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ChinaofechnologyKeywords:edvelyconsensusposed control strategy is illustrated by a numerical example.king.Cooof industrceivedois exethe decoupled dynamics of its neighbors. Although event triggeredMASs, it has to check whether the trigger condition reaches the pre-13,14, the aver-15, the self-controllerl approach isSelf-triggeredems in 1820.works on self-model. In 21,presented for aspecial rst-order MASs and is shown to achieve the rendezvousself-triggered control for MASs are based on state feedback underContents lists available at ScienceDirectNeurocomputingNeurocomputing 190 (2016) 179187/10.1016/j.neucom.2016.01.019scribed threshold. Event-triggered control strategy requires dedicateda restrictive assumption that the states of agents can be measured.Nevertheless, the states of agents are not measured in many cases,this assumption is not often held, control based on state feedbackcannot be used in those cases. For this reason, output feedback0925-2312/& 2016 Elsevier B.V. All rights reserved.nCorresponding author.E-mail address: zhang_ (H. Zhang).control can reduce the frequency of information exchange amongobjective. However, the control strategy proposed in 21 is notheld for general linear systems. It also can be observed that mostthis control mechanism. In order to utilize limited resources effec-tively, so researchers are looking for a more effective control strategythat the states of controller are updated only when required. For thispurpose, event-triggered control is proposed. Works on this subjectcan be found in 48.In event-triggered control strategy, control tasks are executed viathe so-called event condition instead of period control. In 9,adis-tributed event-triggered consensus control strategy is presented for asecond-order multi-agent system. A novel event-triggered consensusprotocol is proposed in 10, where each agent implements a model oftegies are presented for rst order integrator inage consensus can be guaranteed for MASs. Intriggered control for linear systems via state feedbackhas been studied. In 16,17, a self-triggered controstudied for two classes of nonlinear control systems.control is also considered in network control systIt is worth mentioning that most of existingtriggered control focus on the rst-order integratora distributed self-triggered control strategy isthe point of view of resources usage, a certain disadvantage existed inevent instant. There is no need to keep track of the measurementerror. Both centralized and distributed self-triggered control stra-The study on MASs includes ocproblem 2 and distributed ltering 3has many applications in the aspectsthe consensus problem of MASs has reearly study of consensus problem is basedstrategy, under which the control task1. Introduction hardware to monitor the plant. Nevertheless, large scale hardwarerequirements cannot be realized in many cases and unnecessary dataConsensusSelf-triggered controlOutput feedbackproblem 1,consensusperativecontrolofMASsy and military. Thus,extensive attention. Then time-triggered controlcuted periodically. Fromtransmission can increase network load, which will result in the wasteof communication resources.The disadvantage promotes the development of self-triggeredcontrol strategy. Self-triggered control has been proposed in11,12. In self-triggered control strategy, the next update time ofcontroller is calculated depending on the measurement of currentMulti-agent systems& 2016 Elsevier B.V. All rights reserved.Self-triggered output feedback control forof multi-agent systemsMiaomiao Wua, Hao Zhanga,n, Huaicheng Yanb,c, HongliangaDepartment of Control Science and Engineering, Tongji University, Shanghai 200092,bKey Laboratory of Advanced Control and Optimization for Chemical Process of MinistrycSchool of Information Science and Engineering, East China University of Science and TdDepartment of Biomedical Engineering, National University of Singapore, 117575, Singaporearticle infoArticle history:Received 29 August 2015Received in revised form16 December 2015Accepted 12 January 2016Communicated by Guang Wu ZhengAvailable online 2 February 2016abstractThis paper studies the self-trigger(MASs). A novel self-triggerdistributed cases, respectiobserver is adopted. A dynamicresponse. Under this triggeringare updated at triggering time.Moreover, the asymptoticjournal homepage: www.elsevieconsensusRendEducation, Shanghai 200237, China, Shanghai 200237, Chinaed control consensus problem of general linear multi-agent systemscontrol strategy based on output feedback is proposed for centralized and. In consideration of the states of agents that are not available, a stateobserver-based control law is employed to improve the transientstrategy, both the estimated states of MASs and the states of controllerThe next triggering time is predetermined at the last triggering instant.of MASs can be guaranteed. Finally, the effectiveness of the /locate/neucompackets are randomly dropped without acknowledgment to theestimator is investigated.are not available.A state observer is given byM. Wu et al. / Neurocomputing 190 (2016) 179187180Most of the literatures assumes that the dynamics of agent arelinear time-invariant(LTI) systems. The LTI systems means that ifthe initial state and the input are the same, the output waveformwill always be the same. A large number of physical systems canbe modelled as LTI systems. In 26, the paper addresses a problemof decentralized estimation for a class of linear time-invariantsystems affected by stochastic disturbances and deterministicunknown inputs. The H1consensus problem of homogeneous LTIsystems is studied in 27. In this paper, we studied the self-triggered output feedback control for MASs of general LTIdynamics and assumes that the states of the agents are notentirely available.Inspired by 28,29, we investigate the self-triggered outputfeedback control consensus problem of general linear MASsdynamics in this paper, in which the states of the agents are notentirely available. Both centralized and distributed observer-basedself-triggered control approaches are proposed, where the nexttriggering instant of each agent is calculated based on informationfrom its own and neighboring agents at previous sampling instant.The contributions of this paper contain two aspects. Firstly, a stateobserver is used to estimate the states of agents. A dynamicobserved-based control law is adopted. Not only the estimatedstates but also the states of controller are taken into consideration.Thus, the transient response of closed-loop systems can beimproved. Secondly, a novel self-triggered control strategy isproposed. In this control strategy, the next triggering instant ispredetermined at the previous triggering instant. There is no needto monitor the state measurement error continuously any more.The rest of this paper is organized as follows: Section 1 explainsa brief background. Section 2 introduces the relevant knowledge ofgraph theory and describes the consensus problem. Centralizedself-triggered control strategy is proposed in Section 3. Corre-spondingly, distributed self-triggered control is presented in Sec-tion 4. An example is given in Section 5. Section 6 summarizes theresults of this paper.Notations. In this paper, matrices are assumed to have compa-tible dimensions if not explicitly stated. InARnC2nis identity matrix,1 denotes the column vector with all ones. For matrix AARnC2n,AARnC2ndenotes the transpose of matrix A. For vector xcolx1;xnARnand matrix A, JxJ and JAJ denote 2-norms of x and A,respectively. A real matrix P40Po0 denotes P, which is positive(negative)-denite matrix. A C10 B denotes the Kronecker product ofmatrices A and B.2. Problem formulationThe communication topology among agents is represented byan undirected graph 30GV;E;A, whereVfv1;vNg is a setof nodes with the node indices belonging to a nite index set I f1;Ng and EDV C2V is a set of edges. For an edge vi;vj, viandvjare called adjacent or neighboring. A path on G from node vi1tonode vilis a sequence of edges of the form vik;vik1;k1;lC01.A graph is called connected if there exists a path between everynC2napproach should be considered. In 22, a self-triggered controlsystem with a static output-feedback controller is studied, thetrigger condition depends directly on the output of the plant. In23, sufcient LMI conditions for the H1output feedback controldesign of linear discrete-time systems are proposed. In 24, theproblem of H1ltering for nonlinear networked systems withrandomly occurring distributed delays, missing measurements andsensor saturation is considered. In 25, the optimal estimationproblem in lossy networked control systems where the controlpair of distinct nodes. The adjacency matrix AaijC138AR is the_xitAxitBuitGyitC0yit;yitC xit; i 1;N;:C263wherexitARnis the observer state of xi(t),yitARqis themeasured output of observer, and GARnC2qis feedback matrix to bedetermined.Control law is designed as follows:uitFvit;_vitABFvitKCPjANiaijeAtC0tkvitkC0vjtkC0C1C0 xitkC0xjtkC0C1; tAtk;tk1;8it;vitC138;MABF0 ABF#; H00C0KC KCC20C21;DGC0C20C21; LINC10 L; 5then_zitMzitXjANiaijHI2C10 eAtC0tkzitkC0zjtkDhit: 6The state measurement error for the system is dened aseitI2C10 eAtC0tkzitkC0zit; 7substituting (7) into (6), one has_zitMzitXjANiaijH eitzitC0ejtzjtC0C1Dhit; 8which can be written in compact form_ztINC10 MztL C10 HetztINC10 Dht; 9where ete1t;eNtC138, ztz1t;zNtC138, hth1t;hNtC138.DenotewitXjANiaijzitC0zjt; itXjANiaijeitC0ejt; 10thus, wtL C10 I2nzt, tL C10 I2net, where wtw1t;wNtC138and t1t;NtC138.It follows from (9) and (10) that_witMwitXjANiaijHwitC0wjtXjANiaijHitC0jtXjANiaijDhitC0hjt; 11which can be written in compact form_wtINC10 MwtL C10 HwttL C10 Dht: 12DenoteQPQ C0QC0QQ#; T In0C0InIn#;QTQTC01;MTMTC01;D TD;HTHTC01;witTwit;wtw1t;wNtC138;itTit; t1t;NtC138; 13where P40, Q40, thus, Q40. Now, we are ready to propose thefollowing result on the centralized self-triggered control strategybased on output feedback.Theorem 1. Assuming that the communication graphG is connected,pair (A,B) is controllable and pair (A,C) is observable. Given F and Gsatised that ABF and AGC are Hurwitz. If there exists matricesP40 and Q40, scalar 40 such thatPABFABFP PBFPBFAQQAC02CC#oC0I 14is held. is chosen sufciently large such that 2Z1 and the trig-gering instant is chosen as follows:tk1rtk1ln 1C20aJ wtkJC20!; 15JwtkJ JL C10 HwtkJwhere A0;1,C20a a1a, a2JLC10QHJ, JMJC0 JAJ,C20 JMJ JAJ. Then, the consensus of N agents in (1) will be guaranteedunder the control law (4) with K C0QC01C. Furthermore, for anyinitial conditions in Rnand any time tZ0, the inter-execution timeinterval is strictly positive.Proof. Consider a candidate Lyapunov function for the closed-loop system (12)VtXNi 1witQwit; 16where Q is dened in (13). From (1) and (3), one has_hitAGChit: 17If G is designed to make AGC Hurwitz, then hi(t) will approachzero asymptotically. From (12) and (17), one can nd that theestimated error h(t) is decoupled from the consensus dynamics w(t). Thus, the stability of (12) is equivalent to the stability of thefollowing system:_wtINC10 MwtL C10 Hwtt: 18Calculating the time derivative of V(t) according to the closed-loopsystem (18), one has_Vt2XNi 1witQMwitXjANiaijHwitC0wjt01A2XNi 1XjANiaijwitQHitC0jt: 19Analyzing the rst part in (19), one has2XNi 1witQMwitXjANiaijHwitC0wjt01A2 wtINC10QML C10QH wt; 20wherewt,Q,M andH are dened in (13).Since G is connected,zero is a simple eigenvalue of L and all itsnon-zero eigenvalues are positive. Let UARNC2Nbe a unitary matrixsuch that ULU diagf0;2;Ng. The right and left eigen-value are 1 and 1, respectively. One can choose U 1=NpX1C138and U1=NpX2hi, with X1ARNC2NC01and X2ARNC01C2N. Lett1t;NtC138UC10 I2nwt. Notice thatw1twNt0, then, 1t0. Substituting wtU C10 I2nt into(20), noticing K C0QC01C, and choosing sufciently large suchthat 2Z1, one has2wtINC10QML C10QHwtXNi 2itQMMQ2iQHitrXNi 2ititrC0JwtJ2; 21where PABFABFPPBFPBFAQ QAC02CChi, and the last inequalityis derived by using (14).Analyzing the second part in (19), one has2XNi 1XjANiaijwitQHitC0jt2XNi 1XjANiaijwitQHitC0 tr2JL C10QHJJwtJJtJ; 22jtriggering condition (24). It follows from (18) that_wtINC10MwtL C10Hwtt: 28Substituting (26) into (28), one can obtain_wtINC10MwtL C10HwttINC10MRemark 3. In distributed self-triggered control, the sequence oftriggering time instant for agent i is denoted by ti0;ti1;. Thetriggering instant is different for each agent. The execution ofcontroller for agent i need its own estimated states and the statesof controller at current triggering time as well as estimated statesand the states of controller for its neighbors at last triggering_zitMzitjANiaijH I2C10 eAtC0tkzitikC0M. Wu et al. / Neurocomputing 190 (2016) 179187182L C10HI2NC10 eAtC0tk wtkC0INC10Mt; 29thenJ_wtJrJMJJ wtkJ JL C10H wtkJeJAJtC0tk JMJJtJ: 30It follows from (26) and (30) thatJ _tJrJAJJwtkJeJAJtC0tkJ_wtJrJMJJtJJAJJ wtkJ JMJJ wtkJ JL C10HwtkJeJAJtC0tk: 31Meanwhile,ddtJtJrJ _tJ, then one hasddtJtJrJAJJ wtkJeJAJtC0tk J_wtJrJMJJtJJAJJwtkJ JMJJwtkJJL C10H wtkJeJAJtC0tk: 32Solving this differential inequality for tAtk;tk1 with JtkJ 0.Thus, the corresponding solution of (32) is given byJtJ C20JwtkJ JL C10HwtkJC0eJAJtC0tkC0eJMJtC0tkC16C17:33where t is dened in (13). Substituting (21) and (22) into (19),one has_VtrC0J wtJ22JLC10QHJJwtJJtJ C0JwtJJwtJ C02JLC10QHJJtJ: 23Dene the triggering condition asJtJrC20aeJAJtC0tkJwtkJ; 24whereC20a is dened in Theorem 1. it is a new measurement errorsimilar to ei(t), according to (7), one hasitI2C10 eAtC0tkwitkC0wit; 25which can be written in compact formtI2NC10 eAtC0tkwtkC0wt: 26TheneJAJtC0tkJwtkJrJtJ JwtJ: 27It follows from (24) and (27) that JtJraJ wtJ, where a isdened in Theorem 1. Thus,_VtrC01JwtJ2, then_Vto0for A0;1, which can prove that N agents in (1) will reachconsensus under the control protocol (4) with the proposeduitFvit;_vitABFvitKCPjANiaijeAtC0tikvitikC0eAtC0tjk0tC16C17vjtjk0tC16C17! 8:C0 I2C10 eAtC0tjk0tC16C17!zjtjk0tC16C17!Dhit: 37Substituting (36) into (37), one has_zitMzitXjANiaijHeitzitC0ejtzjtDhit; 38which can be written in compact form_ztINC10 MztL C10 HetztINC10 Dht: 39Then,_wtINC10 MwtL C10 HwttL C10 Dht: 40If G is designed to make AGC Hurwitz, then hi(t) will approachzero asymptotically. From (17) and (40), one can nd that theestimation error h(t) is decoupled from the consensus dynamics w(t), thus, the stability of (40) is equivalent to the stability of theinstant.The measurement error for agent i is dened aseit I2C10 eAtC0tikC16C17zitikC0zit: 36It follows from (5) and (35) thatXiC16C17C16From (24) and (33), one has that an upper bound of the time forJtJ to evolve from 0 toC20aeJAJtC0tkJ wtkJ satisesC20J wtkJ JL C10H wtkJC0eJAJtC0tkC0eJMJtC0tkC16C17C20aeJAJtC0tkJ wtkJ:34Thus, an upper bound is given by tk1rtk1lnC181C20aJ wtkJC20 J wtkJ JLC10H wtkJ!, whereandC20 are dened in Theorem 1.Remark 2. Note that the inter-execution time interval is strictlypositive because 40. The next triggering instant tk1is calcu-lated according to the states of system wt at current event timetk, there is no need to monitor the state measurement error todetermine whether the control tasks should be executed.4. Distributed self-triggered controlCompared with the centralized self-triggered control mechan-ism in the previous section, the distributed self-triggered controlmechanism is proposed in this section. The state observer is givenby (3), the distributed self-triggered control protocol is designed aseAtC0tikxitikC0eAtC0tjk0tC16C17xjtjk0tC16C17!; tAtik;tik1:35then,_witM witHXjANi wititC0 wjtjtC16C17: 49Combining (46) with (49), one has_witM I2C10 eAtC0tikC16C17witikC0it HXjANiI2C10 eAtC0tikC16C17witikC16C0 I2C10 eAtC0tjk0tC16C17!wjtjk0tC16C17!M I2C10 eAtC0tikC16C17witikC0MitH I2C10 eAtC0tikC16C17XjANiwitikC0 I2C10 eAtikC0tjk0tC16C17!wjtjk0tC16C17;50M. Wu et al. / Neurocomputing 190 (2016) 179187 183following system:_wtINC10 MwtL C10 Hwtt: 41Now, we are ready to present the following result on the dis-tributed self-triggered control scheme based on output feedback.Theorem 2. Assume that the communication graph G is connected,pair (A,B) is controllable and pair (A,C) is observable. Given F and Gsatised that ABF and AGC are Hurwitz. If there exists matricesP40 and Q40, and scalar 40 such that

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