Stephen AROCKIA SAMY ,Raja RAMACHANDRAN ,Pratap ANBALAGAN ,Yang CAO

1Department of Mathematics,Alagappa University,Karaikudi 630 003,Tamil Nadu,India

2Ramanujan Centre for Higher Mathematics,Alagappa University,Karaikudi 630 003,Tamil Nadu,India

3Research Center for Wind Energy Systems,Kunsan National University,Gunsan 573-701,Republic of Korea

4School of Cyber Science and Engineering,Southeast University,Nanjing 210096,China

Abstract: The main aim of this work is to design a non-fragile sampled data control (NFSDC) scheme for the asymptotic synchronization criteria for interconnected coupled circuit systems(multi-agent systems,MASs).NFSDC is used to conduct synchronization analysis of the considered MASs in the presence of time-varying delays.By constructing suitable Lyapunov functions,sufficient conditions are derived in terms of linear matrix inequalities(LMIs) to ensure synchronization between the MAS leader and follower systems.Finally,two numerical examples are given to show the effectiveness of the proposed control scheme and less conservation of the proposed Lyapunov functions.

Key words: Multi-agent systems (MASs);Non-fragile sampled data control (NFSDC);Time-varying delay;Linear matrix inequality (LMI);Asymptotic synchronization

1 Introduction

In recent times,multi-agent systems (MASs)have emerged as the most important nonlinear model and have been used efficiently (Beard et al.,2002;Fax and Murray,2004;Ren and Sorensen,2008).A system is composed of small subsystems,known as agents,which work together to form a network of other agents.Agents can connect with one another through a network to accomplish global objectives without the need of local contact or supervisors.Acquiring data from all agents is a challenging task,even though the entire system is controlled.In such cases,we use a multichannel system view.Specifically,in agreement with the prearranged values of our problem,data acquisition can be categorized as a leader-follower (Jia et al.,2011;Rakkiyappan et al.,2015;Tang et al.,2015)or a leaderless problem(Meng et al.,2011).In the leader-follower problem,the leader serves as a commander,who creates the necessary path without the knowledge of the followers.A subgroup of followers must directly access the information about the leader in this case.For some applications,MAS integration is not a single phase,but rather a two-part process that includes a balancing physical and communication restrictions.In this context,the research on MASs has progressed dramatically in recent years (Wang CY et al.,2019;Liu JL et al.,2021;Yue et al.,2021).

Artificial neural networks (ANNs) have grown in popularity among the scientists and researchers who worked in the field of artificial intelligence networks since the 1980s.A neural network (NN) is a computer imitation composed of nodes that are connected to each other(or neurons).ANN research has made major progress in biology,automatic control,computer acknowledgement,etc.,providing a better understanding of challenging and practical issues of modern systems (Sompolinsky et al.,1988;Chen and Aihara,1995;Zheng et al.,2017).In this regard,ANNs provide substantial nonlinearities,making it challenging to analyze their dynamical properties.Thus,several researchers have focused their efforts on analyzing the dynamical behavior of ANNs and have produced some noticeable results in Alimi et al.(2019)and Wang WP et al.(2019).

In practice,chaotic synchronization is a critical matter in nonlinear science.The concept of chaotic synchronization involves synchronizing two chaotic systems.However,the projected results indicated that the followers’ system actions had an effect on the leader system,despite the fact that the leader system was self-contained.The fundamental complexity of nonlinear dynamics causes extra challenges in handling synchronization issues in MASs in this paper.Note that the aforementioned literature primarily focused on synchronized MASs,in which all agents have similar dynamics.However,in real-world applications,various dynamics may be more realistic.As a result,studying synchronization in MASs becomes significantly more difficult.A practical synchronization is used in this case because of many different types of agents in the system.Indeed,MAS synchronization issues are diffi-cult to tackle due to the intrinsic complexity of nonlinear dynamics.Jia et al.(2019)addressed synchronization of MASs with time-varying control and delayed communications.Sang and Zhao (2019) studied the chaotic synchronization for NNs under timevarying delays using the Lyapunov stability theory.Kaviarasan et al.(2021) investigated a stochastic non-fragile controller for MASs with a communication delay.In Zhao YS et al.(2019),a widespread synchronization for chaotic NNs was given,and numerous suitable conditions were found by employing a suitable Lyapunov-Krosovskii functional.

To achieve the MAS aim,many control systems have been used,including impulsive control,eventtriggered control(Zhou et al.,2015),pinning control(Wang YL et al.,2014),quantized control(Tan et al.,2019),and adaptive control.Zhang et al.(2018)and Jiang et al.(2020)derived sampled data control(SDC) for MASs.Zhong et al.(2022)discussedH∞load frequency control for a multi-area power system based on non-fragile propositional integral control.Among them,SDC performed well in a task to study the dynamics of MASs due to the improvement of digital technology.The main purpose of SDC is to update the control signals only at sampling instants during the sampling intervals.In sampling intervals,control signals on SDC systems are consistent and may change only during the instants of the sample.Control signals are thus gradually transmitted in SDC systems,and these discontinuous signals cannot be employed directly for stabilizing the control systems.From this perspective,SDC has been increasingly important in control systems,and some remarkable results have been achieved(Liu Y et al.,2019;Xu et al.,2021;Zhao C et al.,2021;Lavanya and Nagarani,2022).On the other hand,non-fragile control can adjust gain fluctuation in the design of the controller and is used to assure system stability when small disruptions occur in the design of the controller.A few noteworthy works on the non-fragile control problem in MASs have been published (Ali et al.,2020b;Saravanakumar et al.,2020).

To a greater extent,the conditions for achieving asymptotic synchronization of nonlinear MASs with non-fragile sampled data control(NFSDC) have not been comprehensively addressed(Zhang et al.,2018;Jia et al.,2019;Jiang et al.,2020),which serves as a motivation for the investigation in this paper.

The contributions of this paper are summarized as follows:

1.In practical applications,NN-based circuit systems are modeled as MAS-based leader and follower systems for analyzing the synchronization issue,which is based on Lyapunov-Krasovskii functional with Kronecker product.

2.For MASs,NFSDC is designed,and then sufficient conditions of synchronization criteria are derived using Lemma 1.The derived conditions assure the asymptotic stability of the MAS error system of MASs.

3.The generality of our strategy for designing a controller distinguishes it from the past work in this area of research.Furthermore,it provides a broad framework for investigating synchronization difficulties that are related to the consistency of control strength over time,such as NFSDC.

4.The approaches in Ma et al.(2016)have been compared to confirm the efficacy of the suggested method in the numerical simulation.

According to Theorem 2,designing NFSDC can achieve leader-following synchronization of MASs with time-varying delay.Some improved inequality techniques have been successfully used for MASs.As a result,the whole analysis procedure has been simplified.Furthermore,less conservative stability requirements are constructed using linear matrix inequalities(LMIs).Finally,numerical validation confirms the effectiveness of the designed control strategy using MASs,indicating the efficacy of our suggested method when time-varying delays occur.

Notations: Rnand Rn×ndenote thendimensional Euclidean space andn×nreal matrices,respectively.0 andIrefer to the zero and identity matrices with suitable dimensions,respectively.Superscript T denotes the transpose of a matrix.?refers to the Kronecker product.‖·‖refers the Euclidean norm of a vector.

2 Problem statement and preliminaries

This section provides the preliminary work for our main study,including some fundamental concepts in algebraic graph theory and mathematical representations of the relevant subject.

LetG=(ν,ε,W) be an interconnection graph that is used to connectNagents,each of which has a nodeν={z1,z2,···,zN}and a directed edgeε ?{ν × ν}.If the edge (zi,zj)∈εfrom agentzjtozi,thenzjis a neighbor ofziandW=[aij]N×Nis the adjacent weighted matrix.The edge of agentzitozjisaijand must be zero.Nidenotes the set of agent neighborsi.A path of agentzjtoward agentziis a string sequence of edges (zi,zl1),(zl1,zl2),···,(zlm,zj).When a directional path from one node to the other node exists,the directional graph is said to be strongly linked.The Laplacian matrixofGis defined bylij=?aij,i≠j,(i,j=1,2,···,N).

Let us consider the leader and followers system as follows:

wherez0(t)=[z01(t),z02(t),...,z0n(t)]T∈Rnrefers to the state of the leader agents,andzi(t)=[zi1(t),zi2(t),...,zin(t)]T∈Rnrefers to the state of theithagent.The control input isui(t)∈Rn.A=diag(a1,a2,....,an) is a diagonal matrix,andB=(bij)n×nandC=(cij)n×nare weight matrices.g(z0(t)) andg(zi(t))∈Rnrepresent the activation functions of the leader system and follower system,respectively.ι(t)is a time-varying delay and satisfies 0≤ι(t)<ιand ˙ι(t)≤λ,whereλ>0 andι>0 are constants.

Here,cooperative synchronization implies that the two dynamic systems with the leader and followers will attain the same partial state simultaneously.Generally,the leader uses channels to broadcast a communication signal that triggers the followers,and this signal induces the coordination between the leader and followers.

Letηi(t)=zi(t)?z0(t).The synchronization error is described as

whereη(t)=[η1(t),η2(t),···,ηN(t)]T,U(t)=[u1(t),u2(t),···,uN(t)]T,h(η(t))=[g(η1(t))?g(η0(t)),g(η2(t))? g(η0(t)),···,g(ηN(t))?g(η0(t))]T,andh(η(t ? ι(t)))=[g(η1(t ?ι(t)))?g(η0(t ?ι(t))),g(η2(t ?ι(t)))?g(η0(t ?ι(t))),···,g(ηN(t ?ι(t)))?g(η0(t ?ι(t)))]T.

Definition 1(Subramanian et al.,2019) The MAS(1)is called asymptotic synchronization if

for alli=1,2,···,N.

It is possible to obtain sufficient conditions with a suitable Lyapunov function using the following lemma:

Lemma 1(Seuret and Gouaisbaut,2013) For a given matrix>0,the following inequality holds forω:[c,d]→Rn:

Assumption 1(Ali et al.,2020a) For allq=1,2,···,nandgq(·),there exist positive constantsandsuch that

The time-step sampled data from agentiare encapsulated in a packet of data,subsequently transferred to its controller and agentjover agenticommunication network.Zero-order hold(ZOH) is used to update the input and output actuators of controlleri.ZOH represents the outcome of keeping each sample value for one sample interval while converting a discrete-time signal to a continuous-time signal.Now,we consider the following NFSDC scheme:

whereδ >0 is the coupling strength,Jis the feedback controller,and ΔJ(t) is an unknown gain matrix that satisfies ΔJ=β1Ψ(t)β2,withΨT(t)Ψ(t)≤I,whereβ1andβ2are two given constant matrices.

To stabilize Eq.(4),we assume that the control has discrete measurementsz(tk) only at sampling instanttkwith 0=t0

The primary goal of this research is to achieve the asymptotic synchronization of MAS (7) using NFSDC with a large upper sampling interval bound,which is detailed as follows:

Problem 1Here,the objective is to achieve synchronization between leader and follower systems:

1.We construct the delay-dependent sufficient conditions in the form of LMIs,which demonstrate that multi-agent error dynamic networks (7) can achieve asymptotic synchronization.

2.To solve the LMIs in the appropriate direction,the control gain matrixJis calculated.

3 Main results

The leader-following synchronization of MAS(1) connected by nodes in a topology is discussed in this section.

Theorem 1Under Assumption 1,for some positive constantsι,λ,?and given gain matrixJ,if there exist positive matricesP1,P2,P3,P4,P5,P6,P7,and any matricesS1,S2,andS3with appropriate dimensions,such that the following equalities

then the considered leader and followers are asymptotically synchronized under the proposed NFSDC scheme (6).

ProofConstruct the Lyapunov-Krosovskii functional as follows:

Using Theorem 1,the following Theorem 2 is derived to obtain the control gain matrix for the NFSDC scheme (6) to asymptotically synchronize the leader and followers system(1):

Theorem 2Under Assumption 1,for some positive constantsι,λ,?,κ1,andκ2,if there exist positive matrices,,,,,,,and any matrixSwith appropriate dimensions,such that the following equalities

then the considered leader and followers are asymptotically synchronized under the proposed NFSDC scheme (6).In addition,the gain matrixJis given byJ=QS?1.

ProofWe defineS1=S?1,S2=κ1S?1,S3=κ2S?1,Q=JS,=SPKST(K=1,2,···,7),=SM1ST,and=SM2ST.Then,both sides of LMI (8) are multiplied with diagandis as expressed in Theorem 1.If<0,in light of Eq.(21),we have(t)<0.This means that,based on the Lyapunov-Krosovskii stability theorem,it is established that the addressed MAS (1) with gainJis asymptotically synchronized.Thus,the proof is completed.

Remark 1The above theorem ensures that the required criteria are obtained using some constructed inequalities and Lemma 1.It is observed that the integral portions which are less delay-dependent within the Writtinger inequality are ignored,and hence we achieve less conservativeness than Ma et al.(2016).Therefore,our results are more general than those in Ma et al.(2016).

Remark 2Based on the derived LMI condition in Theorem 2,Algorithm 1 describes the calculation of the control gain matrix.

Algorithm 1Asymptotic synchronization of MASs

Remark 3Ifδ=1 and ΔJ=0,then Eq.(5)is called nominally sampled data control.It is described as follows:

From Remark 3,we readily arrive at the following corollary for considered synchronization error system(7)under SDC scheme(22):

Corollary 1Under Assumption 1,for some positive constantsι,λ,?,κ1,andκ2,if there exist positive matrices(K=1,2,···,7) and any matrixSwith appropriate dimensions,such that the following equalities

then the considered leader and followers are asymptotically synchronized under the proposed SDC scheme (6).In addition,the gain matrixJis given byJ=QS?1.

ProofThe remaining elements are the same as those given in Theorem 2.

Remark 4If there is no time-varying delay (i.e.,ι(t)=0 in system (7)),then the multi-agent networks will be reconstructed as follows:

Now,using the same procedure as in Theorem 2,the following corollary can be obtained:

Corollary 2Under Assumption 1,for some positive constants?andκ1,if there exist positive matrices(K=1,2,3)and any matrixSwith appropriate dimensions,such that the following equalities

then MAS (24)can be asymptotically synchronized.In addition,the gain matrixJis given byJ=QS?1.

ProofWe defineS1=S?1,S2=κ1S?1,Q=JS,=SPKST(K=1,2,3),and ˇM1=SM1ST.Then,both sides of LMI (25) are multiplied with diag,andis as expressed in Theorem 2.If<0,in light of LMI(25),we have(t)<0.This means that,based on the Lyapunov-Krosovskii stability theorem,it is established that the addressed MAS (1) with gainJis asymptotically synchronized.Thus,the proof is completed.

Remark 5In this paper,the synchronization problem is changed into the delay functional differential equation with error as its dynamics,and thus Lyapunov-Krosovskii stability is employed in the analysis.Note that the computational complexity depends on the number of sampling periods,the number of agents (N),and the state vector dimension of agent dynamics(n).If the number of agents,state vector dimension,and number of sampling periods increase,the computation for solving LMIs will be more complicated,which means that the size of the LMIs increases,and it is more time-consuming to verify the feasibility of the obtained LMI conditions.In this respect,all the results of the condition derived in this paper may be more suitable for small,lowdimensional networks of agents.For higher-order LMI approaches,deriving stable conditions in the maximum sampling period with high computational efficiency becomes future research work.

4 Numerical validation

The validation of the derived sufficient requirements has been carried out.Initial consideration is given to the MASs as a means of demonstrating the superiority of the proposed method over the existing methods.Next,in the context of a design example,the nonlinear NN model is studied.

4.1 Example 1: comparison example

Consider the MASs with time-varying delays similar to those in Ma et al.(2016)as follows:

In this scenario,the initial circumstances for the agent are chosen to bez0(0)=[0.8,?0.1]T,z1(0)=[0.2,?0.6]T,z2(0)=[0.4,?0.2]T,z3(0)=[0.7,?0.3]T,z4(0)=[0.6,?0.4]T,and the network synchronization is shown in Fig.1.The validity of Theorem 2 is confirmed by all supporters entering the area around the leader synchronization.The MAS state trajectories are shown in Fig.2 and the synchronization of MASs with controllers is displaced in Fig.3.The comparison of the time delay for system(7) is presented in Table 1.The corresponding control gain matrix and upper bound are presented in Table 2.The synchronization of MASs with different delays is displayed in Fig.4.

Fig.2 Evolution of MASs zi1(t) (a) and zi2(t) (b)(i=0,1,2,3,4)

Fig.3 Synchronization plots of errors e1i(t) (a),e2i(t) (b),e3i(t) (c),and e4i(t) (d) (i=1,2) under non-fragile sampled data control

Fig.4 Synchronization plots of errors ei(t) (i=0,1,2,3,4) with ι=1.2 (a) or ι=1.5 (b)

4.2 Example 2: design example

The proposed units implemented in the NNs with two neurons are depicted in Figs.5 and 6.System(24)can be expressed based on an analysis of the circuit nodes in Fig.1.The values of the parameters given in Table 3 are taken from Lu(2002)and Wang TC et al.(2016),which are standards available in the existing works for the electronic circuits given in Figs.5 and 6.

Fig.1 Communication of MASs

Table 1 Comparison of the maximum allowable time delays

Table 2 Upper bound ? and control gain matrix J corresponding to Example 1

Table 3 Circuit parameters and their values

Here,we consider an artificial circuit model based on Wang TC et al.(2016).The circuit has an activation function circuit with a time-varying delay and holds a corresponding unit.Each circuit unit contains inductors,resistors,capacitors,some operational amplifiers,and other common electrical components.The circuit implementations are adjusted according to the complexity of the NN model.In this investigation,NNs(27)have two neurons and can be rewritten as

wherez(t)=[z1(t),z2(t)]T∈Rn.

Here,we look into the synchronization analysis of NFSDC in multi-agent networks using delayed NNs.Consider the leader and followers system (28)as follows:

Fig.6 Circuit realization of the tangent hyperbolic

For the present circumstances,the agents are chosen to bez0(0)=[1.4,?1.1]T,z1(0)=[0.3,?0.6]T,z2(0)=[0.2,?0.4]T,z3(0)=[0.6,?0.2]T,z4(0)=[0.5,?0.9]T,and the network synchronization is plotted in Fig.1.The validity of Theorem 2 is confirmed by all supporters entering the area around the leader synchronization.The state trajectories of MASs are shown in Fig.7,and the synchronization of MASs with controllers is displaced in Fig.8.The comparison of the maximum allowable time delay for system(5)is presented in Table 4.The corresponding upper bound and control gain matrix are presented in Table 5.The synchronization of MASs with different delays is displayed in Fig.9.

Fig.8 Synchronization plots of errors e1i(t)(a), e2i(t)(b),e3i(t)(c),and e4i(t)(d) (i=1,2) under non-fragile sampled data control

Fig.9 Synchronization plots of errors ei(t)(i=0,1,2,3,4) with ι=1.4 (a) or ι=1.7 (b)

Table 4 Comparison of the maximum allowable time delays

Table 5 Upper bound ? and control gain matrix J corresponding to Example 2

Fig.7 Evolution of MASs zi1(t) (a) and zi2(t) (b)(i=0,1,2,3,4)

5 Conclusions

In this work,the multi-agent system (MAS)synchronization problem has been investigated using a non-fragile sampled data control (NFSDC)scheme.The suitable Lyapunov-Krosovskii functional has been constructed to derive the synchronization criteria for the proposed MASs,expressed in linear matrix inequality (LMI) forms.The proposed control scheme has achieved synchronization between the leader and follower systems with derived sufficient conditions.Finally,numerical validation has demonstrated the effectiveness of the presented method.As a future work topic,it will be exciting to explore the possibility of extending the NFSDC scheme to event-triggered control when unanticipated disruptions occur in MASs.

Peng et al.(2022) and Yang et al.(2022)discussed the event-triggered optimal control and observer-based fault detection filter methods.Zhu et al.(2022) discussed the novel reconstruction method for temperature distribution measurement based on ultrasonic tomography.Motivated by these studies,the proposed control method can be applied to multi-area nonlinear power systems based on the novel reconstruction approach,which will be considered in future work.

Contributors

Stephen AROCKIA SAMY designed the research.Raja RAMACHANDRAN drafted the paper.Yang CAO and Pratap ANBALAGAN revised and finalized the paper.

Compliance with ethics guidelines

Stephen AROCKIA SAMY,Raja RAMACHANDRAN,Pratap ANBALAGAN,and Yang CAO declare that they have no conflict of interest.

Data availability

Due to the nature of this research,participants of this study did not agree for their data to be shared publicly,so supporting data are not available.