Robust consensus of Lur'e networks with uncertain communications
2017; Institution of Engineering and Technology; Volume: 11; Issue: 6 Linguagem: Inglês
10.1049/iet-cta.2016.1205
ISSN1751-8652
Autores Tópico(s)Distributed Sensor Networks and Detection Algorithms
ResumoIET Control Theory & ApplicationsVolume 11, Issue 6 p. 877-882 Brief PaperFree Access Robust consensus of Lur'e networks with uncertain communications Miao Liu, Miao Liu State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing, 100871 People's Republic of ChinaSearch for more papers by this authorZhongkui Li, Corresponding Author Zhongkui Li zhongkli@pku.edu.cn State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing, 100871 People's Republic of ChinaSearch for more papers by this author Miao Liu, Miao Liu State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing, 100871 People's Republic of ChinaSearch for more papers by this authorZhongkui Li, Corresponding Author Zhongkui Li zhongkli@pku.edu.cn State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing, 100871 People's Republic of ChinaSearch for more papers by this author First published: 03 February 2017 https://doi.org/10.1049/iet-cta.2016.1205Citations: 3AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract This study addresses the robust consensus problem of a network of Lur'e systems coordinating over uncertain communication channels. The authors model each communication channel of the network as an ideal transmission system with the unitary transfer function perturbed by a norm-bounded uncertainty, which is particularly susceptible to describe quantisation errors and communication noises. Based on the uncertain relative state information among neighbouring agents, a distributed consensus protocol is proposed and designed in terms of linear matrix inequalities to ensure robust consensus for both undirected connected graphs and leader-follower graphs containing a directed spanning tree. 1 Introduction Consensus or synchronisation of multi-agent systems has received compelling attention in the systems and control community over the last decade, due to its potential applications in broad areas such as spacecraft formation flying, multi-point surveillance, and sensor networks [1–3]. The essential issue of consensus problems dwells on designing appropriate distributed consensus protocols such that the agents can reach an agreement on certain variables of interest. The consensus problem has been extensively studied by many researchers from different perspectives; see [4–8] and the references therein. In most existing works on consensus, the agent dynamics are restricted to be linear or in some cases simple integrators, which might be restrictive. Consensus in networks with non-linear dynamics is more challenging to study [9, 10]. A typical non-linear system is the Lur'e system, which refers to such a system that consists of a linear dynamical system and a non-linear feedback loop satisfying certain sector condition [11]. Such dynamics model can represent many kinds of non-linearities such as saturation and dead zone [12]. Several chaotic systems, including the Chua's circuits, can be described in the Lur'e form. A few works have been published on consensus of Lur'e systems. The robust consensus problem for networked Lur'e systems is considered in [13]. The global synchronisation problem of a network of coupled Lur'e systems is studied from the perspective of global synchronised region in [14]. A class of consensus protocols are proposed for uncertain Lur'e networks in [15], where a sufficient condition is derived to guarantee robust tracking consensus. Notably, all the aforementioned works assume that in the network, information exchanges over the communication channels in a highly idealised manner, in which case the directed or undirected graph associated with the network is commonly represented by a known constant matrix. In practical applications, however, the communications are usually subject to channel constraints and transmission imperfections, such as data rate limit, quantisation, limited channel capacity, and packet dropout [16–18]. Several works, e.g. [19–21], have been reported to study the robust consensus problem by taking in account uncertain communication channels, where only linear or single-integrator agents are considered. How to solve the robust consensus problem of multi-agent systems with more complex dynamics in the presence of uncertain communications remains unclear and is an interesting work. This paper intends to study the robust consensus problems for Lur'e networks with uncertain communications. Both the cases with undirected and leader-follower communication topologies are considered. Compared with the existing related works, the main challenge here is to handle both the non-linear agent dynamics in the Lur'e form and uncertain communications. Motivated by the logarithmic quantisers in [22], we model each communication channel of the network as an ideal transmission system with the unitary transfer function subject to a norm-bounded uncertainty. Such a communication model is particularly susceptible to describe quantisation errors and communication noises. Based on such an uncertain communication model, a consensus protocol relying on uncertain relative state information is proposed. Sufficient conditions in terms of linear matrix inequalities are derived to design the consensus protocol. It is theoretically demonstrated that the consensus protocol ensures that the Lur'e systems can globally reach consensus for both undirected connected graphs and leader-follower graphs containing a directed spanning tree. Finally, a network with Chua's circuits is presented as a numerical example to illustrate the effectiveness of the theoretical results. Compared with the previous works [13–15] on consensus of Lur'e networks, where the information exchange among the agents is conducted over ideal communication channels, in the current paper we assume that the communication channels are perturbed by additive norm-bounded uncertainties. Such an uncertain communication model is more practical and can describe quantisation errors and communication noises. In contrast with the existing works [19–21, 23] on robust consensus problems over uncertain communication channels, where the agent dynamics are restricted to be linear and even single integrators, the current paper focuses on high-dimensional non-linear Lur'e networks. Due to the interaction of the uncertain communications and the non-linear agent dynamics, it is a highly non-trivial task to design the proposed consensus protocols. The rest of this paper is organised as follows. Some necessary mathematical preliminaries are summarised in Section 2 and the robust consensus problem is formulated in Section 3. The robust consensus problems for undirected graphs and leader-follower graphs are investigated in Section 4 and Section 5, respectively. Section 6 conducts a numerical simulation example. Finally, Section 7 concludes this paper. 2 Mathematical preliminaries 2.1 Notations denotes the set of real matrices. The superscript 'T' represents the transpose of a real matrix. denotes the identity matrix of dimension N and 1 represents a column vector with all entries equal to 1. denotes a diagonal matrix. represents the Kronecker product of matrices A and B and represents the 2-norm of a vector x. 2.2 Graph theory In this subsection, we summarise some relevant results on algebraic graph theory, most of which are standard and mainly adopted from [1, 3, 24]. A directed graph is a pair , where is the set of nodes and is the set of edges in which an edge is represented by a pair of two different nodes. A graph is said to be undirected provided that it satisfies as long as . A directed path from node to node is a sequence of ordered edges , . An undirected path is defined analogously. An undirected graph is connected if there exists a path between every pair of nodes. A directed graph contains a directed spanning tree if there exists a node called the root such that the node has directed paths to all other nodes in the graph. Suppose there are N nodes in the directed graph . The adjacency matrix is defined by , if and 0 otherwise. Positive denotes the weight of the edge (i,j). The Laplacian matrix associated with is defined as and . Lemma 1 [1].Zero is eigenvalue of with as a right eigenvector and all non-zero eigenvalues have positive real parts. Besides, zero is a simple eigenvalue of if and only if has a directed spanning tree. For an undirected graph , assuming that is arbitrarily oriented such that every edge has a head and a tail, the incidence matrix is defined as For an undirected graph , the Laplacian matrix of can be also presented by the incidence matrix . Lemma 2 [24].For a weighted undirected graph, , where is a diagonal matrix whose diagonal items are positive . 3 Problem formulation In this paper, we consider a network of N agents with Lur'e non-linear dynamics. The dynamics of the i th node are described by (1)where is the state vector of the i th agent, is the control input, is the output, and the non-linear function , which belongs to the slope with . Specifically, satisfies (2)In (1), are constant matrices with compatible dimensions and (A,B) is assumed to be stabilisable. The communication topology among the N agents is modeled by a graph , which merely depicts the information flow of the network. In practice, the information exchange among the agents is conducted over communication channels, which are subject to channel noises and communication constraints. Motivated by the logarithmic quantisers in [22], in the current paper we model each communication channel as an ideal transmission system with a unitary transfer function perturbed by an addictive uncertainty. We propose the following consensus protocol: (3)where denotes the feedback matrix, is the (i,j)th entry of the adjacency matrix of the graph , and denotes the uncertainty associated with communication channel between agent i and j. Note that the uncertain communication channel model in the current paper is quite general and can describe many possible transmission errors or information constraints, such as the quantisation errors and multiplicative transmission noises. We assume that the uncertainties are norm-bounded, satisfying the following assumption. Assumption 1.There exist such that , . The objective of this paper is to solve the robust consensus problem for the Lur'e systems in (1) in the presence of uncertain communication channels. Specifically, we will design the consensus protocol (3) to guarantee that the states of the N agents in (1) reach an agreement characterised by , 4 Robust consensus of Lur'e networks with uncertain communications and undirected graphs In this section, we investigate the robust consensus problem for the undirected graphs. Assume that the following assumption holds. Assumption 2.The graph of the N agents is undirected and connected. Due to the bidirectionality of the communication channels in the network, it is natural to suppose that two neighbouring agents exchange information through the same channel, i.e. if . Without loss of generality, assume that there exist l communication channels in the network. Let be the diagonal uncertainty matrix, defined by . Without causing confusions, we write as . Let and . Substituting (3) into (1) and using Lemma 2, we can obtain the closed-loop network dynamics as (4)where is the Laplacian matrix of and . Let , where . It is not difficult to get that 0 is a simple eigenvalue of M with as the eigenvector and 1 is the other eigenvalue with multiplicity . Evidently, if and only if . Therefore, we can refer to e as consensus error. By noting that and , the Lur'e network (4) can be rewritten in terms of e as (5)By letting and , it is easy to see that (6)where and we have used the fact that . Substituting (6) into (5) yields (7)where with , , and with , . Using (2), we can see that belongs to the sector , i.e. (8)Clearly, the consensus problem is solved if and only if the consensus error e in (7) is globally asymptotically stable. Before moving forward, we introduce the following two lemmas. Lemma 3 Schur Complement Lemma, [25].For any constant symmetric matrix the following statements are equivalent: Lemma 4.( -Procedure, [25])Let be symmetric matrices. Then for all satisfying , if there exist scalars such that . Now we are ready to present the main result of this section. Theorem 1.Supposing that Assumptions 1 and 2 hold, the consensus protocol (3) solves the robust consensus problem, if there exist matrices , , and diagonal such that (9)where , , and are the non-zero eigenvalues of . The feedback matrix K can be chosen as . Proof.Consider the Lyapunov function candidate (10)where is positive definite. The time derivative of along (5) can be obtained as (11)Multiplying (8) by , we obtain that (12)Using Lemma 4, can be ensured for all satisfying (8), if there exist scalars , , such that (13)Let , then (13) can be rewritten into (14)Let and . It is not difficult to obtain that (15)In light of (15), we can see that (14) holds, if (16)where . Under Assumption 1, it follows from Lemma 1 that zero is a simple eigenvalue of and all the other eigenvalues are positive. Besides, it is easy to confirm that and are the right and left eigenvectors of both and M corresponding to the zero eigenvalue. Then, we have . Let be such a unitary matrix that and , where are the non-zero eigenvalues of . Moreover, U can be denoted by where . Take the following linear transformations: where , , and . It follows that (16) holds if and only if (17)where . In light of the definitions of e and , it can be verified that . Then, (17) can be ensured, if (18)where , , + , and . Note that we have used the fact that to obtain (18). It is evident that all the matrices in (18) are block diagonal. Therefore, (18) holds, if (19)where . By letting and utilising Lemma 3, (19) holds if and only if (20)where . Since (20) is linear with respect to , we only need to check (20) for and . This gives rise to (9). Up to now, we have proved that if (9) holds, then for all non-zero satisfying (12) and all satisfying Assumption 1. This indicates that (5) is globally asymptotically stable, which implies that the consensus protocol (3) solves the robust consensus problem in presence of uncertain communication channels satisfying Assumption 1. □ Remark 1.Different from the previous works on consensus or synchronisation of Lur'e networks, e.g. [13–15], where neighbouring agents exchange information through ideal communication channels, in this paper we assume that the communication channels as ideal transmission systems perturbed by norm-bounded uncertainties. Such an uncertain communication model is more practical and can describe quantisation errors and transmission noises [22]. It is worth mentioning that the conclusion of Theorem 1 will reduce to the result in [14], when there exist no uncertainties in the communication channels, i.e. . Remark 2.Previous literatures [19, 20] also investigate robust consensus over uncertain communication channels, in which the agent dynamics are restricted to be linear or even single integrators. In contrast, the current paper addresses general high-order agent dynamics with Lur'e non-linearity. This difference in the agent dynamics renders the design of the consensus protocols more troublesome. Moreover, it is challenging to transform the robust consensus problem of high-dimensional Lur'e networks with uncertain communications into solving the LMIs in (9). 5 Robust consensus of Lur'e networks with uncertain communications and leader-follower graphs In this section, we investigate the case where in the network there exists a leader, which receives no information from any followers. Without loss of generality, we assume that the leader is labelled by 1 and the followers are labelled by . The leader's and the followers' dynamics are still described by (1). Furthermore, suppose that the input of the leader is zero, i.e. , and the followers' inputs are given by (3). The communication graph associated with the N agents is assumed to satisfy the following assumption. Assumption 3.The graph contains a directed spanning tree with the leader as the root and the subgraph associated with the followers is undirected. Since the leader has no neighbours, the Laplacian matrix associated with can be partitioned as where and . Under Assumption 3, it follows from Lemma 1 that has only one zero eigenvalue and all other eigenvalues are positive. Assume that there are q followers, which can be labelled by , having access to the leader through q edges in the graph . Suppose that there exist l edges in . Therefore, there are edges in the subgraph . Let denote the uncertainty matrix associated with the l communication channels. For the leader-follower graphs satisfying Assumption 3, we introduce a modified incidence matrix: where is the incidence matrix associated with and is composed of the first q columns of . After some calculations, we can verify that . Let as the leader-follower consensus error, where . Similarly as in the previous section, by letting with , and with , we can obtain the close-loop network dynamics in terms of as (21)where satisfies the sector condition . It is obvious that the robust leader-follower consensus problem is solved if asymptotically converges to zero. Theorem 2.Suppose that Assumptions 1 and 3 hold. The consensus protocol (3) solves the robust leader-follower consensus problem for leader-follower communication topologies, if there exist matrices , , and diagonal such that the LMIs in (9) hold. Proof.The proof is similar to that of Theorem 1 and thus is omitted here for brevity. □ 6 Numerical simulations In this section, we present a numerical example to verify the effectiveness of the theoretical results. Consider a network of six agents with Chua's circuits, described by (1), with being the states of the agents, (22)and , which belongs to the slope [14]. The communication graph of the network is depicted in Fig. 1, which obviously satisfies Assumption 2. Suppose that the communication channels are subjected to norm-bounded uncertainties satisfying Assumption 1. The largest and smallest non-zero eigenvalues of the Laplacian matrix corresponding to the communication graph are and . Select the parameters in (22) as , , , , [14]. Solving the LMIs (9) with gives a solution: Then, we can obtain from Theorem 1 that . For illustration, we choose the channels uncertainties to be , , , , , , , . In this case, the trajectories of the agents' states are depicted in Fig. 2. The consensus error e is shown in Fig. 3, which converges to zero. Obviously, the simulation results verify that consensus is indeed achieved under the consensus protocol (3). Fig. 1Open in figure viewerPowerPoint Communication graph Fig. 2Open in figure viewerPowerPoint Tajectories of the agents' states Fig. 3Open in figure viewerPowerPoint Trajectories of the consensus errors 7 Conclusions In this paper, we have studied the robust consensus problem of Lur'e networks with uncertain communications. We have proposed a distributed consensus protocol based on the uncertain relative state information among neighbouring agents. Sufficient conditions in terms of linear matrix inequalities have been derived to ensure that the Lur'e systems can reach robust consensus for both undirected connected graphs and leader-follower graphs containing a directed spanning tree. A simulation example has also been presented to validate the effectiveness of the theoretical results. 8 Acknowledgments This paper was supported in part by the National Natural Science Foundation of China under grant nos. 61473005, 11332001, and a Foundation for the Author of National Excellent Doctoral Dissertation of PR China. 9 References 1Ren W., and Beard R.W.: 'Information consensus in multivehicle cooperative control', IEEE Control Syst. Mag., 2007, 27, (2), pp. 71– 82 2Antonelli G.: 'Interconnected dynamic systems: an overview on distributed control', IEEE Control Syst. Mag., 2013, 33, (1), pp. 76– 88 3Li Z. Chen M.Z.Q., and Ding Z.: 'Distributed adaptive controllers for cooperative output regulation of heterogeneous agents over directed graphs', Automatica, 2016, 68, pp. 179– 118 4Wen G. Duan Z., and Chen G. et al.: 'Consensus tracking of multi-agent systems with Lipschitz-type node dynamics and switching topologies', IEEE Trans. Circuits Syst. I: Regul. Pap., 2014, 61, (2), pp. 499– 511 5Li Z., and Ding Z.: ''Distributed adaptive consensus and output tracking of unknown linear systems on directed graphs', Automatica, 2015, 55, pp. 12– 18 6Li Z. Duan Z., and Ren W. et al.: 'Containment control of linear multi-agent systems with multiple leaders of bounded inputs using distributed continuous controllers', Int. J. Robust Nonlinear Control, 2015, 25, (13), pp. 2101– 2121 7Meng Z. Dimarogonas D.V., and Johansson K.H.: 'Leader-follower coordinated tracking of multiple hetergeneous Lagrange systems using continuous control', IEEE Trans. Robot., 2014, 30, (3), pp. 739– 745 8Zhang H. Lewis F.L., and Qu Z.: 'Lyapunov, adaptive, and optimal design techniques for cooperative systems on directed communication graphs', IEEE Trans. Ind. Electron., 2012, 59, (7), pp. 3026– 3041 9Yang T. Meng Z., and Shi G. et al.: 'Network synchronization with nonlinear dynamics and switching interactions', IEEE Trans. Autom. Control, 2016, 61, (10), pp. 3103– 3108 10Wen G. Yu W., and Zhao Y. et al.: 'Pinning synchronisation in fixed and switching directed networks of Lorenz-type nodes', IET Control Theory Appl., 2013, 7, (10), pp. 1387– 1397 11Arcak M., and Kokotović P.: 'Feasibility conditions for circle criterion designs', Systems & Control Letters, 2001, 42, (5), pp. 405– 412 12Yang T. Meng Z., and Dimarogonas D.V. et al.: 'Global consensus for discrete-time multi-agent systems with input saturation constraints', Automatica, 2014, 50, (2), pp. 499– 506 13Zhang F. Trentelman H.L., and Scherpen J.M.A.: 'Fully distributed robust synchronization of networked Lur'e systems with incremental nonlinearities', Automatica, 2014, 50, (10), pp. 2515– 2526 14Li Z. Duan Z., and Chen G.: 'Global synchronised regions of linearly coupled Lur'e systems', Int. J. Control, 2011, 84, (2), pp. 216– 227 15Zhao Y. Duan Z., and Wen G. et al.: 'Robust consensus tracking of multi-agent systems with uncertain Lur'e-type non-linear dynamics', IET Control Theory Appl., 2013, 7, (9), pp. 1249– 1260 16Liu S. Li T., and Xie L.: 'Distributed consensus for multi-agent systems with communication delays and limited data rate', SIAM J. Control Optim., 2011, 49, (6), pp. 2239– 2262 17Nair G.N. Fagnani F., and Zampieri S. et al.: 'Feedback control under data rate constraints: An overview', Proc. IEEE, 2007, 95, (1), pp. 108– 137 18Hespanha J.P. Naghshtabrizi P., and Xu Y.: 'A survey of recent results in networked control systems', Proc. IEEE, 2007, 95, (1), pp. 138– 162 19Zelazo D., and Burger M.: 'On the robustness of uncertain consensus networks', IEEE Trans. Control Netw. Syst., 2015, in press, DOI: 10.1109/TCNS.2015.2485458 20Li T. Wu F., and Zhang J.F.: 'Multi-agent consensus with relative-state-dependent measurement noises', IEEE Trans. Autom. Control, 2014, 59, (9), pp. 2463– 2468 21Li Z., and Chen J.: 'Robust consensus of linear feedback protocols over uncertain network graphs', IEEE Trans. Autom. Control, 2017, in press 22Fu M., and Xie L.: 'The sector bound approach to quantized feedback control', IEEE Trans. Autom. Control, 2005, 50, (11), pp. 1698– 1711 23Wang J., and Elia N.: 'Distributed averaging under constraints on information exchange: emergence of lévy flights', IEEE Trans. Autom. Control, 2012, 57, (10), pp. 2435– 2449 24Mesbahi M., and Egerstedt M.: ' Graph theoretic methods in multiagent networks' ( Princeton University Press, 2010) 25Boyd S. Ghaoui L.E., and Feron E. et al.: ' Linear matrix inequalities in systems and control theory' ( SIAM, Philadelphia, PA, 1994) Citing Literature Volume11, Issue6April 2017Pages 877-882 FiguresReferencesRelatedInformation
Referência(s)