Nash Equilibrium Seeking in Multi-Vehicle Systems: A Lie Bracket Approximation-Based Approach

Details Nash Equilibrium Seeking in Multi-Vehicle Systems: A Lie Bracket Approximation-Based Approach Nash Equilibrium Seeking in Multi-Vehicle Systems: A Lie Bracket Approximation-Based Approach

2011-09-28

arxiv.org/abs/1109.6129v2

Mathematics

Dynamical Systems

Scientific paper

We consider problems in multi-agent systems where a network of autonomous vehicles needs to self-organize such that some global objective function is maximized. To deal with the agents' lack of global information we define the problem as a potential game where the agents/players are only able to access local measurements of their individual utility functions whose parameters and detailed analytical forms may be unknown. We then propose a distributed and adaptive continuous-time algorithm, where each agent applies a local sinusoidal extremum seeking feedback adopted to its specific motion dynamics. By interpreting this multi-vehicle system as an input-affine system with periodic excitations and by using geometric methods based on Lie brackets, we calculate a simplified system which approximates the qualitative behavior of the original one better than existing methods, while only requiring twice differentiability of the utility functions. We show that the Lie bracket direction is directly related to the unknown gradient of the global potential function and prove global practical asymptotic stability of a compact invariant set consisting of all local maxima of the potential function (local Nash equilibria). We apply the proposed algorithm to a sensor coverage problem and solve it in a distributed way where the agents do not need any a priori knowledge about the distribution of the events to be detected and about the detection probabilities of the individual agents. The proposed Lie bracket-based methodology and the coverage control schemes are illustrated through simulations.

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