Research On Distributed Optimization Algorithms Based On Hybrid System Model

The interaction among agents plays an important role in distributed algorithms.The dis-tributed convex optimization problems are cooperatively solved by a networked communication of agents which is regarded as a cooperative optimization problem,while the game is a nonco-operative optimization problem.The task of each agent is to seek the optimal solution or Nash equilibrium to an optimization problem by exchanging information with its neighbors.In dis-tributed optimization,the convergence of an algorithm is vital to seek the optimal solution or Nash equilibrium.In practice,however,the effect of communication delays and networked at-tacks on the convergence to the optimal solution is inevitable.The size of communication delays may affect the convergence of a distributed algorithm.The existence of networked attacks may result in unconnected communication topologies and correspondingly affect the convergence of the algorithms.In addition,multi-agents may also be subject to some unknown factors,such as external disturbances and unmodeled terms.The proposed ideal algorithms may be not ap-plied to such cases.Therefore,it is significant to study the convergence to the optimal solution or Nash equilibrium under communication delays,networked attacks,external disturbances and unmodeled terms.The main work of this thesis consists of the following points.To proceed the study on the effect of time-varying delays on distributed algorithms,we first investigate the stability of switched systems with constant delay and it is used to learn/train how to analyze the stability of such switched delay systems.Then based on this idea,we consider a class of nonlinear time-varying delay systems allowing the large delays to occur.In addition,the large delays and small delays are allowed to occur in an alternating manner and we model this problem into a switched delay system to analyze the stability.To obtain exponential stability,the allowable occurrence total activation time and the number of large delays are given.Such an idea can be used to analyze the effect on convergence of the optimal solution to a distributed convex optimization problem with large delays.For distributed optimization problems with small delays,some work has been done to inves-tigate such optimization problems.While large delays occur,that is,when the bound of delays exceeds the maximum allowable delay bound,the existing algorithms may be unstable and the optimal solution may be not found easily.To the best of our knowledge,few results have been reported about this problem.To explore this problem in this thesis,the alternating manner of small and large delays can be applied to solve distributed optimization resource allocation prob-lem.Based on switching techniques,the proposed algorithm is modeled as a switched delay system.Finally,the allowable occurrence total activation time and the number of large delays are given to guarantee that the optimal solution is exponentially stable.In the ideal case,the existing distributed algorithms assume that the communication chan-nels are connected.In practice,however,making all communication networks connected or secure is impossible since the existence of networked attacks may lead to communication fail-ures.When attacks occur,we allow attackers to be able to switch topologies at will,until the attacks are shut down and normal operation resumes.To investigate this problem,we first model the proposed algorithm with attacks into a switched system.Then,we use an average dwell-time automaton and time-ratio monitor constraint to tackle attacks.Finally,a switched algorithm is modeled as a hybrid dynamical system and a Lyapunov function is constructed to show the optimal solution can be achieved exponentially under persistent attacks.To consider the effect of external disturbances and unmodeled terms on Nash equilibrium,robust distributed Nash equilibrium seeking for aggregative games under persistent attacks is investigated in this thesis.Combining the above considered attacks,we introduce two auxiliary variables to tackle the behaviors of attacks.In addition,we consider each player has the double-integrator dynamic influenced by unknown time-varying disturbances and unmodeled terms.The whole closed-loop system is generalized to a hybrid system.To analyze the convergence of the Nash equilibrium,a Lyapunov function is constructed.Finally,uniform asymptotic stability and uniform global asymptotic stability results are obtained under different mild assumptions.