【作者】 万鹏；

【导师】 孙棣华；

【作者基本信息】 重庆大学 ， 控制理论与控制工程， 2020， 博士

【摘要】 人工神经网络,是从信息处理角度对生物神经网络进行抽象而建立的数学模型。随着人工神经网络的研究工作不断深入,其在模式分割、智能机器人、自动控制、预测估计、故障诊断、系统辨识等领域已成功地解决了许多现代计算机难以解决的实际问题,显示出了良好的智能特性,这些智能特性主要取决于神经网络的动力学行为。多稳定性是描述多个稳定平衡态或周期解共存的概念。这种动力学行为在神经网络的一些应用中是必不可少的,包括图像处理、模式识别和联想记忆存储。Hopfield型神经网络,已经成为吸引大量多稳定性研究兴趣的主要模型。在实际生活中,周期函数能很好地描述系统的发展过程,比如生态系统、机械震动、市场供需、交通系统、生物活动中的心跳和记忆等等,而这些实际问题都可以总结为讨论微分方程周期解的稳定性。基于此,本文研究了Hopfield神经网络多稳定性和产生全局稳定周期解的控制策略问题。在神经网络的理论研究中,神经网络的动力学行为与时滞、不确定性、随机噪声和扩散现象关系密切。近二十年来,众多学者考虑在这些因素下,如何保证Hopfield神经网络的全局稳定性或者局部稳定性,相关的研究成果层出不穷。然而,针对带有反应扩散项、脉冲效应和混合时滞的神经网络,如何利用矩阵凸组合和线性矩阵不等式技巧获得保守性更低的全局稳定周期解的存在唯一性条件,仍需深入研究。当分段线性、非饱和、非连续非单调激活函数出现在离散时间、连续时间、分数阶、Takagi-Sugeno模糊神经网络中,如何分析其单稳定性和多稳定性是一个难题。对于不稳定的时滞神经网络,如何设计脉冲控制器使得神经网络产生全局稳定周期解。针对这些问题,本文以离散时间、连续时间、分数阶、TakagiSugeno模糊、随时间切换、惯性反应扩散神经网路为研究对象。从分段线性,非饱和分段线性和非连续非单调激活函数的几何属性角度出发,充分运用严格对角占优矩阵、收缩映射、不动点定理、Ascoli-Arzela定理和凸组合方法,构造适当的Lyapunov-Krasovskii泛函,本文完成的主要工作包括:(1)对离散神经网络和四元数神经网络进行了多稳定性分析。一类分段线性激活函数,使神经网络的存储容量大大提高。根据分段线性激活函数的几何性质,将n维欧式空间划分为许多超矩形区域。利用Schauder不动点定理和严格对角占优矩阵,给出了神经网络在各超矩形区域内平衡点存在唯一性的几个充分条件,证明了保证神经网络平衡点的局部渐近稳定性和其它平衡点的不稳定性的充分条件,估计了局部稳定平衡点的吸引域。估计得到的离散神经网络局部稳定平衡点的吸引域是超球形区域,可以比原矩形区域大。在没有其他条件的情况下,估计得出的四元数神经网络局部稳定平衡点的吸引域是超矩形区域,而且肯定比原来的矩形区域要大。(2)不饱和分段线性激活函数具有计算简单快速和避免梯度消失等优点,这种激活函数是许多成功的前馈神经网络的重要组成部分。针对具有不饱和分段线性激活函数的分数阶神经网络,研究了其概周期解的单稳定性和多稳定性,给出了一些全局Mittag-Leffler吸引集,并通过Ascoli-Arzela定理证明了全局Mittag-Leffler稳定概周期解的存在唯一性。利用局部正不变集,给出了保证概周期解的局部Mittag-Leffler稳定性的充分条件,证明了在每个正不变集内都存在一个局部Mittag-Leffler稳定的概周期解,所有轨迹都收敛于该正不变集内的这个周期轨迹。(3)讨论了具有非单调不连续激活函数和时变时滞的Takagi-Sugeno模糊神经网络概周期解的多稳定性问题。根据非单调不连续激活函数的几何性质,利用Ascoli-Arzela定理和不等式技术,证明了在一定条件下,该网络在某些超矩形区域具有局部指数稳定的概周期解,还估计了局部稳定概周期解的吸引域。理论成果包括有界性、全局吸引性、多稳定性、吸引域等,可推广到具有非单调不连续激活函数的Takagi-Sugeno模糊神经网络概周期解的单稳定性和多稳定性,弥补多稳定性在模糊神经网路领域的空白。(4)针对具有离散和有限分布时变时滞的惯性反应扩散神经网络和随时间切换的神经网络,提出了一种新的周期脉冲控制策略。为了降低全局一致指数收敛准则的保守性,提出了利用可调参数和矩阵二次、三次凸组合方法,研究了两种网络的有界性和Lagrange稳定性。利用压缩映射定理和脉冲时滞相关的LyapunovKrasovskii泛函方法,给出了周期解存在性、唯一性和全局指数稳定性的充分条件。需要指出的是,所述的Lyapunov-Krasovskii泛函包括三重积分项和新的四重积分项,将减少神经网络稳定性条件的保守性。即使原始神经网络模型是不稳定的,甚至发散的,两类神经网络也可以通过脉冲控制生成全局指数稳定的周期解。更多还原

【Abstract】 Artificial neural network is a mathematical model of biological neural network abstracted from the perspective of information processing.With the development of artificial neural networks,they have successfully solved many practical problems that are difficult to be solved by modern computers in the fields of pattern segmentation,intelligent robot,automatic control,predictive estimation,fault diagnosis and system identification,etc.,showing good intelligent characteristics,which mainly depends on the dynamic behavior of neural network.Multistability is a concept to describe the coexistence of multiple stable equilibrium states or periodic solutions.Such dynamical behavior is essential in some applications of neural networks,including image processing,pattern recognition,and associative memory storage.Hopfield neural networks have become the main model attracting a lot of interest in multistability research.In real life,periodic function can well describe the development process of systems,such as ecosystem,mechanical vibration,market supply and demand,transportation system,heartbeat and memory in biological activities,etc.,and these practical problems can be summarized as discussing the stability of periodic solutions of differential equations.Based on these,we study the multistability and control strategy for generating globally stable periodic solutions of Hopfield neural networks.In theoretical study of neural networks,the dynamic behavior of neural networks is closely related to time delay,uncertainty,random noise and diffusion.In the past two decades,many scholars have been devoted to the research on how to guarantee the global or local stability of Hopfield neural network under these factors.However,for the neural networks with impulse,time delay,and reaction-diffusion terms,how to obtain the existence and uniqueness conditions of globally stable periodic solutions with less conservatism by using matrix-based convex combinations and linear matrix inequality methods still need to be studied in depth.When piecewise linear,unsaturating piecewise linear and discontinuous nonmonotonic activation functions appear in discrete-time,continuous-time,fractional-order,Takagi-Sugeno fuzzy neural networks,how to analyze their monostability and multistability is a difficult problem.For unstable neural networks with time-varying delays,how to design impulsive controllers to generate globally stable periodic solutions need to be discussed.Aiming at these problems,we take discrete-time,continuous-time,fractional-order,TakagiSugeno fuzzy,time-dependent switching,inertial reaction-diffusion neural networks as research objects.From the geometric properties of the piecewise linear,unsaturating piecewise linear and discontinuous nonmonotonic activation function perspective,using strictly diagonally dominant matrix,contraction mapping,fixed point theorem,AscoliArzela theorem and convex combination method fully,constructing appropriate Lyapunov-Krasovskii functional,the main works accomplished in this paper are presented as follows:(1)Multistability results are developed for discrete-time neural networks and quaternion-valued neural networks.A class of piecewise linear activation function make the storage capacity for neural networks increase greatly.According to the geometric properties of piecewise linear activation functions,the n-dimensional Euclidean space is divided into many hyperrectangular regions.By using Schauder’s fixed point theorem and strictly diagonally dominant matrix,some sufficient conditions for the existence and uniqueness of equilibrium points in hyperrectangular regions of neural networks are given.Sufficient conditions are derived to guarantee the local asymptotic stability of the equilibrium points of the neural network and the instability of the other equilibrium points.The estimated attractive basins for discrete-time neural networks are hyperspherical regions,which can be larger than the original rectangular regions.Without any other condition,the attraction basins for quaternion-valued neural networks,given as rectangular regions,are larger than originally rectangular regions.(2)The unsaturating piecewise linear activation function has the advantages of simple fast calculation and avoiding gradient disappearance,and it is an important part of many successful feedforward neural networks.For fractional-order neural networks with unsaturating piecewise linear activation functions,we study the monostability and multistability results of almost-periodic solutions,some globally Mittag-Leffler attractive sets are given,and the existence and uniqueness of globally Mittag-Leffler stable almost-periodic solutions are demonstrated by using Ascoli-Arzela theorem.Sufficient criteria are demonstrated to guarantee the local Mittag-Leffler stability of almost-periodic solutions by using local positive invariant sets,it is proved that there exists a locally Mittag-Leffler stable almost-periodic solution in each positive invariant set,and all trajectories converge to the periodic trajectories in this positive invariant set.(3)The multistability problem of almost-periodic solutions of Takagi-Sugeno fuzzy neural networks with nonmonotonic discontinuous activation functions and time-varying delays is discussed.Based on the geometrical properties of the nonmonotonic discontinuous activation functions,by using Ascoli-Arzela theorem and the inequality techniques,it is demonstrated that under some reasonable conditions,the addressed networks have unique locally exponentially stable almost-periodic solution in some hyperrectangular regions,and we also estimate the attraction basins of the locally stable almost-periodic solutions.These results here,which include boundedness,globally attractivity,multiple stability,attraction basin,can be extended to monostability and multistability for almost-period solutions of Takagi-Sugeno fuzzy neural networks with nonmonotonic discontinuous activation functions,which fill the gap of multistability in fuzzy neural networks.(4)A new periodic impulsive control strategy is designed for inertial reaction-diffusion neural networks and time-dependent switching neural networks with discrete and finite distributed time-varying delays to generate globally exponentially stable periodic solutions.In order to reduce the conservatism of the global uniform exponential convergence criterion,a method based on adjustable parameters and matrix-based quadratic,cubic convex combination are proposed to study the boundedness and Lagrange stability of the addressed neural networks.Sufficient conditions for the existence,uniqueness,and globally exponential stability of periodic solutions are developed by utilizing contraction mapping theorem and impulse-delaydependent Lyapunov-Krasovskii functional method.It should be pointed out that the addressed Lyapunov-Krasovskii functionals include triple integral terms and new quadruple integral terms,which will reduce the conservatism of the stability conditions of the neural networks.Even if the original neural network model is unstable or even divergent,the two types of neural networks can generate globally exponentially stable periodic solutions through impulsive control.更多还原