多智能体系统的脉冲一致性及其动力学研究
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摘要
多智能体系统的一致性问题具有广泛的应用前景,是当前国际控制理论与应用领域的前沿课题之一。本文主要基于图谱理论、随机矩阵理论与脉冲控制理论等研究多智能体系统的脉冲一致性问题,力求解决此领域的相关问题。脉冲耦合非线性系统的同步及其动力学是当前国际动力学与控制领域中的重要研究内容之一。本文主要基于压缩原理研究一类脉冲耦合振子的同步问题,并研究一类脉冲耦合Duffing振子的复杂动力学行为及其分岔机理。本文的研究对于加强多智能体系统、脉冲系统的理论及其在实际中的应用具有重要意义。本文的主要内容有：
第一章介绍多智能体系统的一致性问题、非光滑系统、脉冲系统等的产生、发展与研究现状等,在此基础上给出本课题研究的意义和论文的主要工作与结构安排。
当前连续控制协议研究得较多,但其他形式控制协议研究得较少。第二章针对一类线性多智能体系统,提出脉冲控制协议,该脉冲控制协议具有控制量小、实现方便等优点。基于脉冲系统理论给出线性多智能体系统达到一致的充分条件,并进一步给出脉冲时刻及脉冲矩阵的设计方法。最后研究网络拓扑动态切换的情形。
第三章在第二章的基础上研究非线性多智能体系统的一致性问题。根据网络结构分为两个部分。第一部分研究无向网络上非线性多智能体系统的脉冲一致性问题,其中每个智能体均为同样的非线性动力系统。网络拓扑固定时利用Laplacian矩阵的最大、最小特征值给出离散时刻和脉冲常数的设计方案。网络拓扑动态切换时给出设计步骤。第二部分研究有向切换网络上非线性多智能体系统的脉冲一致性问题。基于随机矩阵理论给出系统达到一致的充分条件,并当网络拓扑动态切换且网络结构图均为强连通平衡图时给出脉冲控制协议的设计方案。
无源性方法是关联系统稳定性分析的有效方法之一。第四章研究一类多智能体系统的脉冲输出一致性,其中每个智能体均为无源的。基于无源性原理分别给出多智能体系统在网络拓扑固定、切换两种情形下达到输出一致的充分条件。第
二、三章中构成多智能体系统的各智能体的方程需要相同,但在智能体是无源的情形下,各智能体的方程无需相同。
脉冲耦合振子是指仅在离散时刻相互作用的耦合振子,已在图像处理等领域具有广泛应用。第五章基于压缩原理研究脉冲耦合振子的同步问题。基于所提出的脉冲系统的部分压缩原理分别对两脉冲耦合振子和网络脉冲耦合振子进行同步分析,并给出了脉冲耦合振子达到同步的充分条件。
周期脉冲作用下非线性系统为非光滑系统,具有丰富的动力学行为。第六章研究周期脉冲作用下Chen系统的复杂动力学行为,并通过Floquet理论揭示该系统周期解的非光滑分岔机理。周期脉冲作用下Chen系统主要通过两种途径到达混沌,即经鞍结分岔到达混沌和倍周期分岔到达混沌。
由于脉冲耦合系统本质上为高维非光滑系统,因而会产生丰富而复杂的动力学行为。第七章研究环形脉冲耦合Duffing振子的复杂动力学行为。通过构造Poincare映射,给出该系统的分岔条件,并得到Poincare映射的Jacobi矩阵的解析表达式,结合打靶法及龙格-库塔方法求出系统的分岔集及Floquet乘子。若脉冲作用周期固定,在耦合强度变化时,系统经历稳定解、周期解、概周期解、超混沌等复杂动力学过程,利用Floquet理论研究该系统的周期解的稳定性和一些经典的分岔。
第八章对本文的结果进行总结,并对今后的工作提出展望。
The consensus problem of multi-agent systems has wide application prospect and is one of current international topics in the field of control theory and application. This dissertation investigated the consensus problem of multi-agent systems mainly based on graph theory, random matrix theory and control theory and devoted to solve the related isslues. The synchronization and dynamics of impulsive coupled nonlinear systems are important research contents in the current international dynamics and control field. This dissertation investigated the synchronizaiton of a class of impulsive coupled systems based on contraction theory and studied the complex dynamics and its bifurcation mechanism of a class of impulsive coupled Duffing oscillators. This work has important significance in improving the theory of multi-agent systems and impulsive systems and its application in practice. The basic contents of this dissertation are given as following:
The research background, current research status of the consensus problem of multi-agent systems, non-smooth systems and impulsive systems are given firstly in the chapter 1. Then the significance of this research and the structure of the dissertation are introduced.
Now continuous control protocols are received much attention, however other forms of control protocols have received relatively little attention. In chapter 2, we introduce impulsive control protocols for multi-agent linear dynamic systems. The impulsive control protocols need low-cost and can easily implement. Sufficient conditions are given to guarantee the consensus of the multi-agent linear dynamic systems by the theory of impulsive systems. Furthermore, how to select the discrete instants and impulsive matrices is discussed. The case that the topologies of networks are switching is also considered.
The third chapter investigates the impulsive consensus of multi-agent nonlinear systems on the basis of chapter 2. The third chapter is divided into two parts based on the topology of the network. In the first part, we investigate the problem of impulsive consensus of networked multi-agent systems, where each agent can be modeled as an identical nonlinear dynamical system. Firstly, an impulsive control protocol is designed for network with fixed topology based on the local information of agents. Then sufficient conditions are given to guarantee the consensus of the networked nonlinear dynamical system by using algebraic graph theory and impulsive control theory. Furthermore, how to select the discrete instants and impulsive constants is discussed. The case that the topologies of the networks are switching is also considered. In the second part, we investigate the problem of impulsive consensus of multi-agent systems for directed networks with switching topologies, where each agent can be modeled as an identical nonlinear system. Then sufficient conditions are given to guarantee the consensus of the multi-agent system based on the stochastic matrices theory. When the topologies of the networks are switching and each graph is strongly connected and balanced, the scheme to design the impulsive control protocol is proposed.
The passivity approach is a nice tool for controlling interconnection systems. In chapter 4, we study the problem of impulsive output consensus of multi-agent dynamical systems, where each agent is a passive system. Based on the passive theory of impulsive systems, sufficient conditions are given to guarantee the output consensus of the multi-agent systems in two cases that the network is fixed and the topologies of networks are switching. The equations of agents need to be identical in chapter 2 and chapter 3, however, The equations of agents need not to be identical when the agents are passive.
Impulsively coupled oscillators which are assumed to interact with each other only at discrete times have been utilized for various image processing applications and so on. In chapter 5, we investigate the synchronization problem of impulsively coupled oscillators based on the contraction theory. Contraction analysis of two impulsively coupled oscillators and networked impulsively coupled oscillators is provided based on the proposed partial contraction theory of impulsive systems, respectively. Sufficient conditions for synchronization of impulsively coupled oscillators are derived.
The nonlinear systems with periodic impulsive forces are non-smooth systems and have complex dynamics. In chapter 6, the complex dynamics of Chen system with periodic impulsive forces is investigated and the Floquet theory is used to explore the non-smooth bifurcation mechanism for the periodic solutions. The non-smooth bifurcation of Chen system with periodic impulsive forces is analyzed. The system can evolve to chaos by a cascading of period-doubling bifurcations. Besides, the system can evolve to chaos immediately by saddle-node bifurcations from periodic solutions.
Impulsively coupled systems are high dimensional non-smooth systems and have rich and complex dynamics. In chapter 7, the complex dynamics of the non-smooth system which is unidirectionally impusively coupled by three Duffing oscillators in ring structure is investigated. By constructing a Poincar6 mapping, we get the bifurcation condition, and give analytical expression of Jacobi mapping matrix of Poincare map. Then we obatin the bifurcation set and Floquet characteristic multipliers by the shooting method and the Runge-Kutta method. When the period is fxed and the coupling strength changes, the system experiences stable solution, periodic solution, Quasi-periodic solutions, hyper-chaotic, etc. The Floquet theory is used to study the stability of the periodic solutions of the system and some classic bifurcation.
In chapter 8, some meaningful results are summarized. Also some existing problems as well as the future work are pointed out.
第一章介绍多智能体系统的一致性问题、非光滑系统、脉冲系统等的产生、发展与研究现状等,在此基础上给出本课题研究的意义和论文的主要工作与结构安排。
当前连续控制协议研究得较多,但其他形式控制协议研究得较少。第二章针对一类线性多智能体系统,提出脉冲控制协议,该脉冲控制协议具有控制量小、实现方便等优点。基于脉冲系统理论给出线性多智能体系统达到一致的充分条件,并进一步给出脉冲时刻及脉冲矩阵的设计方法。最后研究网络拓扑动态切换的情形。
第三章在第二章的基础上研究非线性多智能体系统的一致性问题。根据网络结构分为两个部分。第一部分研究无向网络上非线性多智能体系统的脉冲一致性问题,其中每个智能体均为同样的非线性动力系统。网络拓扑固定时利用Laplacian矩阵的最大、最小特征值给出离散时刻和脉冲常数的设计方案。网络拓扑动态切换时给出设计步骤。第二部分研究有向切换网络上非线性多智能体系统的脉冲一致性问题。基于随机矩阵理论给出系统达到一致的充分条件,并当网络拓扑动态切换且网络结构图均为强连通平衡图时给出脉冲控制协议的设计方案。
无源性方法是关联系统稳定性分析的有效方法之一。第四章研究一类多智能体系统的脉冲输出一致性,其中每个智能体均为无源的。基于无源性原理分别给出多智能体系统在网络拓扑固定、切换两种情形下达到输出一致的充分条件。第
二、三章中构成多智能体系统的各智能体的方程需要相同,但在智能体是无源的情形下,各智能体的方程无需相同。
脉冲耦合振子是指仅在离散时刻相互作用的耦合振子,已在图像处理等领域具有广泛应用。第五章基于压缩原理研究脉冲耦合振子的同步问题。基于所提出的脉冲系统的部分压缩原理分别对两脉冲耦合振子和网络脉冲耦合振子进行同步分析,并给出了脉冲耦合振子达到同步的充分条件。
周期脉冲作用下非线性系统为非光滑系统,具有丰富的动力学行为。第六章研究周期脉冲作用下Chen系统的复杂动力学行为,并通过Floquet理论揭示该系统周期解的非光滑分岔机理。周期脉冲作用下Chen系统主要通过两种途径到达混沌,即经鞍结分岔到达混沌和倍周期分岔到达混沌。
由于脉冲耦合系统本质上为高维非光滑系统,因而会产生丰富而复杂的动力学行为。第七章研究环形脉冲耦合Duffing振子的复杂动力学行为。通过构造Poincare映射,给出该系统的分岔条件,并得到Poincare映射的Jacobi矩阵的解析表达式,结合打靶法及龙格-库塔方法求出系统的分岔集及Floquet乘子。若脉冲作用周期固定,在耦合强度变化时,系统经历稳定解、周期解、概周期解、超混沌等复杂动力学过程,利用Floquet理论研究该系统的周期解的稳定性和一些经典的分岔。
第八章对本文的结果进行总结,并对今后的工作提出展望。
The consensus problem of multi-agent systems has wide application prospect and is one of current international topics in the field of control theory and application. This dissertation investigated the consensus problem of multi-agent systems mainly based on graph theory, random matrix theory and control theory and devoted to solve the related isslues. The synchronization and dynamics of impulsive coupled nonlinear systems are important research contents in the current international dynamics and control field. This dissertation investigated the synchronizaiton of a class of impulsive coupled systems based on contraction theory and studied the complex dynamics and its bifurcation mechanism of a class of impulsive coupled Duffing oscillators. This work has important significance in improving the theory of multi-agent systems and impulsive systems and its application in practice. The basic contents of this dissertation are given as following:
The research background, current research status of the consensus problem of multi-agent systems, non-smooth systems and impulsive systems are given firstly in the chapter 1. Then the significance of this research and the structure of the dissertation are introduced.
Now continuous control protocols are received much attention, however other forms of control protocols have received relatively little attention. In chapter 2, we introduce impulsive control protocols for multi-agent linear dynamic systems. The impulsive control protocols need low-cost and can easily implement. Sufficient conditions are given to guarantee the consensus of the multi-agent linear dynamic systems by the theory of impulsive systems. Furthermore, how to select the discrete instants and impulsive matrices is discussed. The case that the topologies of networks are switching is also considered.
The third chapter investigates the impulsive consensus of multi-agent nonlinear systems on the basis of chapter 2. The third chapter is divided into two parts based on the topology of the network. In the first part, we investigate the problem of impulsive consensus of networked multi-agent systems, where each agent can be modeled as an identical nonlinear dynamical system. Firstly, an impulsive control protocol is designed for network with fixed topology based on the local information of agents. Then sufficient conditions are given to guarantee the consensus of the networked nonlinear dynamical system by using algebraic graph theory and impulsive control theory. Furthermore, how to select the discrete instants and impulsive constants is discussed. The case that the topologies of the networks are switching is also considered. In the second part, we investigate the problem of impulsive consensus of multi-agent systems for directed networks with switching topologies, where each agent can be modeled as an identical nonlinear system. Then sufficient conditions are given to guarantee the consensus of the multi-agent system based on the stochastic matrices theory. When the topologies of the networks are switching and each graph is strongly connected and balanced, the scheme to design the impulsive control protocol is proposed.
The passivity approach is a nice tool for controlling interconnection systems. In chapter 4, we study the problem of impulsive output consensus of multi-agent dynamical systems, where each agent is a passive system. Based on the passive theory of impulsive systems, sufficient conditions are given to guarantee the output consensus of the multi-agent systems in two cases that the network is fixed and the topologies of networks are switching. The equations of agents need to be identical in chapter 2 and chapter 3, however, The equations of agents need not to be identical when the agents are passive.
Impulsively coupled oscillators which are assumed to interact with each other only at discrete times have been utilized for various image processing applications and so on. In chapter 5, we investigate the synchronization problem of impulsively coupled oscillators based on the contraction theory. Contraction analysis of two impulsively coupled oscillators and networked impulsively coupled oscillators is provided based on the proposed partial contraction theory of impulsive systems, respectively. Sufficient conditions for synchronization of impulsively coupled oscillators are derived.
The nonlinear systems with periodic impulsive forces are non-smooth systems and have complex dynamics. In chapter 6, the complex dynamics of Chen system with periodic impulsive forces is investigated and the Floquet theory is used to explore the non-smooth bifurcation mechanism for the periodic solutions. The non-smooth bifurcation of Chen system with periodic impulsive forces is analyzed. The system can evolve to chaos by a cascading of period-doubling bifurcations. Besides, the system can evolve to chaos immediately by saddle-node bifurcations from periodic solutions.
Impulsively coupled systems are high dimensional non-smooth systems and have rich and complex dynamics. In chapter 7, the complex dynamics of the non-smooth system which is unidirectionally impusively coupled by three Duffing oscillators in ring structure is investigated. By constructing a Poincar6 mapping, we get the bifurcation condition, and give analytical expression of Jacobi mapping matrix of Poincare map. Then we obatin the bifurcation set and Floquet characteristic multipliers by the shooting method and the Runge-Kutta method. When the period is fxed and the coupling strength changes, the system experiences stable solution, periodic solution, Quasi-periodic solutions, hyper-chaotic, etc. The Floquet theory is used to study the stability of the periodic solutions of the system and some classic bifurcation.
In chapter 8, some meaningful results are summarized. Also some existing problems as well as the future work are pointed out.
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