若干非线性系统的稳定性与混沌同步
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摘要
非线性科学是研究不同学科中非线性现象共性的一门前沿学科,是在以非线性为特征的各门分支学科基础上发展起来的综合性学科.非线性动力学行为的研究不仅具有重要的理论意义,还能进一步提高非线性控制理论向工程领域应用的可能性,具有重要的实际意义,因此非线性动力学是非线性科学主要的研究内容之一.稳定性问题是控制系统设计中的首要问题,混沌是非线性系统的常见特性,因此非线性系统的稳定性与混沌同步近年来成为了非线性动力学研究的热点问题.在简要地介绍该领域的研究背景与发展现状后,本文的工作主要集中在两个部分:第一部分讨论一类重要的非线性系统――时滞神经网络的稳定性问题;第二部分研究几类非线性系统的混沌同步问题.
针对时滞神经网络的稳定性问题,本文主要从考虑干扰因素和扩展模型两方面,对其进行了深入地研究.一方面,在时滞神经网络模型中考虑各种可能具有干扰因素,如参数摄动、随机干扰、脉冲效应等.利用Lyapunov泛函、Young不等式和Halanay不等式,分别研究了不确定BAM神经网络的鲁棒指数稳定性和脉冲细胞神经网络的指数稳定性.另一方面,根据实际需要对时滞神经网络模型进行了扩展,建立了Markov跳变神经网络模型、T-S模糊神经网络模型和高阶神经网络模型,基于Lyapunov稳定性理论、随机分析方法和状态空间分解方法,分别研究了Markov跳变Cohen-Grossberg神经网络的鲁棒稳定性、T-S模糊细胞神经网络的均方鲁棒稳定性和高阶Cohen-Grossberg神经网络的多周期性.
针对几类非线性系统的混沌同步问题,本文主要利用驱动――响应同步原理,从以下三个方面研究几类非线性混沌系统的同步控制.一是改进了混沌同步的控制方法.将自适应控制、鲁棒H∞控制、定位控制等现代控制理论中的方法应用到混沌同步控制中,分别研究了时滞混沌神经网络的自适应同步、时滞混沌系统的H∞同步和广义复杂动态网络的定位控制同步.二是拓展了混沌同步概念的扩展.除完全同步之外,还研究了混沌系统的其他同步方式.在同时考虑传输信道时间延迟和不确定参数的情况下,研究了统一混沌系统的自适应混合延迟投影同步.该同步方式包含了许多已知的同步概念,如完全同步、反同步、延迟同步、投影同步等.三是放宽了混沌同步外部条件.在存在外部干扰的条件下研究混沌系统的同步问题.通过分析驱动系统与响应系统之间误差系统的指数稳定性,建立了一类具有脉冲干扰的神经网络指数同步的充分性判据,以输出耦合和状态耦合形式给出了两种简易的同步控制器设计方法.
Nonlinear science is a frontier discipline which studies the common nonlinear phe-nomena in di?erent disciplines. It is also a comprehensive discipline which developedon the basis of subjects characterized by nonlinear behavior. Since the study on thenonlinear dynamics has not only theoretical significance but also practical significance,it is very important in the research of nonlinear science. Stability is the preconditionfor the control system design, while chaos is one of common characteristics of nonlinearsystems. Recently, the stability and chaos synchronization of nonlinear systems receivemuch attention in nonlinear dynamics. After a brief introduction to the background andresearch progress in the filed, the research work focus mainly on two parts. The first partdiscusses the stability of delayed neural networks (DNNs), since DNN is an importantclass of nonlinear systems. The second part investigates chaos synchronization of severaldi?erent nonlinear systems.
The main contributions on the stability of delayed neural networks are consideringinterference factors and extending neural network models. On the one hand, there mayexist some interference factors in delayed neural networks, for instance, parameter un-certainties, stochastic disturbance, impulse e?ects, etc. By using Lyapunov functional,Young and Halanay inequalities, the robust exponential stability for uncertain BAM neu-ral networks and the exponential stability for impulsive cellular neural networks are stud-ied. On the other hand, the thesis establishes some new delayed neural networks models,such as Markov jump neural networks, T-S fuzzy neural networks and higher-order neuralnetworks. Based on Lyapunov stability theory, stochastic analysis approaches and decom-position of the state space, the robust stability of Markov jump Cohen-Grossberg neuralnetworks, the robust stability of T-S fuzzy cellular neural networks and the multiperiod-icity of higher-order Cohen-Grossberg neural networks are investigated.
The main contributions on chaos synchronization of several di?erent nonlinear sys-tems in the thesis include the following three points. Firstly, the control methods of chaossynchronization are improved. The methods in modern control theory, such as adaptivecontrol, robust H∞control and pinning control are applied to the chaos synchroniza-tion. Moreover, the adaptive synchronization of chaotic delayed neural networks, theH∞synchronization of delayed chaotic neural networks and the pinning control synchro-nization of generalized complex dynamical networks are analyzed. Secondly, the chaossynchronization notations are extended. In addition to complete synchronization, thelag synchronization and projective synchronization of chaotic systems are studied. Theadaptive hybrid lag projective synchronization (AHLPS) problem of a class of unifiedchaotic systems with channel time-delay and parameter uncertainty is investigated. TheAHLPS is a new type of chaos synchronization, which includes complete synchroniza-tion, anti-synchronization, lag synchronization and projective synchronization. Thirdly,the external conditions of chaos synchronization are relaxed. The synchronization of aclass of neural networks with impulsive noise is investigated by analyzing the exponential stability of error system. Furthermore, designing laws for the controlling gain matrix inthe synchronization of neural networks are proposed via output or state coupling.
针对时滞神经网络的稳定性问题,本文主要从考虑干扰因素和扩展模型两方面,对其进行了深入地研究.一方面,在时滞神经网络模型中考虑各种可能具有干扰因素,如参数摄动、随机干扰、脉冲效应等.利用Lyapunov泛函、Young不等式和Halanay不等式,分别研究了不确定BAM神经网络的鲁棒指数稳定性和脉冲细胞神经网络的指数稳定性.另一方面,根据实际需要对时滞神经网络模型进行了扩展,建立了Markov跳变神经网络模型、T-S模糊神经网络模型和高阶神经网络模型,基于Lyapunov稳定性理论、随机分析方法和状态空间分解方法,分别研究了Markov跳变Cohen-Grossberg神经网络的鲁棒稳定性、T-S模糊细胞神经网络的均方鲁棒稳定性和高阶Cohen-Grossberg神经网络的多周期性.
针对几类非线性系统的混沌同步问题,本文主要利用驱动――响应同步原理,从以下三个方面研究几类非线性混沌系统的同步控制.一是改进了混沌同步的控制方法.将自适应控制、鲁棒H∞控制、定位控制等现代控制理论中的方法应用到混沌同步控制中,分别研究了时滞混沌神经网络的自适应同步、时滞混沌系统的H∞同步和广义复杂动态网络的定位控制同步.二是拓展了混沌同步概念的扩展.除完全同步之外,还研究了混沌系统的其他同步方式.在同时考虑传输信道时间延迟和不确定参数的情况下,研究了统一混沌系统的自适应混合延迟投影同步.该同步方式包含了许多已知的同步概念,如完全同步、反同步、延迟同步、投影同步等.三是放宽了混沌同步外部条件.在存在外部干扰的条件下研究混沌系统的同步问题.通过分析驱动系统与响应系统之间误差系统的指数稳定性,建立了一类具有脉冲干扰的神经网络指数同步的充分性判据,以输出耦合和状态耦合形式给出了两种简易的同步控制器设计方法.
Nonlinear science is a frontier discipline which studies the common nonlinear phe-nomena in di?erent disciplines. It is also a comprehensive discipline which developedon the basis of subjects characterized by nonlinear behavior. Since the study on thenonlinear dynamics has not only theoretical significance but also practical significance,it is very important in the research of nonlinear science. Stability is the preconditionfor the control system design, while chaos is one of common characteristics of nonlinearsystems. Recently, the stability and chaos synchronization of nonlinear systems receivemuch attention in nonlinear dynamics. After a brief introduction to the background andresearch progress in the filed, the research work focus mainly on two parts. The first partdiscusses the stability of delayed neural networks (DNNs), since DNN is an importantclass of nonlinear systems. The second part investigates chaos synchronization of severaldi?erent nonlinear systems.
The main contributions on the stability of delayed neural networks are consideringinterference factors and extending neural network models. On the one hand, there mayexist some interference factors in delayed neural networks, for instance, parameter un-certainties, stochastic disturbance, impulse e?ects, etc. By using Lyapunov functional,Young and Halanay inequalities, the robust exponential stability for uncertain BAM neu-ral networks and the exponential stability for impulsive cellular neural networks are stud-ied. On the other hand, the thesis establishes some new delayed neural networks models,such as Markov jump neural networks, T-S fuzzy neural networks and higher-order neuralnetworks. Based on Lyapunov stability theory, stochastic analysis approaches and decom-position of the state space, the robust stability of Markov jump Cohen-Grossberg neuralnetworks, the robust stability of T-S fuzzy cellular neural networks and the multiperiod-icity of higher-order Cohen-Grossberg neural networks are investigated.
The main contributions on chaos synchronization of several di?erent nonlinear sys-tems in the thesis include the following three points. Firstly, the control methods of chaossynchronization are improved. The methods in modern control theory, such as adaptivecontrol, robust H∞control and pinning control are applied to the chaos synchroniza-tion. Moreover, the adaptive synchronization of chaotic delayed neural networks, theH∞synchronization of delayed chaotic neural networks and the pinning control synchro-nization of generalized complex dynamical networks are analyzed. Secondly, the chaossynchronization notations are extended. In addition to complete synchronization, thelag synchronization and projective synchronization of chaotic systems are studied. Theadaptive hybrid lag projective synchronization (AHLPS) problem of a class of unifiedchaotic systems with channel time-delay and parameter uncertainty is investigated. TheAHLPS is a new type of chaos synchronization, which includes complete synchroniza-tion, anti-synchronization, lag synchronization and projective synchronization. Thirdly,the external conditions of chaos synchronization are relaxed. The synchronization of aclass of neural networks with impulsive noise is investigated by analyzing the exponential stability of error system. Furthermore, designing laws for the controlling gain matrix inthe synchronization of neural networks are proposed via output or state coupling.
引文
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