Hybrid event-triggered synchronous control on master-slave neural networks with deception attacks

LIU Yang,ZHANG Zhenzhen,BAI Chunjiang,CHEN Hao

(1.College of Electrical Engineering,Southwest Minzu University,Chengdu 610041,China; 2.National Key Laboratory of Science and Technology on space Microwave,China Academy of Space Technology(Xi’an),Xi’an 710000,China)

Abstract：This paper presents a hybrid event-triggered synchronous control method for a master-slave neural network subject to deception attacks.The proposed method employs a hybrid event-triggered scheme based on Bernoulli distribution to reduce the communication burden and handle the unpredictable network environment by transmitting data only when necessary.Notably,the memory event trigger mechanism can transmit more data during the peak and bottom of the system’s response state.To account for uncertainties in the system,including network-induced delay and randomly occurring deception attacks are developed within a unified framework and the corresponding Lyapunov-Krasovskii functional (LKF) is constructed.A stabilization criterion is then derived using Lyapunov theory.Numerical simulations demonstrate the effectiveness of the proposed method.

Key words：neural network; time delay; hybrid event-triggered scheme; deception attacks

0 Introduction

As one of the complex networks,neural networks (NNs) have aroused considerable interests for many years owing to the successful applications and potential capacities in space technology,image processing,quadratic optimization and secure communication and so on[1-5].Along with some dynamic behaviors appearing in complex NNs,synchronization is one of the most significant and attractive ones,which has been widely studied by scholars[6,7].Thus far,quantities of control methods have been proposed to realize the synchronization of NNs,such as impulsive control[7],sampled-data control[8],adaptive control[9]and output feedback control[10].

In addition,time delay is a common phenomenon in many systems,and it is crucial to study the stability of systems with time delay.In[11],the authors conduct a study on the stability of a micron microwave ranging system by performing a time delay stability test.

In the recent a few decades,with the rapid development of the modern high-speed communication networks,the research focusing on networked systems has been a hot topic[12].In virtue of its high efficiency,networked systems are extensively used in various fields.At the same time,in the process of its development,a few challenges have emerged,especially the limitation of bandwidth between communication channels.For this purpose,how to design the appropriate transmission mechanism to ensure system performance is a challenging research issue while saving communication resources.

Recently,the time-triggered mechanism (TTM) based on sampled-data control has shown the superiority in achieving synchronization of NNs[13].Under this trigger mechanism,signals will be transmitted at a fixed interval frequently,which will result in unnecessary waste of transmission resources when the system is in an ideal running state.Encouraged by this,event-triggered mechanism (ETM) is proposed to reduce the communication frequency[14].A periodic ETM that samples at fixed instants is proposed in[15],where the occurrence of Zeno phenomenon can be avoided.[16]proposes a discrete ETM,which co-devises the trigger threshold and the control gain.Most of the proposed trigger mechanisms have constant trigger thresholds,namely,the thresholds do not involve with dynamic changes in the system.To address this situation,a dynamic ETM is proposed in[17],in which the trigger threshold is time-varying and depends on some set parameters or variables.In the research of load frequency control,[18]proposes an adaptive ETM,where the trigger threshold can be adjusted spontaneously to an appropriate value based on the system state.In addition,[19]proposes a memory event-triggered mechanism (METM) that takes into account not only the most recent released packet,but also the last few packets,which increases the number of released packets when the system state response at peaks and bottoms.The ETMs mentioned are more suitable for the ideal network operation environment.Inspired by the above,this paper comes up with a hybrid-triggered mechanism (HTM) contains static event-triggered mechanism (SETM) and METM,trying to address the impact of uncertain system operation environment and approach a balance between ensuring system performance and saving communication resources.

Recently,another emerging and noticeable challenge in networked systems is the potential occurrence of cyberattacks,which are usually inserted into the transmission network to manipulate the smooth transmission,aiming to destroy or disrupt the system[20].Cyberattacks can be generally divided into denial-of-service (DoS) attacks,deception attacks and replay attacks,which are widely studied in various networked systems.[21]addresses the stability of networked system under deception attacks.Instead,design of NNs filter with HTM and deception attacks are investigated in[22].

Motivated by aforementioned discussions,this work aims to design an output feedback controller with HTM under deception attacks to realize the master-slave synchronization of NNs.The contributions of this paper are summarized as follows:a hybrid event-triggered scheme coordinated by Bernoulli distribution is proposed in this paper,which does not only reduce the network burden effectively,but also can improve the system performance.In this paper,deception attacks and HTM are integrated into a single framework.The corresponding neural network system model is established and a new,efficient control method is introduced.

1 Preliminaries

Considering the following system:

Master system:

(1)

Slave system:

(2)

By definee(t)=x(t)-y(t),the error system can be depicted:

(3)

wherer(e(t-d(t)))=g(x(t-d(t)))-g(y(t-d(t))),x(t)=col{x1(t),x2(t),…,xn(t)}∈nis the state vector andu(t)∈mis the control input;z(t)∈pdenotes the controlled output,g(x(t))=[g1(x1(t)),g2(x2(t)),…,gn(xn(t))]Tis the neuron activation function.Wrepresents connection weight matrix with appropriate dimensions; d(t) is the time-varying delay satisfyingis constant.Jis an external input vector andω(t) is the disturbance.A,B,C,Dis constant matrix with appropriate dimensions.

Assumption1.Fori∈{1,2,…,n},the activation error functionr(x) satisfiesr(0)=0 and ∀s1≠s2

(4)

Assumption2.The deception attack functionf(x) satisfies the following inequality

(5)

Wheres1≠s2∈andare known constant scalars.

1.1 Hybrid-triggered transmission mechanism

Considering the unpredictability of network conditions,an HTM containing SETM and METM is introduced to achieve a better balance between the limited transmission resource and the stability of the NNs.SETM and METM are employed in different network situations.

CaseA.SETM.When the system is operated in a non-ideal condition,the SETM with more trigger times is activated.In the SETM,the next instant that signal released to the controller can be computed as:

(6)

where Δ(t)=e(tkh)-e(tkh+ih),which represents the error between the last transmitted data and the currently sampled data,σ∈[0,1]is a given threshold scalar and Ω>0 is a positive definite diagonal matrix to be determined,h>0 is the sampling interval.

The zero-order-hold (ZOH) with the holding interval of[tkh+τtk,tk+1h+τtk+1) is introduced to store the latest transmitted data from the controller.Defineτ1(t)=t-tkh-ih,the controller can be presented as:

ua(t)=Ke(tkh)=Ke(t-τ1(t))+Δ(t)

t∈[tkh+τtk,tk+1h+τtk+1)

(7)

CaseB.METM.The METM with less trigger times is activated when the system works in an ideal environment.Comparing with the conventional triggering mechanism,not only the last released packet is taken into account,but also the previous released packets are utilized under METM.In general,the current sampled signal is released if the error between the sampled signal and the last transmitted is large enough.But the sampled signals with small state variation at the peak and bottom might not meet the trigger condition.In this situation,system state response is actually far from zero.Thus,signals are expected to be released frequently to stabilize the system state.It can be solved effectively in METM.The next instant that signal is released to the controller can be computed as:

(8)

where Δtk-j+1(t)=e(tk-j+1h)-e(tkh+ih),j=1,2,…,m.μ(t)=μ0+μ1e-v‖x(tkh+ih)‖.j∈[0,1]represent the weighting parameters andj=1.μ0,μ1andvare given positive constants.

Defineτ2(t)=t-tkh-ih,the controller can be presented as:

(9)

Remark1.In actual network systems,communication resources are often limited due to the limitations of hardware conditions and network environment.In order to achieving better system performance,an SETM (6) is applied into the system.Note that when the threshold scalarσis small enough,the number of released packets will increase considerably,in particular,in (6),whenσ=0,the corresponding trigger mechanism reduces to TTM.

Remark2.In METM (8),mrepresents the number of packets released recently utilized by the triggering mechanism.In particular,whenm=1,this triggering mechanism is conventional that only uses the last released packet.jrepresents the weights of the previous triggered packets.The newer packet is supposed to be more important than the previous ones.Therefore,the following inequality holds:j>j+1,j=2,…,m-1.

Remark3.An adaptive variableμ(t) is introduced in (8),which depends on the currently sampled datae(tkh+ih).μ(t) decreases as ‖e(tkh+ih)‖ increasing.Thus,when the system state is unstable,more packets are released to maintain the system performance.It is easy to know thatμ0≤μ(t)≤μ0+μ1,villustrates the rate that the system state affects onμ(t).

Similarly,according to the method of HTM proposed in[23],combining (7) with (9),the controller can be represented as:

uh(t)=α(t)ua(t)+(1-α(t))ub(t)

(10)

Mode switching between the two triggering mechanisms can be achieved by a random Bernoulli variableα(t),which satisfies the Bernoulli distribution andIfα(t)=1,the SETM is chosen to transmit the signal,conversely,ifα(t)=0,the METM is activated.

Deception attack may occur when the triggered packet is transmitted into the network,which will replace the previous transmission data.Signal transmission can only operate when the deception attack is vanished.Considering the above situation,the controller can be reformulated as:

u(t)=Kβ(t)f(e(t))+(1-β(t))uh(t)

(11)

Combining (7)-(11),the controller u(t) can be designed as:

u(t)=K{β(t)f(e(t))+(1-β(t))

Then,the system is expressed as:

(12)

In order to derive the main results,the following definition and lemmas are employed.

Definition1[24].The guaranteedH∞performance controller designing problem is stated as follows.For a prescribedγ>0,it is to find a suitableu(t) such that:

1) Forω(t)=0,the system (12) is asymptotically stable.

Lemma1.[25]:Forx(t) andg(x(t)) satisfying Assumption 1,x(t) andf(x(t)) satisfying Assumption 2,if there exist positive semi-definite diagonal matricesUandV,then the following inequalities hold:

where

2 Main results

(13)

Ψ21=[∏1∏20 ∏301×3∏4∏5

∏21∏3…m∏3∏6]

Proof.Choose the following Lyapunov-Krasovskii functional for the closed-loop system:

(14)

Taking the time derivative ofv(t) along the trajectory of the system and calculate the mathematical expectation,one obtains:

v2(t)=eT(t)(R1+R2+R3)e(t)-

(15)

in which

BK[e(t-τ2(t))+Δtk-u+1(t)]+Wr(e(t-d(t)))+Dω(t)

BK(e(t-τ1(t)+Δ(t)))

where

From the event-triggered conditions,one obtains:

σeT(t-τ1(t))Ωe(t-τ1(t))-ΔT(t)ΩΔ(t)≥0

(16)

and

ρeT(t-τ2(t))Ωe(t-τ2(t))-

(17)

By utilizing Lemma 1,it is obtained:

(18)

WhereUandVare positive semi-definite.Then,the following inequality holds:

σxT(t-τ1(t))Ωx(t-τ1(t))-

ρxT(t-τ2(t))Ωx(t-τ2(t))+

(19)

By the Schur complement,we can derive that (13) is equivalent toThis completes the proof.

3 Simulation

In this section,a numerical example is given to illustrate the effectiveness the designed control strategy.Consider the master system and slave system with the following parameters:

and controller gain can be set as:

K=[5183 8,2.503 2,3.438 9,6.867 6]

Fig.1 state response of x(t) under K

In order to cope with the extremely hostile operating environment,we setσ=0,that is,SETM becomes TTM.According to the different values ofα(t),the trigger mechanism will switch between TTM and METM.When the system running environment is ideal,METM is activated,conversely,TTM is activated.It can be seen from the Figure 2 that the number of triggers in METM is significantly less than in TTM,which achieves the desired effect.

Fig.2 Released instants under varying switching signal α(t)

4 Conclusion

TheH∞hybrid event-triggered synchronization problem of master-slave neural networks has been investigated in this paper.In order to reduce the unnecessary data transmission between the communication channels,a hybrid event-triggered mechanism is proposed.In addition,the deception attacks occurring randomly is considered for the control method on the system.Moreover,sufficient synchronization criteria are derived to guarantee the system asymptotically stable by using Lyapunov stability theory.Finally,a numerical example is provided to demonstrate the effectiveness of the designed control method.