A theoretical explanation for the observed behavior of the Vicsek model, which 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.
Abstract:
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.
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Q1. What have the authors contributed in "Coordination of groups of mobile autonomous agents using nearest neighbor rules" ?
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.
Q2. What is the meaning of the union of a collection of graphs?
By the union of a collection of simple graphs, , each with vertex set , is meant the simple graph with vertex set and edge set equaling the union of the edge sets of all of the graphs in the collection.
Q3. What is the meaning of the definition of a simple graph?
By the intersection of a collection of simple graphs, , each with vertex set , is meant the simple graph with vertex set and edge set equaling the intersection of the edge sets of all of the graphs in the collection.
Q4. What is the form of the update equations?
The explicit form of the update equations exemplified by (38), depends on the relationships between neighbors which exist at time .
Q5. What is the simplest way to show that a group of agents is linked to its leader?
Then(40)The theorem says that the members of the -agent group all eventually follow their leader provided there is a positive integerwhich is large enough so that the -agent group is linked to its leader across each contiguous, nonempty time-interval of length at most .
Q6. What is the definition of a tight condition for a set of matrices?
This condition is tight in the sense that one can find a finite set of matrices with joint spectral radius , whose infinite products converge to zero despite the fact that there does not exist common quadratic Lyapunov function for the set.