Research On Noncooperative Games Via Multi-Agent Systems

In recent years,game theory has been widely applied in engineering scenarios,such as smart grids,mobile sensor networks and communication networks.The distributed decision of multi-agent systems based on game theory has attracted more and more attention in various fields.A key problem in multi-agent games is to seek the Nash equilibrium(NE),which corresponds to the control objective of multi-agent systems.The aim of distributed NE seeking algorithms is to design strategy-updating rules for agents,who update their strategies by following rules to arrive at the NE of games.Since physical systems in practical engineering scenarios usually have complex dynamics,more and more researchers pay attention to the design of NE seeking algorithms for agents with inherent dynamics.Therefore,the study for the design of strategy-updating rules for multi-agent systems with complex dynamics is practical significant.In the framework of multi-agent systems,this paper studies the design of strategy-updating rules(i,e.,NE seeking algorithms)for noncooperative games.The main contents of this paper are summarized as follows.1)Aggregative games in a network of general linear systems subject to external disturbances are explored in this paper.Taking the outputs of general linear systems as the strategies in the game,an aggregative game with multiple disturbed general linear systems is established.Distributed continuous-time NE seeking algorithms are proposed for the case with perfect and imperfect information,respectively.In the case with imperfect information,to seeking the NE of the aggregative game in a distributed manner,a dynamic average consensus estimator is used to estimate certain global information.Based on the internal model to deal with external disturbances and by introducing the integral of gradient of cost functions for the first time on the basis of passivity theory,NE seeking algorithms are proposed to force the strategies of all agents to evolve to NE regardless of the effect of disturbances.The convergence of the two algorithms is analyzed via Lyapunov stability theory,passivity theory and singular perturbation theory.Simulations are performed to illustrate the effectiveness of the proposed methods.2)A class of noncooperative games in the network of double-integrator agents with the discrete-time interactive scheme is explored.The existing distributed continuous-time NE seeking algorithms require that agents communicate with their neighbors in real-time.In this paper,we propose two discrete-time communication schemes based on the designed distributed continuous-time NE seeking algorithm for the efficient use of communication resources.In specific,we propose respectively periodic and event-triggered communication schemes for the discrete-time interactions among agents.The rule in the periodic case is implemented synchronously and easily.The rule in the event-triggered case is executed asynchronously without Zeno behaviors.All agents in both cases can reach to the NE asymptotically by interacting with neighbors at discrete times.Simulations are performed in the network of Cournot competition to illustrate the effectiveness of the proposed methods.3)The design of a distributed method for double-integrator agents to seek the generalized Nash equilibrium(GNE)of a class of aggregative games,which is subject to both local and coupled constraints,is studied in this paper.According the KKT condition of the formulated problem,a novel distributed GNE seeking algorithm is proposed to force double-integrator agents to regulate their strategies in real-time.The designed algorithm integrates the projected output feedback to deal with local convex constraints,with the coordination of Lagrange multipliers to handle the global equality constraints and the dynamic average consensus estimator to estimate the aggregator.Under the assumption that the communication topologies are strongly connected and weight-balanced directed graphs,the convergence of the designed algorithm is obtained.Numerical examples are given to illustrate the results.4)A class of constrained noncooperative games with unknown cost functions is studied in this paper.In the game,the strategy of each agent is subject to both the coupled equality constraint and local inequality constraints.By virtue of the exact penalty method,an auxiliary cost function is constructed with the cost function and the local constraints.With only knowing the values of the costs,the extremum seeking control is used to estimate the gradient information of auxiliary cost functions.The optimal Lagrange multiplier corresponding to the global equality constraint is estimated in a distributed malner.As a result,a novel distributed GNE seeking algorithm is proposed.Moreover,a diminishing dither signal is introduced in the seeking algorithm to renlove undesirable steady-state oscillations occurring in the classic extremum seeking control.The nonlocal convergence of the designed seeking algorithm to the GNE of the game is obtained by the singular perturbation theory,averaging analysis and Lyapunov stability theory.Numerical examples are given to verify the effectiveness of our proposed method.