Guaranteed-Cost Consensus for Singular Multi-Agent Systems With Switching Topologies

Transcription

1 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL 61, NO 5, MAY Guaranteed-Cost Consensus for Singular Multi-Agent Systems With Switching Topologies Jianxiang Xi, Yao Yu, Guangbin Liu, Yisheng Zhong Abstract Guaranteed-cost consensus problems for high-order singular multi-agent systems with switching topologies are investigated Firstly, to achieve a trade-off design object between consensus regulation performances control energy consumptions, a quadratic cost function is constructed by state errors among agents control inputs of all agents guaranteed-cost consensus problems are introduced Then, based on linear matrix inequality techniques, sufficient conditions for guaranteed-cost consensus consensualization are presented respectively, which can guarantee the scalability of singular multi-agent systems since the dimensions of all the variables in these conditions are independent of the number of agents Moreover, an upper bound of the cost function is determined, explicit expressions of consensus functions are given on the basis of the Second Equivalent Form, it is shown that consensus functions are dependent on the average of initial states of all agents but are independent of switching topologies Finally, the applications of theoretical results in multi-agent supporting systems are shown Index Terms Consensualization, consensus function, guaranteed-cost consensus, linear matrix inequality, multi-agent system I INTRODUCTION I N the past few years, the study of multi-agent systems has attracted considerable attention from scientific communities This is partly due to potential applications of multi-agent systems in various fields, such as biology (eg, flocking [1] [3]), physics (eg, collective motions of self-propelled particles [4], [5]), system engineering (eg, synchronization of complex networks [6], [7], formation control [8] [10], attitude synchronization of multiple spacecrafts [11] congestion control of distributed sensor networks [12]), etc Information consensus is a basic problem of cooperation control of multi-agent systems, which means that multiple agents Manuscript received April 13, 2013; revised August 04, 2013, September 08, 2013; accepted September 27, 2013 Date of publication January 17, 2014; date of current version April 24, 2014 This work was supported in part by the National Natural Science Foundation of Chinaunder Grants , , , in part by the Shaanxi Province Natural Science Foundation Research Projectionunder Grants 2013JQ8038 This paper was recommended by Associate Editor M Frasca J Xi is with High-Tech Institute of Xi an, Xi an , China, also with the Department of Automation, TNlist, Tsinghua University, Beijing , China ( xijianxiang933@gmailcom) G Liu is with High-Tech Institute of Xi an, Xi an , China Y Yu (corresponding author) is with School of Automation Electrical Engineering, University of Science Technology Beijing, Beijing , China Y Zhong is with Department of Automation, TNlist, Tsinghua University, Beijing , China Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TCSI achieve an agreement on some common variables of interest, for example, the phase of multiple oscillators, the heading direction of self-propelled particles, the attitude of multiple spacecrafts, etc Due to the motions of agents the fault of communication equipments or circuits, interaction topologies among agents may be time-varying Moreover, information delays may appear in the process of information transmission because of the congestion of communication channels the variations of physical characteristics of transmission medium Consensus behaviors of self-propelled particles with time-varying topologies were revealed by numerical simulations in [4], but no any theoretical explanations were presented In [13] [14], based on graph theory, theoretical explanations for consensus behaviors shown in [4] were given, where it was assumed that interaction topologies are switching First-order multi-agent systems with time delays were dealt with in [15] [16] Consensus problems for first-order multi-agent systems with both switching topologies time delays were investigated in [17] [18] For the cases with external disturbances topology uncertainties, an consensus control approach was proposed in [19] In [20] [21], it was pointed out that the multi-agent system is a special kind of complex networks some very interesting novel results on synchronization of complex networks were given, where nonlinear protocols were proposed In [13] [21], the dynamics of each agent was modeled as a first-order integrator, but many practical multi-agent systems in the real world are of high order Xiao Wang [22] presented necessary sufficient conditions for consensus of high-order linear time-invariant (LTI) multi-agent systems with directed topologies, but did not deal with consensus design problems Both consensus analysis design problems for multi-agent systems with state feedback consensus protocols were investigated in [23] Static output feedback consensus protocols were proposed in [24] [25] Li et al [26] presented consensus consensualization criterions for multi-agent systems with dynamic output feedback consensus protocols, where protocol states of neighboring agents are required to construct consensus protocols By the low gain approach, Seo et al [27] constructed dynamic output feedback consensus protocols without using protocol state information In [28], output consensus for high-order LTI multi-agent systems was studied In [22] [28], it was assumed that interaction topologies among agents are fixed Wang et al [29] gave sufficient conditions for consensualization, where interaction topologies are undirected frequently connected In [30] [31], high-order linear multi-agent systems with switching undirected topologies IEEE Personal use is permitted, but republication/redistribution requires IEEE permission See for more information

2 1532 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL 61, NO 5, MAY 2014 were considered In [32], multi-agent systems with switching directed topologies were dealt with, where the dynamics of each agent has a special structure It should be mentioned that control energy consumptions were not considered in the above literatures However, in many practical multi-agent systems, consensus regulation performances control energy consumptions should be considered simultaneously For example, geese flocks often fly in a V-shaped formation to guarantee lower energy consumptions In the literature, there is not a general approach to realize a trade-off design objective between state regulation performances control energy consumptions Moreover, in aforementioned literatures, the dynamics of each agent was modeled as a normal system, but it should be pointed out that singular systems are more nature representations of dynamic systems Especially, each agent can only be modeled as a singular system rather than a normal system if there exist algebraic constraints among coordinated variables As shown in [33], the transistor circuit in Fig 1(a) can be equivalently described by the circuit in Fig 1(b) which can be modeled by a singular system as follows: where The singularity of the coefficient matrix reflects the fact that unless,therewillexistan impulse when the circuit is turned on at Moreover, multi-agent supporting systems with each block supported by several pillars three-link manipulator networks can only be modeled by singular multi-agent systems rather than normal ones, as shown in [34] [35] respectively In [34], admissible consensus analysis design problems were investigated in terms of linear matrix inequality (LMI) tools By the Riccati equation method, necessary sufficient conditions for admissible consensualization of singular multi-agent systems with static output feedback consensus protocols were given in [35] Xi et al [36] proposed singular dynamic output feedback consensus protocols In [34] [36], it was supposed that interaction topologies are fixed To the best of our knowledge, admissible consensus analysis design problems for singular multi-agent systems with switching topologies are still not comprehensively investigated The current paper investigates the impacts of switching topologies on consensus for singular multi-agent systems proposes a distributed guaranteed-cost consensus approach to realize a trade-off design between consensus regulation performances control energy consumptions, where a quadratic cost function is bounded by some constant Sufficient conditions for guaranteed-cost consensus are given which only include two LMI constraints independent of the number of agents, an approach is shown to determine an upper bound of the cost function It should be pointed out that the bound Fig 1 Simple circuit system of the guaranteed cost may be loose in some cases By the changing variable methods, LMI criterions for guaranteed-cost consensualization are presented Moreover, based on the Second Equivalent Form, an approach is proposed to determine consensus functions it is found that switching movements do not impact consensus functions Compared with the existing works about consensus, the current paper has the following three novel features Firstly, the current paper gives an approach to achieve a trade-off design between consensus regulation performances control energy consumptions Control energy consumptions were not considered in most literatures about consensus Secondly, the current paper investigates the influences of switching topologies on consensus for singular multi-agent systems In [34] [36], the assumption that interaction topologies are fixed was required, their approaches cannot be used to deal with the cases with switching topologies Thirdly, the current paper gives a general approach to determine the consensus function shows that switching movements do not impact the consensus function The state projection approach in [34] is no longer valid when switching topologies are considered no general approach was given to determine the consensus function in [35] This paper is organized as follows In Section II, basic concepts results on graph theory singular systems are presented respectively, the problem description is given In Section III, guaranteed-cost consensus consensualization conditions are proposed respectively in terms of LMIs, an approach is shown to determine the upper bound of the cost function, an explicit expression of the consensus function is given The theoretical results are used to deal with cooperative control problems of multi-agent supporting systems in Section IV Finally, concluding remarks are stated in Section V In the current paper, for simplicity of notation, 0 is applied to denote zero matrices of any size with zero vectors zero number as special cases denotes an -dimensional column vector with all components 1 represent -dimensional column vectors with the th component 1 0elsewhere

3 XI et al: GUARANTEED-COST CONSENSUS FOR SINGULAR MULTI-AGENT SYSTEMS 1533 II PRELIMINARIES AND PROBLEM DESCRIPTION In this section, basic concepts conclusions about graph theory singular systems are presented the problem description is given A Basic Concepts Results on Graph Theory Singular Systems An undirected graph consists of a vertex set, an edge set, a symmetric adjacency matrix with,where if only if Furthermore, it is supposed that The neighbor set of is denoted by The in-degree matrix of the undirected graph is defined as is said to be the Laplacian matrix of If there does not exist an isolated vertex in the undirected graph,then is said to be connected More details about graph theory can be found in [37] The basic properties of the Laplacian matrix are given as follows Lemma 1 ([37]): be the Laplacian matrix of an undirected graph,then i) at least has a zero eigenvalue, is the associated eigenvector; that is, ; ii) If is connected, then 0 is a simple eigenvalue of, all the other eigenvalues are positive The following basic concepts results on singular systems are also required Definition 1 ([38]): The pair is said to be regular if is not identically zero for some Definition 2 ([39]): The pair is said to be impulse-free if for Definition 3 ([34]): The pair is said to be admissible if it is regular, impulse-free asymptotically stable Lemma 2 ([40]): The pair is admissible if only if there exists a matrix such that Lemma 3 ([38]): The pair is regular impulse-free if only if the following equation holds B Problem Description Consider the following singular multi-agent system where with, are the state the control input respectively The interaction topology of a singular multi-agent system is described by an undirected graph, where each vertex represents an agent, the edge between (1) two vertices sts for the interaction channel between them, the weight of the edge denotes the interaction strength Consider a distributed consensus protocol as follows where is a gain matrix denotes the time-varying neighbor set of agent The current paper mainly focuses on the case with switching interaction topologies the finite set with an index set st for all possible interaction topologies, where is the set of natural numbers denotes a switching signal, whose value at time is the index of the interaction topology at time Itissupposed that Assumption 1: All interaction topologies in are connected; Assumption 2: The switching movements satisfy, then under consensus protocol (2), the dynamics of the singular multi-agent system can be rewritten as where is the Laplacian matrix of the interaction topology For given, consider a quadratic cost function as follows Associated with the cost function, the definitions of the guaranteed-cost consensus consensualization are respectively presented as follows Definition 4: System (3) is said to achieve guaranteed-cost consensus if for any given admissible initial state,itis regular impulse-free there exist a vector-valued function dependent on a positive scalar such that,where is called a consensus function is said to be a guaranteed cost Definition 5: System (1) is said to be guaranteed-cost consensualizable by protocol (2) if there exists a gain matrix such that it achieves guaranteed-cost consensus Remark 1: In (1), the pair represents the intrinsic dynamics of each agent Because may be singular, each agent may include three types of modes; that is, dynamic modes, static modes impulsive modes However, normal systems only include dynamic modes Hence, the methods for investigating consensus analysis design problems of normal multi-agent systems cannot be directly applied to deal with the ones for singular multi-agent systems Furthermore, consensus protocol (2) describes the attraction interaction among agents, which can propel the states of all agents to be identical Moreover, when is singular, there exist some algebraic constraints among states (2) (3) (4)

4 1534 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL 61, NO 5, MAY 2014 The admissible initial state means that satisfy those algebraic constraints Remark 2: In (4), the terms can be regarded as the evaluation indexes of the control energy consumptions dynamic performances of agent respectively, denote weights of the control energy consumption dynamic performance respectively In this sense, by choosing, a trade-off design object between control energy consumptions consensus regulation performances can be achieved Moreover, for guaranteed-cost control of linear isolated systems, quadratic cost functions constructed by states control inputs can be applied to achieve a trade-off design object between state regulation performances control energy consumptions in the literature (eg, [47], [48]) However, the approaches in [47] [48] cannot be used to investigate guaranteed-cost consensus for multi-agent systems since states of multi-agent systems may diverge even if consensus is achieved The current paper will investigate guaranteed-cost consensus analysis design problems; that is, under what conditions system (3) achieves guaranteed-cost consensus how to determine the gain matrix such that system (3) achieves guaranteed-cost consensus Moreover, the approaches are presented to determine the consensus function the guaranteed cost respectively III MAIN RESULTS This section gives LMI conditions for guaranteed-cost consensus consensualization respectively, determines an upper bound of the cost function presents an explicit expression of the consensus function on the basis of the Second Equivalent Form From Lemma 1 Assumption 1, there exists an orthogonal matrix with the first column such that where is a symmetric matrix where,thenby(5)(6), system (3) can be transformed into (5) (6) (7) (8) Actually, subsystem (7) (8) describe the consensus dynamics disagreement dynamics of system (3) respectively denote the eigenvalues of the Laplacian matrix of the th interaction topology in, then one has from Lemma 1 Assumption 1 The following theorem presents LMI criterions for guaranteed-cost consensus shows an approach to determine an upper bound of the cost function Theorem 1: System (3) achieves guaranteed-cost consensus if there exists page) where where such that (see equation at the bottom of the In this case, the guaranteed cost satisfies Proof: be linearly independent -dimensional column vectors, then there exist such that then it can be shown by (9) (10) that (9) (10) (11) (12) (13) (14)

5 XI et al: GUARANTEED-COST CONSENSUS FOR SINGULAR MULTI-AGENT SYSTEMS 1535 Due to (13) (14) that, one can show by In the following, we discuss the problem of determining the guaranteed cost One can show that (15) Since is nonsingular, from (13) (14), are linearly independent Hence, by (11) (15), system (3) achieves consensus if only if subsystem (8) is asymptotically stable; that is, Consider the following Lyapunov function cidate (16) Due to, by taking the derivative of with respect to time along the solution of subsystem (8), it can be obtained by Assumption 2 that (17) Moreover, it can be obtained by (18) (19) that (21) (22) (23) denote the eigenvalues of, then one can see by (5) that the eigenvalues of are Since is symmetric, there exists an orthogonal matrix such that Due to, from (21) to (24), one has (24) (18) (19) then it can be shown by (17) (18) that (25) where For the case that,byschur complement in [49] the convex property of LMIs, one can show from (25) that The LMI has the convex property, so the inequalities (26) (20) can guarantee that then due to, one can see that which means that if only if In this case, one has by (19) Since is nonsingular, it can be shown that if inequalities (20) hold (27) (28)

6 1536 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL 61, NO 5, MAY 2014 Since (28), one has, by (16), (27) of simulations, it is shown that LMI conditions in Theorem 1 are usually feasible From the proof of Theorem 1, the following corollary can be obtained directly Corollary 1: System (3) achieves consensus if (29) From (26) (29), let, then one has there exists such that The inequalities can guarantee that subsystem (8) is regular impulse-free by Lemma 2, the equation can ensure that subsystem (7) is regular impulse-free by Lemma 3 Hence, one can see that system (3) is regular impulse-free since nonsingular transformations do not change the regularity impulsive properties is nonsingular By the above analysis, the conclusion of Theorem 1 can be obtained Remark 3: The condition By the changing variable method, the following theorem presents sufficient conditions for guaranteed-cost consensualization Theorem 2: System (1) is guaranteed-cost consensualizable by protocol (2) if there exist such that (see equation at the bottom of the page), where In this case, Proof: If, then one has is applied to guarantee that system (3) is regular impulsefree By Schur complement, are equivalent to It can be found that is invertible,thenpre post-multiplying by respectively, one can obtain that It is obvious that are necessary for Actually, can ensure that system (3) achieves consensus the terms are used to guarantee the required performances shown in (4) Moreover, are feasible if only if is stabilizable, which means that it is necessary for system (3) to achieve consensus that is stabilizable there always exists such that are feasible Furthermore, for a required trade-off design object between consensus regulation performances energy consumptions, which can be determined by choosing,byanumber Pre- post-multiplying by respectively setting, it can be obtained that The proof of Theorem 2 is completed From Corollary 1, by a similar analysis to Theorem 2, the following corollary can be obtained, which presents sufficient conditions for consensualization Corollary 2: System (1) is consensualizable by protocol (2) if

7 XI et al: GUARANTEED-COST CONSENSUS FOR SINGULAR MULTI-AGENT SYSTEMS 1537 there exist such that In this case, The following algorithm shows a procedure to determine the gain matrix Algorithm: Step 1) Choose positive symmetric matrices to obtain the required trade-off objective between consensus regulation performances energy consumptions Step 2) Check the condition LMI tools in the literature (eg, [17], [18]), where the dimensions of all the variables of LMI consensus criterions are dependent on the number of agents In this case, it is time-cost memory-cost to check those criterions if multi-agent systems consist of numerous agents It is not difficult to find that LMI conditions in Theorems 1 2 have lower calculation complexity Moreover, the Riccati equation was applied to deal with consensualization problems of high-order linear normal multi-agent systems with switching topologies in [30] [31], where consensualization criterions have lower calculation complexity, but their approaches are no longer valid when the guaranteed cost is considered When system (3) achieves guaranteed-cost consensus, the states of all agents will tend to be identical The consensus functionisoftenusedtodescribetheidentical state in the literature be invertible matrices such that (30) If the condition does not hold, then system (1) is not guaranteed-cost consensualizable by protocol (2) Stop Step 3) Check Step 4) Set to obtain The LMI conditions in the above conclusions are dependent on the second small maximum eigenvalues of Laplacian matrices of interaction topologies in the switching set When is huge, the eigenvalues of the Laplacian matrix of each interaction topology may be difficult to be obtained Actually, many approaches were proposed to estimate the second small maximum eigenvalues in the literature For example, the approaches to estimate the second small eigenvalue were presented in [41] [42] the Gersgorin disc theorem in [43] can be used to estimate the maximum eigenvalue Hence, it is not necessary to determine the precise values of Remark 4: By the above analysis, the consensus properties of system (3) are jointly determined by consensus protocols, the dynamics of each agent interaction topologies Furthermore, system (3) can be regarded as a high-dimensional switching system with a specific structure; that is, interaction topologies are switching but the dynamics of each agent is time-invariant, consensus problems of system (3) are converted into admissible problems of reduced-order switching singular subsystems It should be pointed out that the approaches in the literature about switching isolated systems (eg, [44] [46]) cannot be directly used to deal with consensus problems of singular multi-agent systems with switching topologies due to this specific structure By the structure property of Laplacian matrices of interaction topologies, we give LMI conditions for consensus consensualization of system (3) Remark 5: Consensus problems for multi-agent systems with switching topologies were extensively investigated by which is called the Second Equivalent Form of the pair as shown in [38] The following theorem gives an approach to determine the consensus function of system (3) Definition 3: If system (3) achieves guaranteed-cost consensus, then consensus function satisfies Proof: By (6), one can see that Since the first column of is, it can be shown that Due to one can obtain by (31) (32) that From the proof of Theorem 1, by (11) (15), one has (31) (32) (33) (34) which means that subsystem (7) determines the consensus function, then it can be shown by (7) (30) that (35)

8 1538 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL 61, NO 5, MAY 2014 Due to is nonsingular Hence, it can be ob- it can be shown that tained by (35) that (36) From (33), (34) (36), the conclusion of Theorem 3 can be obtained By Theorem 3, since consensus functions are dependent on the dynamics of each agent the initial states but are independent of the variances of interaction topologies, multi-agent systems with the same initial states but different switching movements have the same consensus function Moreover, the Second Equivalent Form always exists can be found by some nominal procedures as shown in [38], so the expression in Theorem 3 is easily obtained Remark 6: Intuitional speaking, the consensus function of a multi-agent system describes its macroscopic property as a whole Olfati-Saber Murray [14] proposed the -consensus problem for determining the consensus functions of first-order multi-agent systems showed that consensus functions are the average value of the initial states of all agents if interaction topologies are undirected,thenit is not difficult to find that the conclusions in [14] are special cases of ours Xiao Wang [22] gave explicit expressions of consensus functions of high-order LTI multi-agent systems with switching topologies, where it was assumed that consensus functions are time-invariant In [23], an initial state projection approach was presented to deal with the case with time-varying consensus functions It should be pointed out that each agent was modeled as a normal system in [22] [23], their methods are no longer valid for singular cases Based on the Second Equivalent Form, Theorem 3 gives a general approach to determine consensus functions of singular multi-agent systems with switching topologies Remark 7: In [33], it was shown that some circuits can only be modeled as singular systems rather than normal ones Synchronization problems of networks consisting of those circuits can be investigated by our approaches Moreover, multi-agent systems with many three-link manipulators can be applied to clean the facade of a large building as shown in [35] Since there are specific constraints on the motion of each manipulator, the dynamics of each agent is singular In this case, the theoretical results in the current paper can also be used IV APPLICATIONS IN MULTI-AGENT SUPPORTING SYSTEMS (MASS) In [50], the model for the MASS with each agent supporting by one pillar was introduced it was shown that the MASS has potential applications in earthquake damage-preventing buildings, water-floating plants large-diameter parabolic antennae or telescopes In [34], it was shown that each agent in a MASS can only be modeled by a singular system when this MASS consists of many independent blocks each Fig 2 Model of each agent in a MASS [34] block is supported by several pillars For the case that each agent in a MASS is supported by two pillars called Unit I Unit II respectively, as shown in Fig 2, where the mass, the damping coefficient the stiffness coefficient, let denote the heights velocities of Unit I Unit II respectively, then agent can be described by where Example 1 (Scalability): Consider a MASS consisting of agents, where the parameters of each agent are chosen as Fig 3 shows an undirected interaction topology with the edges labeled from 1 to, where the weight of edge is Since the topology has a link structure, it is not difficult to determine the eigenvalues of its Lapalcian matrix Consider the following two cases: Case 1: LMIs in Theorem 1 with ; Case 2: LMIs in Theorem 1 with Under the same experiment conditions, for different,the calculation time to check the feasibility of LMIs is shown in Table I It can be found that the method in Theorem 1 significantly improves the calculation efficiency for a large Example 2 (Water-Floating Plant): All agents in a waterfloating plant try to keep the plant horizontal The plant will

9 XI et al: GUARANTEED-COST CONSENSUS FOR SINGULAR MULTI-AGENT SYSTEMS 1539 Fig 3 Undirected interaction topology TABLE I CALCULATION TIME TO CHECK FEASIBILITY keep horizontal if all agents achieve consensus on the heights velocities Since the water disturbs the plant all the time, the consensus function oscillates continuously Fig 4 Switching topology set Thus, one can obtain that where is a self-feedback matrix to adjust the dynamics of each agent The parameters of each agent are chosen as, then the dynamic modes of each agents are placed into at with Fig4 gives four undirected interaction topologies in the switching set, it is assumed that the weights of edges of all interaction topologies are 1 without loss of generality The interaction topologies of the water-floating plant are romly chosen from with s then one can see that One can choose that,where is symmetric positive In this case, it can be found that The state trajectories of the water-floating plant with the switching signal shown in Fig 6 are given in Fig 5, where the trajectories marked by circles describe the curves of the consensus function in Theorem 3, the trajectory of the cost function is depicted in Fig 7 One can see that the water-floating plant keeps horizontal state trajectories converge to the curves of the consensus function Furthermore, by repeating this simulation with different switching signals, it is found that state trajectories always converge to the same curves This shows that switching movements do not impact the consensus function Moreover, For the case that,due to, by (17), (23), (24) (26), it can be obtained that From Theorem 2, by replacing by, it can be obtained that which is coincident with Fig 7 V CONCLUSION Guaranteed-cost consensus analysis design problems for high-order linear singular multi-agent systems with switching topologies were dealt with Linear matrix inequality conditions for guaranteed-cost consensus consensualization were presented respectively sufficient conditions for consensus consensualization were also given respectively These conditions are independent of the number of agents, so they can guarantee the scalability of singular multi-agent systems Moreover,

10 1540 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL 61, NO 5, MAY 2014 Fig 6 Switching signal Fig 7 Trajectory of the cost function Since all the eigenvalues of the Laplacian matrices of undirected topologies are real, the convex property can be used to decrease calculation complexity, However, for multi-agent systems with switching directed topologies nonlinear consensus protocols as discussed in [6] [51], the convex condition does not hold Further work should be done on guaranteed-cost consensus analysis design problems for singular multi-agent systems with nonlinear consensus protocols directed topologies Moreover, optimal control approaches may be used to deal with the cases that the bound on the guaranteed cost is specified a priori by system designers Fig 5 State trajectories it was shown that an upper bound of the cost function is dependent on initial states of all agents consensus functions are dependent on the average of initial states of all agents but are independent of switching movements REFERENCES [1] H G Tanner, A Jadbabaie, G J Pappas, Stable flocking of mobile agents, part I: Fixed topology, in Proc IEEE Conf Decision Control, Dec 2003, pp [2] H G Tanner, A Jadbabaie, G J Pappas, Stable flocking of mobile agents, part II: Dynamic topology, in ProcIEEEConfDecision Control, Dec 2003, pp [3] R Olfati-Saber, Flocking for multi-agent dynamic systems: Algorithms theory, IEEE Trans Automat Contr, vol 51, no 3, pp , Mar 2006

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(MASS): Control with centralized estimator of disturbance, in Proc IEEE/RSJ Int Conf Intelligent Robots Syst, June 1994, pp [51] Y Chen, J Lü, Z Lin, Consensus of discrete-time multi-agent systems with transmission nonlinearly, Automatica, vol 49, no 2, pp , Feb 2013 Jianxiang Xi was born in 1982 He received the BS MS degree from the High-Tech Institute of Xi an, Xi an, China, in , respectively He received the PhD degree in control science engendering from High-Tech Institute of Xi an, Xi an, China in 2012 by a coalition form with Tsinghua University He is currently a Lecturer in Control Engendering Department of High-Tech Institute of Xi an His research interests include complex systems control, switched systems swarm systems

12 1542 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL 61, NO 5, MAY 2014 multi-agent systems Yao Yu received the BS degree from Department of Control Science Engineering, Huazhong University of Science Technology, China in 2004, the MS PhD degrees from Department of Automation, Tsinghua University, China in 2010 She was a Postdoctoral Researcher with Tsinghua University Currently, she is a lecturer with the School of Automation Electrical Engineering, University of Science Technology Beijing, Beijing, China Her research interests include nonlinear control, robust control, time-delay systems Guangbin Liu was born in 1963 He received the BS degree from the High-Tech Institute of Xi an, Xi an, China, in 1984 the MS PhD degrees from Xi an Jiaotong University, Xi an, China, in , respectively He is currently a Professor in Control Engendering Department of High-Tech Institute of Xi an His research interests include GNSS signal processing, complex systems control swarm systems Yisheng Zhong received the BE degree in control engineering from Harbin Institute of Technology, Harbin, China, in 1982, the ME degree in electronic engineering from the University of Electro-Communications, Tokyo, Japan, in 1985, the PhD degree in electrical engineering from Hokkaido University, Sapporo, Japan, in 1988 He was a Postdoctorate Scholar at Tsinghua University, Beijing, China, from 1989 to 1990, since 1991, he has been with the Department of Automation, Tsinghua University, where he is currently a Professor also with the Tsinghua National Laboratory for Information Science Technology His research interests include robust control, nonlinear control, electromechanical system control