Algorithm Design For Distributed Nash Equilibrium Seeking And Optimization Problems

Game theory is a mathematical model to study the conflict and cooperation between rational decision makers.It can be used to model and analyze some large scale optimization problems.In the game,using the designed distributed algorithm,each player can adjust its decision variable to Nash equilibrium and then optimize its cost function.The existing distributed Nash equilibrium seeking algorithms are designed in the cases that the cost functions' mathematical models are known and the players can exchange their information over the secure communication network.However,in practice,the players' cost functions can be affected by some uncertain factors and their mathematical expressions are unknown.This means that the accurate gradient information of cost functions can not be utilized to design distributed gradient-based algorithms.To address this case,the extremum seeking control method is introduced into the designing of distributed strategies.It is an online or real-time optimization method.The study of the extremum seeking scheme is helpful to design the non-model-based distributed optimization.In addition,since the existence of networked attacks can destroy the edges of communication topology,break down the information exchange and then affect the ability of distributed strategy to seek Nash equilibrium or optimal solution,it is worth studying the distributed algorithm under networked attacks.In general,the main work of the thesis consists of the following aspects:For the extremum seeking control problems in the presence of actuator dead-zone,this thesis designs a new extremum seeking algorithm with a fast learning mechanism.This new scheme can rapidly compensate the effect of input dead-zone and regulate the output of the system to an arbitrarily small neighborhood of the cost function's extremum.The closed-loop system of this scheme can be modeled as a singularly perturbed model whose boundary layer system is a fast learning dynamic and reduced system is the classic extremum seeking scheme.In theoretical analysis,based on the stability of the limit systems and the generalized singular perturbation theorem,we can obtain that the closed-loop system is semiglobally practically asymptotically stable.For the case that the players' dynamics are affected by actuator dead-zone,the proposed dead-zone compensator with a fast learning mechanism can be utilized to design the distributed Nash equilibrium seeking algorithm.This algorithm can make the input jump out of dead-zone intervals and drive the players' decision variables to the Nash equilibrium of the considered game problem.By introducing an auxiliary dynamical system into this new algorithm,we relax the constraint on the eigenvalues of the Laplace matrix.Based on Lyapunov stability theorem and singular perturbation method,exponential stability of the proposed algorithm is obtained.For the case in which the cost functions' mathematical expressions are unknown,this thesis studies the generalized Nash equilibrium seeking problem for the game with equality constraints.Employing extremum seeking control method and multi-agent consensus protocol,a non-modelbased distributed algorithm is proposed.In this algorithm,estimation dynamics based consensus are used to estimate all players' decision variables,and extremum seeking-based optimization dynamics can steer players' decisions to Nash equilibrium.The standard singular perturbation theorem and Lyapunov stability theorem are employed to analyze the convergence of the proposed strategy.For the case that communication network is subject to potential attacks,this thesis models two distributed switched algorithms to study the influence of networked attacks on Nash equilibrium seeking and resource allocation problems,respectively.Under the constraints on the attacks duration and frequency,by constructing appropriate Lyapunov functions,the designed distributed switched algorithms are able to seek the Nash equilibrium and optimal resource allocation.