1 + τl−1 ) + (eα (t−hl−1 −τl−1 ) − 1 ) α

 

where γ1 = , γ2 = αξ . μ(α + lnT μ ) b

lnμ

(α + x˜(hl ) + γ1 (e Tb

)(t−h0 )

− 1)

lnμ

+ γ1 eα (t−h0 −τ0 ) (e

)τ0

x˜(h0 )

(α + lnT μ )τ0 b

(16)

− 1 ) + γ2 (eα (t−h0 −τ0 ) − 1 )

1 α (t−h0 −τ0 ) (α + lnTμ )(t−h1 +τ0 ) b e e

x˜(h0 )

(α + lnT μ )t−h1

− 1)

μ

+γ1 eα (t−h0 −τ0 ) e +γ2 e

(α + lnT μ )(t−h1 ) b

b

(e

(α + lnT μ )τ0 b

(20)

(α

γ2 e(α

1 (α e

μ

)(t−hl +τ0 +···+τl−1 ) α (a0 +···+al−1 )

e

(α

+ lnT μ b

+ lnT μ b

+ lnT μ b

x˜(h0 )

+ lnT μ b

)(t−hl ) α al−1

e

)(t−hl )

(e

(α

+ lnT μ b

(e

)τl−1

(α + lnT μ )τ0 b

− 1)

− 1)

− 1)

)(t−hl +τ1 +···+τl−1 ) α a1

e

(α + lnT μ )(t−hl ) α al−1 b

e

( eα a0 − 1 )

(eαal−1 − 1 )

)(t−hl +τ0 +···+τl−1 )+α (a0 +···+al−1 )

x˜(h0 ) + χ1

1 (α + lnTμ )(t−Ta )+α Ta b e x˜(h0 ) + χ1

(21)

μ

(22)

x˜(t ) 

(eα (h1 −h0 −τ0 ) −1 ) + γ1 (e

(α + lnT μ )(t−h1 ) b

−1 )

1

μ

e−ηt x˜(h0 ) + χ1

(23)

 lnμ α+ τmin + α amax < 0

(24)

Tb

Then, the bound is as follows: lnμ

χ1 =

γ1 eαal−1 (e(α+ Tb 1−e

(α

+ lnT μ b

)τl−1

− 1)

)τmin +α amax

+ γ1 (eα (hl+1 −hl ) − 1 ) +

γ2 eαal−1 (eαal−1 − 1 ) 1−e

(α + lnT μ )τmin b

−γ1 eα amax 1−e

(α + lnT μ )τmin +α amax b

+

γ2 eαamax (eαamax − 1 ) 1−e

(α + lnT μ )τmin

(25)

b

Case 2: When γ 2 < 0, one has

χ1 

−γ1 1−e

(α + lnT μ )τmin +α amax b

− γ1 +

−γ2 1−e

(α + lnT μ )τmin

(26)

b

(17)

It indicates that the quasi-synchronization of heterogeneous nonlinear multi-agent systems is achieved. So the proof is completed. 

(18)

Remark 6. According to the impulsive control theorem, the quasisynchronization heterogeneous nonlinear of multi-agent systems can be achieved by choosing an appropriate impulsive gain c within a limited interval. When the impulsive gain c is chosen

For t ∈ [h1 , h1 + τ1 ), based on (14) and (16), one has

x˜(t ) 

− 1 ) + γ2 (eα (t−h1 −τ1 ) − 1 )

Tb

χ1 

For t ∈ [h0 + τ0 , h1 ), according to (15), we have

1

)τ1

Based on (10), we can get γ 1 < 0. Case 1: When γ 2 > 0 we can get

When t ∈ [h0 , h0 + τ0 ), based on (14), one has

(α + x˜(t ) ≤ eα (t−h0 −τ0 ) e Tb μ

(eα (h1 −h0 −τ0 ) − 1 )

 lnμ (t − Ta ) + α Ta  −ηt α+

ξ

1 (α + lnTμ )(t−h0 ) b e

− 1)

where Ta = a0 + a1 + · · · + al−1 denotes attack length, and t − Ta shows communication length. According to (10), it holds that

(15)

μ

b

Based on (11), we have

− 1)

When t ∈ hk−1 + τk−1 , hk , because of existing DOS attacks, The impulsive effect loses function. Then, the error state can be described as

 eα (t−hl−1 −τl−1 )

(α + lnT μ )τ0

which implies that

Then, error state can be derived that



+ lnT μ b

+ · · · + γ2 e

When t ∈ [hk , hk + τk ), based on (9), by mathematical induction, we can get the state transfer matrix of error systems as follows:



(α

+ γ1 (e

(12)

(α

+ lnT μ b

(e

(19)

)(t−hl +τ0 +···+τl−1 )+α (a0 +···+al−1 )

+ · · · + γ1 e

m

x˜(t ) + mx˜˙ (t ) − x˜(t )

stk t

lnμ

+ γ1 e

x˜(t + m ) − x˜(t )

x˜(tk ) = IN − c(L + D )  In  x˜(tk− )  μ x˜(tk− )

x˜(t ) 

e

1

where Q = A + BL(t ), the matrix measure ϕ (Q) is related to the systems instrict. When t = tk , we have

x˜(t )

(α + lnT μ )τ1 α (t−h1 −τ1 ) b

(α + x˜(t )  e Tb μ

m I + mQ  − 1  lim sup x˜(t ) + W (t ) m→0+ m  ϕ (Q )x˜(t ) + ξ  ∂ x˜(t ) + ξ



b

x˜(h0 )

Then, according to the above analysis, by using mathematical induction, For t ∈ [hl , hl + tl ), one can get

m→0+

x˜(t ) = W (t, hl ) +

(α + lnT μ )τ1

+ γ1 eα (t−h1 −τ1 ) (e

)

Proof. To get the result, we evaluate the change of the norm of x(t). For t = tk , taking the derivative of x(t) with t along the trajectory yields

W (t, s ) = eα (t−s )

1 α (t−h0 −τ0 ) (α + lnTμ )τ0 +τ1 b e e

μ

+ γ1 eα (t−h0 −τ0 ) e

)

ln μ Tb

For t ∈ [h1 + τ1 , h2 ), according to (15) and (17), we have

D. Ye and Y. Shao / Neurocomputing 366 (2019) 131–139

within some limited intervals, the impulsive effect plays a positive role in the synchronization of systems. Different from [29,30], our controller only operates at sampling instants tk , which will reduce the communication consumption of controller. Corollary 1. Consider the heterogeneous nonlinear multi-agent systems with an undirected topology under DOS attacks. It is assumed that Assumption 1,2,3 and 4 holds, 0 < μ < 1. With the proposed impulsive control strategy (4), all the agents can achieve the quasisynchronization if the following conditions are satisfied:

c<

1+μ λmax (L + D )

−η − (α + Ta  ln μ t

τmin



− (α +

α

W (t, s ) = eα (t−s )

ln μ Tb

)

(28)

(29)

IN − c(L + D )  In  < μ

 μI0 e

λmax (GT G ) =

(30)

W (t, s )ξ ds

(α + lnT μ )(t−hl )

lnμ

b



(α + x˜(hl ) + γ1 (e Tb

)(t−hl )

− 1)



x˜(t ) eα (t−hl−1 −τl−1 ) x˜(hl−1 + τl−1 ) +

(37)

ξ α (t−hl−1 −τl−1 ) (e − 1) α

Similar to the proof of Theorem 1, when t ∈ [hl , hl + τl ), by using mathematical induction, we have (α + x˜(t )  μkI0 e Tb

+μ

γ1 e

+ γ1 (e

(α

)(t−hl +τ0 +τ1 +···+τl−1 )+α (a0 +a1 +···+al−1

(α + lnT μ )(t−hl +τ1 +···+τl−1 ) b

+ lnT μ b

(α

+ lnT μ b

)(t−hl )

)(t−hl ) α al−1

e

lnμ

μkI0 e(α+ Tb



(α

+ lnT μ b

− 1 ) + μ(k−1 )I0 γ2 e

)(t−Ta )+α Ta

x˜(h0 )

(α + eα (a0 +a1 +···+al−1 ) (e

(e

× eα a1 (eα a0 − 1 ) + · · · + μγ2 e

(α

+ lnT μ b

)τl−1

lnμ Tb

)τ0

−1 )

− 1)

(α + lnT μ )(t−hl +τ1 +···+τl−1 ) b

)(t−hl ) α al−1

e

(eαal−1 − 1 )

x˜(0 ) + χ2

Based on (30) and (32), it holds that





lnμ (t − Ta ) + α Ta  −η1t Tb

α+

This completes the proof.

kI0

+ · · · + μγ1 e

λ2max (G ) = |λmax (G )| < μ

1+μ c< λmax (L + D ) 

(39)

kI0 ln μ  (η1 − η )t

Theorem 2. Suppose that Assumption 1,3 and 4 holds. It is assumed that μ > 1. Under the proposed control strategy (6), the heterogeneous nonlinear multi-agent systems subject to impulsive disturbances and DOS attacks can achieve the quasi-synchronization if the following conditions are satisfied:

ln μ α+ + η1  0 h

(31)

(40)

Then, we can get

x˜(t )  e−ηt x˜(h0 ) + χ2

(41)

According to (33), one has

ln μ +

 lnμ α+ τmin + α amax  0

(42)

h

Then, we have

ln μ η1 − η

(32)

lnμ

μI0 γ1 eαal−1 (e(α+ Tb

χ2 =

ln μ − (α +

t hl

lnμ

|1 − cλmax (L + D )| < μ

τmin 



x˜(t ) = W (t, hl ) +

where G = IN − c (L + D )  In . Then, we get

τmin 

(36)

b

stk t

then, error state can be derived that

Based on Assumption 3, all eigenvalues of the matrix L + D are positive. Since topology is undirected, the matrix L + D is symmetric. Then, one has

(α + lnT μ )(t−s )

(38)

)

Proof. According to Theorem 1, we get

G  =

 μ  μI0 e

When t ∈ hk−1 + τk−1 , hk , because of existing cyber attacks, there is no impulsive effects. Then, the error state can be described as

Tb

amax

When t ∈ [hk , hk + τk ), according to μ > 1 in Theorem 2, by mathematical induction, we can get state transfer matrix of error systems as follows:

(27) ln μ Tb

135

ln μ Tb

(33)

)

Proof. To get the result, we evaluate the change of the norm of x(t). For t = tk , taking the derivative of x(t) with t along the trajectory yields

D+ x˜(t ) = lim sup m→0+

= lim sup m→0+

x˜(t + m ) − x˜(t ) m

x˜(t ) + mx˜˙ (t ) − x˜(t ) m

+

b 1 − μe μI0 γ2 eαal−1 (eαal−1 − 1 )

1 − μe

+ γ1 (eα (hl+1 −hl ) − 1 )

(α + lnT μ )τmin b

Since that γ 1 < 0 and γ 2 < 0, we can get

χ2 

−μI0 γ1 1 − μe

(α + lnT μ )τmin +α amax b

− γ1 +

μI0 γ2 1 − μe

(α + lnT μ )τmin

(43)

b

4. Numerical examples

m→0+

(34)

When t = tk , we have

x˜(tk ) = IN − c(L + D )  In  x˜(tk− )  μ x˜(tk− )

− 1)

(α + lnT μ )τmin +α amax

Then, the quasi-synchronization of heterogeneous nonlinear multiagent systems under impulsive disturbances and DOS attacks is satisfied. So the proof is completed. 

I + mQ − θ I ) − 1

x˜(t ) + W (t ) m  ϕ (Q − θ I )x˜(t ) + ξ  ∂ x˜(t ) + ξ  lim sup

)τl−1

(35)

In this section, a simulation example is given to show the effectiveness of the above results. The two cases of the example contain the positive impulsive effect and negative impulsive effect under DOS attacks, which is distinguished by impulsive gain c. When

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D. Ye and Y. Shao / Neurocomputing 366 (2019) 131–139

2

2

1

1

0

0

-1

-1

-2

-2 0

2

4

6

8

10

0

2

4

Time(s)

6

8

10

6

8

10

Time(s)

2

2

1

1

0

0

-1

-1

-2

-2 0

2

4

6

8

10

0

2

4

Time(s)

Time(s) Fig. 1. State strajectories xij (t) with impulsive control under DOS attacks.

c < 0.535, the impulsive effect plays a positive role. When c > 0.535, the impulsive effect plays a negative role. Assume that the nonlinear multi-agent systems with three followers and one leader. It is assumed that the initial topology is connected and the pinning matrix D = diag{2, 0, 0, 0}. Then the Laplacian matrix L and matrix H are

⎡

−1 2 0 −1

1

⎢−1 L=⎣ 0 0

0 0 1 −1

⎤

⎡

0 −1⎥ −1⎦ 2

3

⎢−1 H=⎣ 0 0

−1 2 0 −1

0 0 1 −1

⎤

0 −1⎥ −1⎦ 2

The heterogeneous nonlinear multi-agent systems (1) have the following systems parameters



−2.5 1 0

A1 =



A3 = A4 =

 A0 =

−2.5 1 0

10 −1 −18 −2.6 1 0 10 −1 −18



0 1 , −0.5 10 −0.9 −23



0 1 , 0

0 1 0



A2 =



−2.5 1 0

10 1 −18



0 1 , −0.5

 35 B0 = B1 = B2 = B3 = B3 =

6

0 0

0 0 0

0 0 0



It is assumed that the nonlinear function f (xi (t )) = (0.5(|xi1 + 1| − |xi1 − 1| ) 0 0 )T . The initial states are chosen

as x0 = [3.3 0.66 0.1]T , x1 = x2 = x3 = x4 = [0.33 0.66 0.1]T . CASE 1: DOS Attacks with Positive Impulsive Effect Based on Corollary 1, we can get c < 0.535. Let c = 0.5. It is found that μ = 0.866 and α = 22.056. Choose h = 0.02 and η = 20. As given in (10), the attack length rate satisfies

Ta  t

η + (α + ln μ h

ln μ h

)

 0.515

It follows from (11), the maximum attack ratio satisfies:

amax

τmin



− (α +

α

ln μ h

)

 1.56

When impulsive gain c = 0.5 satisfies c < 0.535, the impulsive effect can facilitate the quasi-synchronization of heterogeneous nonlinear multi-agent systems. As shown in Fig. 1, the quasisynchronization of heterogeneous nonlinear multi-agent systems subject to DOS attacks is achieved via impulsive control. Fig. 2(a) dedicates the error state e(t) is divergent without impulsive control. By comparing results in Fig. 2(a) and (b) verifies the efficiency of synchronizing impulsive subject to DOS attacks.

2500

5

2000

4

1500

3

||e(t)||

||e(t)||

D. Ye and Y. Shao / Neurocomputing 366 (2019) 131–139

1000

137

2

1

500

0

0 0

2

4

6

8

0

10

2

4

6

8

10

Time(s)

Time(s) Fig. 2. (a) ζ (t)2 without impulsive control. (b) ζ (t)2 with impulsive control.

2

2

1

1

0

0

-1

-1

-2

-2 0

2

4

6

8

10

0

2

4

Time(s)

6

8

10

6

8

10

Time(s)

2

2

1

1

0

0

-1

-1

-2

-2 0

2

4

6

8

10

0

2

4

Time(s)

Time(s) Fig. 3. State strajectories xij (t) with impulsive disturbance and DOS attacks.

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D. Ye and Y. Shao / Neurocomputing 366 (2019) 131–139

105

3.5

5

3 4

2

||e(t)||

||e(t)||

2.5

1.5

3

2

1 1 0.5 0

0 0

2

4

6

8

10

0

2

4

Time(s)

6

8

10

Time(s)

Fig. 4. (a) ζ (t)2 without control strategy (6). (b) ζ (t)2 with control strategy (6).

CASE 2: DOS Attacks with Negative Impulsive Effect According to Theorem 2, when c > 0.535. Assume that c = 0.54. It is found that μ = 1.052. Choose the control gain θ = 200. Then, we can get α = −7.943. Choose h = 0.02 and η1 = 5 and η = 2. Then α + lnhμ + η1 = − 0.38 satisfies. As given in (32) and (33), the minimum communication length satisfies

τmin 

ln μ = 0.0172 η1 − η

τmin 

ln μ − (α +

ln μ h

)

= 0.009

When impulsive gain c = 0.54 satisfies c > 0.535, the impulsive effect does harm to the quasi-synchronization of heterogeneous nonlinear multi-agent systems. Based on Fig. 3, the quasisynchronization of heterogeneous nonlinear multi-agent systems under impulsive disturbance and DOS attacks is satisfied with the control strategy (6). Fig. 4(a) indicates that the error state e(t) diverges under impulsive disturbance and DOS attacks without the control. With the control strategy (6), the bounded error state e(t) is achieved under impulsive disturbance and DOS attacks, as shown in Fig. 4(b). 5. Conclusion In this paper, the quasi-synchronization of heterogeneous nonlinear multi-agent systems under DOS attacks has been addressed. The DOS attacks can destroy the communication topology. The quasi-synchronization is caused by heterogeneous characteristic. Both the positive impulsive effect and negative impulsive effect are considered. The relationship among control strength, average impulsive interval and attacks strength is analyzed to satisfy the secure quasi-synchronization. Considering the positive or negative impulsive effect, the attacks strength is represented by attacks length rate, maximum attack rate and minimum communication strength, respectively. The average impulsive interval is proposed to unify the synchronization criteria of heterogeneous nonlinear multi-agent systems with two impulsive effects. Declaration of competing interest No conflict of interest exits in the submission of this manuscript, and manuscript is approved by all authors for publication. I would like to declare on behalf of my co-authors that the work described was original research that has not been published previously. All the authors listed have approved the manuscript that is enclosed.

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[30] D. Ye, X. Yang, Distributed event-triggered consensus for nonlinear multi-agent systems subject to cyber attacks, Inf. Sci.s 473 (2019) 178–189. [31] X. Yang, J. Cao, J. Lu, Synchronization of delayed complex dynamical networks with impulsive and stochastic effects, Nonlinear Anal. Real World Appl. 12 (4) (2011) 2252–2266. Dan Ye received the B. S. and M. S. degrees in mathematics and applied mathematics from Northeast Normal University, China in 2001 and 2004, respectively, and the Ph.D. degree in control theory and engineering from the Northeastern University in 2008. She was a lecturer with the Northeastern University from 2008 to 2010. Currently, she is a professor of the College of Information Science and Engineering, Northeastern University, China. Her research interest includes fault-tolerant control, robust control, adaptive control and security of cyber-physical systems.

Yuyuan Shao received the B.S. degree in College of Electrical Information Science and Engineering, Northeast Petroleum University, China, in 2017. Now, he is pursuing the M.S. degree at the College of Information Science and Engineering, Northeastern University. His research interests include multi-agent systems and impulsive control.