summary:Recent years have witnessed an increasing interest in coordinated control of distributed dynamic systems. In order to steer a distributed dynamic system to a desired state, it often becomes necessary to have a prior control over the graph which represents the coupling among interacting agents. In this paper, a simple but compelling model of distributed dynamical systems operating over a dynamic graph is considered. The structure of the graph is assumed to be relied on the underling system's states. Then by following a proper protocol, the state-dependent dynamic graph is driven to a pre-specified structure. The main results are derived via Lasalle's Invariant Principle and numerical examples that find very good agreements with the analytical results are also included

Chen, Fei

Chen, Zengqiang

Liu, Zhongxin

Yuan, Zhuzhi

Czech digital mathematics library

Control a state-dependent dynamic graph to a pre-specified structure

Institute of Mathematics AS CR, v. v. i.

KybernetikaFei Chen; Zengqiang Chen; Zhongxin Liu; Zhuzhi YuanControl a state-dependent dynamic graph to a pre-specified structureKybernetika, Vol. 45 (2009), No. 5, 801--808Persistent URL: http://dml.cz/dmlcz/140044Terms of use:© Institute of Information Theory and Automation AS CR, 2009Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitizeddocuments strictly for personal use. Each copy of any part of this document must contain theseTerms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czKYBERNET IK A — VOLUME 4 5 ( 2 0 0 9 ) , N U MBE R 5 , P AG E S 8 0 1 – 8 0 8CONTROL A STATE–DEPENDENT DYNAMIC GRAPHTO A PRE–SPECIFIED STRUCTUREFei Chen, Zengqiang Chen, Zhongxin Liu and Zhuzhi YuanRecent years have witnessed an increasing interest in coordinated control of distributeddynamic systems. In order to steer a distributed dynamic system to a desired state, it oftenbecomes necessary to have a prior control over the graph which represents the couplingamong interacting agents. In this paper, a simple but compelling model of distributeddynamical systems operating over a dynamic graph is considered. The structure of thegraph is assumed to be relied on the underling system’s states. Then by following a properprotocol, the state-dependent dynamic graph is driven to a pre-specified structure. Themain results are derived via Lasalle’s Invariant Principle and numerical examples that findvery good agreements with the analytical results are also included.Keywords: state-dependent graph, Lasalle’s Invariant Principle, dynamic systemAMS Subject Classification: 37F991. INTRODUCTIONIn recent years, the coordination of distributed dynamic systems has attracted theattention of researchers from system theory [4, 5, 6, 8], biology [1, 3] and physics[9]. Particularly, it has witnessed an increasing interest in the interplay betweeninformation flows and system dynamics [8]. It is recognized that communicationconstraints may have a considerable impact on the performance of a distributedsystem.In [9], Vicsek et al. proposed a simple but compelling model of n autonomousagents moving in a plane at the same speed but with different headings. Theydemonstrated that all agents could move in the same direction eventually. ThenJadbabaie, Lin, and Morse gave a theoretical explanation for this observed behaviorand derived sufficient conditions under which the system can reach a consensus onheadings [5]. This result was extended by Moreau who provided the necessary andsufficient condition for the convergence of individual agents’ states to a commonvalue [8]. The consensus problem over random networks was considered by Hatanoand Mesbahi via notions from stochastic stability [4]. In addition, the relationbetween the rate of convergence and the algebraic connectivity of random graphswas established. In [6], Lu and Chen proposed a general method for synchronizing802 FEI CHEN, ZENGQIANG CHEN, ZHONGXIN LIU AND ZHUZHI YUANtwo chaotic systems based upon Kalman filtering and provided sufficient conditionsfor a driven system to track the states of a drive system asymptotically. Moreover,the consensus problems have received a lot of attention from biology researchers.Franks et al. demonstrated a speed versus accuracy trade-off in consensus decisionmaking. They showed that house-hunting ant colonies choose a new nest morequickly in harsh conditions than in benign ones and are less discriminating [3]. Athrough review of the empirical and theoretical studies of consensus decision makingcan be found in [1]. For these systems, in order to steer the systems to a desiredstate or an objective, it often becomes necessary to have a prior control over thedynamic graph which represents the interaction among agents [7, 10].The problem considered henceforth is obtained by tying the network structure tothe dynamic states residing at nodes. Specifically, we study a scenario where theexistence of an information channel between a pair of agents is determined by theirstates, i. e., if ||xi − xj || ≤ ri, agent i can access the information of agent j. Here,xi denotes the state of agent i and ri represents the sensing radius of agent i. Thesymbol || · || is the Euclidean norm. The network structure has a blend of dynamicand combinatorial features and is called a state-dependent dynamic graph. Thenthe objective is to steer the dynamic graph to a pre-specified structure by designinga proper control law.In fact, we are by no means the first one who have noticed the significance ofcontrolling graphs. In [7], Meshbahi et al. considered graphs with incidence relationsthat are dictated by underlying dynamic states. Then the relation between thecontrollability of a distributed dynamic system and the corresponding graph wasderived, pointing to a new research direction in system and control theory. Later,in [10], the problem of preserving the k-hop connectivity was considered. The mainidea was to model connectivity as an invariance problem and transform it into aset of constraints on the control variables. Then the control law for a connectivityproblem was obtained by minimizing a cost function.The rest of this paper is organized as follows. In Section 2, we develop a generalframework for our problem. The mathematical derivations of main results are pre-sented in Section 3. Some numerical examples, including both one dimension andtwo dimensions, are presented in Section 4. Finally, Section 5 summarizes the mainconclusions.2. PRELIMINARIES AND PROBLEM SETUPLet N, R and C denote the sets of all natural numbers, all real numbers and allcomplex numbers respectively. We denote by In the n × n real identity matrix andthe subscript variable n is omitted when no misunderstanding arises.Graphs are a good choice to represent the relationships among group members(agents). Each agent is denoted by a node, the relation between two nodes is de-scribed by an arc1. It has been found that the dynamic behavior of a group ofagents is closely related to the properties of the graph that represented the relationsamong agents [5, 8]. Next we will first survey some basic notions from graph theory.1When the relation is bidirectional, it is described by an edge.Control a State-Dependent Dynamic Graph to a Pre-Specified Structure 803Then the definition of state-dependent graphs is given. Especially, we will focus ondirected graphs.A directed graph G consists of a non-empty finite set V of elements called nodesand a finite set E of ordered pairs of nodes called arcs. We call V the node set and Ethe arc set of G. We write G = (V, E) to indicate that V and E are the node set andthe arc set of G respectively. For a node i in a directed graph G = (V, E), the set ofits neighbors is defined as {j|j ∈ V and (j, i) ∈ E}. Figure 1 shows an example of adirected graph.Fig. 1. An example of a directed graph. The node set is V = {1, 2, 3, 4}and the arc set is E = {(1, 2), (2, 3), (3, 4), (4, 1)}.A state-dependent graph is a mapping Gs from the distributed system state spaceX to the set of all labeled graphs of order N , GN . Here N is the number of agentsin the system.Gs : X → GN . (1)It is assumed that the node set V of these graphs is fixed; their edge set E(x(t))however is a function of the system state x(t). Especially, in current paper, E(x(t))is specified as:(j, i) ∈ E iff ||xi(t) − xj(t)|| ≤ ri, (2)where xi(t) is the state of agent i at time t and ri denotes the sensing radius ofagent i. Then the structure of a state-dependent graph is totally determined bythe state of the distributed system. For further analysis, we define the state matrixA = (ai,j) as follows: ai,j = xi −xj where ai,j denotes the (i, j)th element of matrixA and xi is the state of agent i.Suppose we want to steer the dynamic state-dependent graph to a pre-specifiedstructure Ḡ = {V̄, Ē} where V̄ = V and Ē is pre-specified. For any such structure,we can construct a matrix Ā = (āi,j), such that āi,j = x̄i − x̄j and (i, j) ∈ Ēiff ||āi,j || ≤ ri. Here x̄i denotes the expected state of agent i. Then the state-dependent graph Gs(x(t)) converges to the pre-specified topology Ḡ iff A convergesto the matrix Ā. Here Gs is defined by (1).To close this section, we formally define the notion of the convergence of dynamicstate-dependent graphs.Definition 1. Convergence of dynamic state-dependent graphs. Considera distributed dynamic system which consists of n agents and let Ḡ be the prescribedtopology. We say the dynamic state-dependent graph, defined by (2), converges toḠ, iff limt→∞ Gs(x(t)) = Ḡ.804 FEI CHEN, ZENGQIANG CHEN, ZHONGXIN LIU AND ZHUZHI YUAN3. MATHEMATICAL ANALYSISIn this section, we aim at giving a protocol to ensure the convergence of a state-dependent graph to a pre-defined topology Ḡ.The protocol is defined as follows:ẋi = −∑j 6=i(xi − xj − āi,j). (3)For the sake of brevity, we defineri,j = xi − xj − āi,j , (4)then system (3) is rewritten asẋi = −∑j 6=iri,j . (5)For further analysis, we define a matrix R = (ri,j), where ri,j is defined by Eq.(4).Here comes a lemma about ri,j .Lemma 1. ∀ i, j, k ∈ {1, 2, · · · , n}, ri,j = ri,k + rk,j , where n is the number ofagents in the system.P r o o f . The derivation of Lemma 1 is straightforward by taking the definitionsof ri,j and āi,j into the lemma. ¤Next we will give a lemma which will be used in the stability analysis of system (3).Lemma 2. Consider matrix R, ∀ i ∈ {1, 2, · · · , n}, ∑j ri,j = 0 if and only if∀ i, j ∈ {1, 2, · · · , n}, ri,j = 0.P r o o f . Necessity (Proof by contradiction): Without loss of generality, assumethat there is a pair (i, j) satisfying ri,j < 0. Since∑j ri,j = 0, the sum of the entries,except for ri,j , in the ith row∑k 6=j ri,k > 0. By Lemma 1, we have(n − 1)ri,j =∑k 6=j(ri,k + rk,j) =∑k 6=jri,k +∑k 6=jrk,j .Then rk,j = −rj,k, rj,j = 0, and∑k rj,k = 0 yield∑k 6=jrk,j = 0.Moreover,∑k 6=j ri,k > 0 indicates that ri,j > 0. Now a contradiction arises.Sufficiency: The proof of sufficiency is rather straightforward, hence it is omittedhere. ¤Next, we will show that even by designing a semi-definite positive Lyapunovfunction, it is possible to use a blend of Lasalle’s Invariant Principle and Lemma 2to establish the stability results.The main result of the current note is given below.Control a State-Dependent Dynamic Graph to a Pre-Specified Structure 805Theorem 1. For system (3), the dynamic state-dependent graph will converge toa pre-specified structure Ḡ.P r o o f . Consider the following functionV ,∑i,j,i6=jVi,j ,with each Vi,j defined byVi,j ,14||xi − xj − āi,j ||2. (6)Thus∂V∂xi=∑j 6=i12(xi − xj − āi,j) +∑j 6=i−12(xj − xi − āj,i). (7)Since āi,j = −āj,i,−(xj − xi − āj,i) = xi − xj − āi,j . (8)Then we have∂V∂xi=∑j 6=i(xi − xj − āi,j). (9)Sinceẋi = −∑j 6=i(xi − xj − āi,j),we haveV̇ =∑i(∂V∂xi)T ẋi = −∑i∥∥∥∑j 6=iri,j∥∥∥2≤ 0. (10)According to Lasalle’s Invariant Principle, system (3) will converge to the followinginvariant set{x| V̇ (x) = 0}, (11)which implies that ∀ i ∑j 6=i ri,j = 0. Thus according to Lemma 2, ∀ i, j, ri,j = 0,which indicates that the pre-defined topology is reached asymptotically. ¤Note that the statement V̇ = 0 is equal to V = 0. Hence, in the subsequentsimulations, we will draw the curve of V instead of V̇ .Corollary 1. For system (3), if initially the state-dependent graph achieves thedesired topology Ḡ, in any subsequent time the desired topology is always kept.P r o o f . If the initial state achieves the desired topology, we have V = 0. By(10), we have V̇ = 0, which implies that the desired topology is always kept. ¤806 FEI CHEN, ZENGQIANG CHEN, ZHONGXIN LIU AND ZHUZHI YUANRemark 1. The significance of Corollary 1 is due to the fact that in many cases weneed the state-dependent graphs to achieve some prescribed topologies, for instanceconnected, at all times other than asymptotically. In these cases, Corollary 1 willmeet our requirements.The main results in this section can find applications, for instance, in the ren-dezvous problem. Consider a group of unmanned vehicles which aims at reachinga rendezvous point. The information flows among vehicles can be described by astate-dependent graph, called communication graph [2]. To get to the rendezvouspoint, a sufficient condition is to guarantee connectivity maintenance. According toCorollary 1, if the communication graph is initially connected, it remains connectedthroughout the system evolution under the proposed protocol, which indicates thatrendezvous is reached.4. NUMERICAL SIMULATIONSIn this section, we verify the stability results obtained in Section 3 by numericalexamples. In our simulations, both one dimension and two dimensions cases areconsidered.There are three nodes, each with one dimension, in the first example. The stateequations of the nodes are described by:ẋi = −∑j 6=i(xi − xj − āi,j) (12)for i = 1, 2, 3, where āi,j is the (i, j)th entry of a pre-defined matrix Ā which isdefined as follows:Ā =0 −1 11 0 2−1 −2 0 .In this case, function V defined in the proof of Theorem 1 isV =∑i∈{1,2,3},j∈{1,2,3}, i 6=jVi,j , (13)where Vi,j is defined by (6). Figure 2 illustrates the curve of the function V . It isclear that as time goes by, V (t) asymptotically reduces to 0, which indicates thatthe pre-defined topology is reached asymptotically.In the second example, the pre-defined topology is defined to be the name ofour university which consists of two characters in Chinese. In this example, thereare 51 nodes, 30 nodes for the first character and 21 nodes for the second charac-ter. For each node, its state is represented by a two-dimensional vector, depictingthe x coordinate and y coordinate of the node moving in the plane. The step-size(a parameter specified in Runge Kutta method in approximating solutions to dif-ferential equations) in the example is 0.0005 seconds. Figure 3 shows consecutivesnapshots of the proximity structure for 51 agents in free-space using Protocol (3).Control a State-Dependent Dynamic Graph to a Pre-Specified Structure 8070 2 4 6 8 10 12 14 16 18 20050100150200250300350400t(sec)VFig. 2. The plot of function V vs time t.(a) (b)(c) (d)Fig. 3. The example of forming the Chinese name of our university.The initial positions, which is highly skew from the desired topology and illustratedby Figure 3 (a), are chosen randomly. And we note that the pre-specified topologyis reached in Figure 3 (d) and maintained thereafter.5. CONCLUSIONMotivated by a class of problems associated with control of distributed dynamicsystems, we considered graphs with incidence relations that are dictated by the un-derlying dynamic states, state-dependent graphs. In particular, we considered theproblem of controlling the structure of dynamic graphs so that the resulting mo-tion always make the structure of dynamic state-dependents graphs converge to apre-defined topology. The potential application of the main results includes: flocksof mobile agents equipped with sensing and communication devices, rendezvous inspace, distributed sensor fusion in sensor networks, synchronization of coupled os-cillators and etc.808 FEI CHEN, ZENGQIANG CHEN, ZHONGXIN LIU AND ZHUZHI YUANACKNOWLEDGMENTThe authors would like to thank the anonymous referees for their comments and suggestions.This work was partly supported by the National Natural Science Foundation of P.R.China (No. 60774088), the Program for New Century Excellent Talents of High Educationof China (No. NCET-2005-229), the Application Base and Frontier Technology ResearchProject of Tianjin (08JCZDJC21900), and the Science & Technology Research Key Projectof Education Ministry of China (No. 107024).(Received June 19, 2008.)REFERENC ES[1] L. Conradt and T. J. Roper: Consensus decision making in animals. Trends Ecol. Evol.20 (2005), 450–456.[2] D.V. Dimarogonas and K. J. Kyriakopoulos: On the rendezvous problem for multiplenonholonomic agents. IEEE Trans. Automat. Control 52 (2007), 5, 916–922.[3] N.R. Franks, A. Dornhaus, J. P. Fitzsimmons, and M. Stevens: Speed versus accuracyin collective decision making. Proc. Roy. Soc. London Ser. B 270 (2003), 270, 2457–2463.[4] Y. Hatano and M. Mesbahi: Agreement over random networks. IEEE Trans. Automat.Control 50 (2005), 11, 1867–1872.[5] A. Jadbabaie, J. Lin, and A. S. Morse: Coordination of groups of mobile autonomousagents using nearest neighbor rules. IEEE Trans. Automat. Control 48 (2003), 6, 988–1001.[6] S. J. Lu and L. Chen: A general synchronization method of chaotic communicationsystems via Kalman filtering. Kybernetika 44 (2008), 1, 43–52.[7] M. Mesbahi: On state-dependent dynamic graphs and their controllability properties.IEEE Trans. Automat. Control 50 (2005), 3, 387–392.[8] L. Moreau: Stability of multiagent systems with time-dependent communication links.IEEE Trans. Automat. Control 50 (2005), 2, 169–182.[9] T. Vicsek, A. Cziroäok, E. Ben-Jacob, I. Cohen, and O. Shochet: Novel type of phasetransition in a system of self-deriven particles. Phys. Rev. Lett. 75 (1995), 6, 1226–1229.[10] M.M. Zavlanos and G. J. Pappas: Controlling connectivity of dynamic graphs. In:IEEE Conf. on Decision and Control and European Control Conference, Seville 2005,p. 6.Fei Chen, Zengqiang Chen, Zhongxin Liu and Zhuzhi Yuan, Department of Automation,Nankai University, Tianjin 300071. China.e-mails: fchen@live.com, chenzq@nankai.edu.cn, lzhx@nankai.edu.cn

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Control a state-dependent dynamic graph to a pre-specified structure

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