In a recent Physical Review Letters article, Vicsek et al. propose a simple but compelling discrete-time model of n autonomous agents (i.e., points or particles) all moving in the plane with the same speed but with different headings. Each agent's heading is updated using a local rule based on the average of its own heading plus the headings of its "neighbors. " In their paper, Vicsek et al. provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models. The Vicsek model proves to be a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function. KeywordsCooperative control, graph theory, infinite products, multiagent systems, switched systems This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.This journal article is available at ScholarlyCommons: http://repository.upenn.edu/ese_papers/29 provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models. The Vicsek model proves to be a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function.

It is shown that switching among stable linear systems results in a stable system provided that switching is "slow-on-the-average." In particular, it is proved that exponential stability is achieved when the number of switches in any finite interval grows linearly with the length of the interval, and the growth rate is sufficiently small. Moreover, the exponential stability is uniform over all switchings with the above property. For switched systems with inputs this guarantees that several input-to-state induced norms are bounded uniformly over all slow-on-the-average switchings. These results extend to classes of nonlinear switched systems that satisfy suitable uniformity assumptions. In this paper it is also shown that, in a supervisory control context, scale-independent hysteresis can produce switching that is slow-on-the-average and therefore the results mentioned above can be used to study the stability of hysteresis-based adaptive control systems.

In a recent Physical Review Letters article, Vicsek et al. propose a simple but compelling discrete-time model of n autonomous agents (i.e., points or particles) all moving in the plane with the same speed but with different headings. Each agent's heading is updated using a local rule based on the average of its own heading plus the headings of its "neighbors. " In their paper, Vicsek et al. provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models. The Vicsek model proves to be a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function. KeywordsCooperative control, graph theory, infinite products, multiagent systems, switched systems This material is posted here with permission of the IEEE. Such permission of the IEEE does not in any way imply IEEE endorsement of any of the University of Pennsylvania's products or services. Internal or personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution must be obtained from the IEEE by writing to pubs-permissions@ieee.org. By choosing to view this document, you agree to all provisions of the copyright laws protecting it.This journal article is available at ScholarlyCommons: http://repository.upenn.edu/ese_papers/29 provide simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent's set of nearest neighbors change with time as the system evolves. This paper provides a theoretical explanation for this observed behavior. In addition, convergence results are derived for several other similarly inspired models. The Vicsek model proves to be a graphic example of a switched linear system which is stable, but for which there does not exist a common quadratic Lyapunov function.

No abstract

report documentation page____________ 7b. RESTRICTIVE MARKINGS None '3. o istr ib u t1 o n 7a v a ! 3 b1 u t y "o f r e p o r t Approved for public release; distribution unlimited y MONITORING ORGANIZATION REPORT NUMBtKW forni Approved OMB No. 0704-01M 6b. OFFICE SYMBOL (If applkabla) 8b. OFFICE SYMBOL (If applkabta) 7 i. NAMEOF m o n it o r in g ORGANIZAllUN

Abstract-In this paper we provide a theoretical foundation for the problem of network localization in which some nodes know their locations and other nodes determine their locations by measuring the distances to their neighbors. We construct grounded graphs to model network localization and apply graph rigidity theory to test the conditions for unique localizability and to construct uniquely localizable networks. We further study the computational complexity of network localization and investigate a subclass of grounded graphs where localization can be computed efficiently. We conclude with a discussion of localization in sensor networks where the sensors are placed randomly.

This paper uses recently established properties of compositions of directed graphs together with results from the theory of non-homogeneous Markov chains to derive worst case convergence rates for the headings of a group of mobile autonomous agents which arise in connection with the widely studied Vicsek consensus problem. The paper also uses graph theoretic constructions to solve modified versions of the Vicsek problem in which there are measurement delays, asynchronous events, or a group leader. In all three cases the conditions under which consensus is achieved prove to be almost the same as the conditions under which consensus is achieved in the synchronous, delay-free, leaderless case.

In this paper we provide a theoretical foundation for the problem of network localization in which some nodes know their locations and other nodes determine their locations by measuring the distances to their neighbors. We construct grounded graphs to model network localization and apply graph rigidity theory to test the conditions for unique localizability and to construct uniquely localizable networks.We further study the computational complexity of network localization and investigate a subclass of grounded graphs where localization can be computed efficiently. We conclude with a discussion of localization in sensor networks where the sensors are placed randomly.