时滞递归神经网络稳定性分析及网络化同步控制
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摘要
人工神经网络(Neural Networks)是一种高度非线性的动力学系统,若进行适当的参数配置其能够表现出大量复杂的动态行为。近年来,由于神经网络在信号处理、模式识别、动态图像处理、保密通信以及全局优化等领域的成功应用,对神经网络动力学行为的探讨引起了人们的广泛关注。神经网络系统是自适应自组织的,其在工程中的很多应用和系统本身的稳定性密切相关,而系统中信号传输时滞的存在往往会导致网络的持续振荡甚至出现混沌现象,因此,对于时滞神经网络稳定性(包括渐近稳定、指数稳定、绝对稳定等)的研究具有一定的理论意义和实际应用价值,近年来也出现了不少研究成果。与此同时,对时滞神经网络的同步控制,尤其是对混沌神经网络同步控制问题的研究也取得了较大的进展。就研究的内容而言本论文可分为两部分,前一部分以Lyapunov稳定性理论、S-过程、线性矩阵不等式(Linear Matrix Inequalities)技术等为主要研究工具,针对含有各种类型时滞(如常数时滞,区间时滞,概率分布已知的时变时滞等)的递归神经网络(RecurrentNeural Networks)分别研究他们的稳定性,并给出了相应的稳定性判据;论文的后一部分探讨了时滞神经网络网络化的同步控制。与现有结果相比,论文在这两个方面均得到了一些较深刻的结果。
从研究对象上看,本论文建立了两个新模型:(1)提出了一种基于时滞概率分布的递归神经网络模型并给出了相应的稳定性判别方法。在现有文献中,神经网络系统中所包含的时滞都是确定性的,所得稳定性判据可分为时滞依赖的(delay-dependent)和非时滞依赖的(delay-independent)两种。一般而言,时滞依赖的稳定性判据比非时滞依赖的能提供更小的保守性,特别是当时滞相对较小时。然而,现有的时滞依赖的稳定性判据只和时滞的变化范围有关,若对于一些神经网络中含有较大的时滞,但是发生较大时滞的概率却非常小的情况,这些确定性的方法显得比较保守。事实上,在许多实际应用系统中,时滞的存在往往具有一定的概率分布规律。本论文中充分利用这种时滞的分布规律对时滞神经网络系统进行建模,通过引入合适的随机变量将时滞的概率分布特性转化到新系统的系数矩阵中。进而利用Lyapunov函数法和线性矩阵不等式技术,得到了一些使时滞递归神经网络稳定的时滞分布依赖(delay-distribution-dependent)的判据。(2)提出了基于一般通信网络的时滞神经网络网络化同步控制(Networked Synchronization Control)模型,并给出了相应的控制设计策略。在同步控制系统的应用过程中,有时候信号的传输需要依赖于一般的通信网络。两个或多个神经网络之间通过公共网络平台协同工作,传递信息的同步系统被称为神经网络网络化同步控制系统。由于网络中信号的传递过程可能会经受网络诱导时滞、数据包丢失、外部随机干扰以及高误码率等不利因素的影响,从而使现有的神经网络同步控制策略不能直接应用到网络化同步控制系统中。针对这一问题,本论文提出了神经网络网络化同步控制的数学模型,并利用Lyapunov函数法以及线性矩阵不等式技术给出了相应的控制设计方法。
从研究方法上看,本论文所做的主要工作有:(1)提出了三个基于线性矩阵不等式的引理。这些引理是现有线性矩阵不等式规则集的补充,同时对改进现有的Lyapunov函数法与自由权值矩阵技术(Free-weighting Matrix Technique)相结合的稳定性分析方法起着重大的作用,因此具有一定的理论意义和实际应用价值.(2)提出了基于分段系统(Piecewise)分析方法的时滞区间中点法。通过引入时滞区间的中点,构造合适的Lyapunov函数,利用分段系统的研究方法探讨含有区间时滞的神经网络鲁棒全局渐近稳定性问题,改进了现有的判据。(3)通过构建新颖的Lyapunov-Krasovskii泛函并利用改进的自由权值技术,得到了使时滞随机神经网络全局指数稳定的新判据。
具体地说本论文研究的内容包括如下几方面:
(1)区间时滞递归神经网络鲁棒全局渐近稳定性分析
首先研究区间时滞神经网络的一个特例,即时滞的下界为零时的情况。通过构建合适的Lyapunov-Krasovskii泛函分别研究在慢变时滞和快变时滞情况下递归神经网络的鲁棒全局渐近稳定性。利用Lyapunov函数法进行稳定性研究,所得判据的性能不仅仅和Lyapunov函数的选取有关,而且和Lyapunov函数导数上界估计的方法有关。在现有的Lyapunov函数法和自由权值矩阵技术相结合的分析方法中,当进行Lyapunov函数导数上界估计时,有一些影响判据性能的项被丢弃掉。本论文通过深入发掘矩阵不等式中隐含的凸关系,提出了一个基于线性矩阵不等式的引理,该引理是现有线性矩阵不等式规则集的有力补充。通过这一引理,本文改进了现有时滞神经网络系统稳定性研究的方法,得到了具有较小保守性的判据。
另一方面,在现有的时滞神经网络稳定性分析的研究中,时滞的下界往往假设为零。现实生活中存在着许多重要的时滞系统,其时滞区间的下界是非零的。本论文针对含有一般区间时滞的神经网络,分别研究了连续型和离散型神经网络的鲁棒全局渐近稳定性。在分析过程中,通过引入时滞区间的中点,建立合适的Lyapunov-Krasovskii泛函。在分析方法上,根据时滞所取的不同区间将递归神经网络系统视为一个非线性分段时滞系统,从而提出了基于分段系统分析方法的时滞区间中点法,得到了使区间时滞神经网络鲁棒全局渐近稳定的新判据。
(2)随机时滞递归神经网络鲁棒全局指数稳定性分析
由于实现神经网络的电容、电阻、放大器等电子器件本身就存在一定的参数不确定性,以及神经网络运行过程中不可避免地会受到外部的随机干扰,因此建立的神经网络模型往往为含参数不确定项的随机模型。本论文通过构建合适的Lyapunov-Krasovskii泛函并利用改进的自由权值技术,分析一类含有参数不确定性的随机神经网络鲁棒全局指数稳定性,得到了使所述神经网络鲁棒全局指数稳定的新判据。
(3)时滞分布依赖的时滞递归神经网络稳定性判据
根据确定性时滞所得稳定性判据往往只和时滞区间的上界和下界有关。针对时滞出现概率分布规律的神经网络系统,探讨新颖的神经网络建模方法和相应的稳定性判别方法。首先给出了这类神经网络的数学建模。通过引入满足Bernoulli二项分布的随机变量,论文将时滞的概率分布转化到系统状态方程的系数矩阵中。根据神经网络类型的不同建立相应的Lyapunov函数,分别研究了离散型递归神经网络、带参数不确定项的离散型随机神经网络以及连续型递归神经网络的稳定性问题,得出了使相应的神经网络稳定的时滞分布依赖的判据,并将所得判据和现有结果进行了比较。这种建模方法能更细致地描述系统中时滞的存在对系统稳定性的影响,同时,也可以推广到其他动力系统或控制系统中。
(4)时滞递归神经网络的网络化同步控制
针对时滞递归神经网络的网络化同步问题进行了研究。首先建立了时滞递归神经网络网络化同步控制的系统模型。在该模型中,给出了同步控制系统中用于信号传输的网络的一般数学描述,并对网络的性能指标进行了定义。分析了随机神经网络网络化同步控制的基本思想,并利用Lyapunov函数法和线性矩阵不等式技术得到了网络化指数同步的充分条件以及相应的控制设计。论文给出了一个网络化混沌神经网络同步的数值例子,并通过TrueTime仿真平台验证了所得结论。进而,论文中根据网络性能指标的概率分布规律,改进了网络的描述方法。在前面章节所得研究成果的基础上,论文得到了使时滞递归神经网络网络化同步控制的网络QoS(Quality of Service)分布依赖的判据和相应的控制设计。
Neural networks(NNs)are a kind of nonlinear dynamic systems.They can exhibit a great of complex dynamic behaviors when the parameters are properly selected.Since a lot of successful applications have been witnessed in many fields, such as signal processing,pattern recognition,dynamic image processing,secure communication,global optimization,and so on,the study of dynamic behaviors of NNs have attracted considerable attention in recent years.NNs are adaptive and self-organizing systems and a lot of their engineering applications are strongly relative to their stability.On the other hand,time delays inevitably exist in NNs and are frequently important sources of oscillation or even chaos.Consequently,the studies of stability,including asymptotical stability,exponential stability,absolute stability,etc.,of NNs with time delays are of profound theoretical and practical significance.Also,a lot of research results on them have emerged in recent years. Meanwhile,many efforts have also been performed on the synchronization control of delayed NNs,especially on the synchronization of chaotic NNs.This thesis contains two parts in broad outline.In the former,it mainly focuses on the studies of the stability problems for recurrent neural networks(RNNs)with various time delays, including constant time delays,interval time delays and time delays with known probability distribution.By utilizing many useful tools,such as Lyapunov stability theory,S-procedure,linear matrix inequalities(LMIs)and so on,the corresponding stability criteria are achieved.In the latter,the networked synchronization control problems for RNNs with time delays are considered.
As far as the research subjects are concerned,we develop two new models in this thesis.(1)Establish a probability-distribution-dependent model for a class of delayed RNNs and provide corresponding stability criteria.In the existing references, the time delays in NNs are deterministic and the achieved stability criteria can be divided into two categories:delay-independent and delay-dependent.Generally speaking,the delay-dependent criteria can lead to less conservative results than these by the delay-independent criteria,especially when the time delays are small enough.However,these stability criteria were derived based only on the information of variation range of the time delays.When it comes to the case that some values of the time delays are very large but the probabilities of the delays taking such large values are very small,the existing methods may lead to more conservative results. Actually,the time delays in some NNs are often existent in a stochastic fashion and their probabilistic characteristics can often be measured by statistical methods. Considering these probability distributions,a new modeling method is introduced by translating the distribution probabilities of the time delays into parameter matrices of the transferred systems.Consequently,by combining the Lyapunov method and LMI technique,some delay-distribution-dependent stability criteria of the delayed RNNs are achieved.(2)Introduce a new model for networked synchronization control of delayed RNNs and provide the corresponding control design method.In real applications of many synchronization control systems,the signal transmissions sometimes have to relay on the common communication networks.Two or more NNs which are either chaotic or periodic sharing a common dynamic behavior by coupling or external forcing and transmitting signals mutually through common communication networks fall into the category of networked synchronization control of NNs. Since the communication will suffer from network-induced time delays,high rate of frame losses,high rate of bit errors,environment disturbance,and so on,the existing synchronization control schemes can not work well in the networked synchronization control systems.On this issue,a new mathematic model of networked synchronization control is introduced,in which the network performances are indexed and the stochastic fluctuations are described in term of a Brownian motion.By using Lyapunov method and some well known inequalities,the exponential mean-square networked synchronization control problems of delayed RNNs are concerned about and the corresponding control design approaches are provided.
As far as the analysis methods are concerned,this thesis contains the following: (1)Develop some LMI-based lemmas.These lemmas are important complements of the existing rule sets of LMI theory.Also,they play a great role in improving the existing stability criteria of delayed RNNs.(2)Introduce a new kind of delay-centralpoint method which is based on the piecewise analysis method.By introducing the center point of the time delay variation interval,developing appropriative Lyapunov functions and employing the piecewise analysis method,some sufficient conditions are achieved which ensure the stability of a class of RNNs with interval time delays.(3) By developing a new Lyapunov-Krasovskii functional and employing the improved LMI technique,the robust global exponential stability problem of a class of stochastic RNNs is investigated.
More specifically,the main contents of this thesis are as follows:
(1)Robust global asymptotical stability of RNNs with interval time delays
Firstly,a special case is considered,where the lower bound of the time delay interval is assumed to be zero.By constructing an appropriative Lyapunov-Krasovskii functional,this thesis studies the robust global asymptotical stability problem of a class of RNNs with slow time-varying delays and fast time-varying delays,respectively. It is well known that the performance of a gained criterion depends not only on the construction of Lyapunov functional,but also on the approach to estimate the upper bound of derivative of the Lyapunov functional.However,in the existing methods,some convexities in certain matrix equation are often ignored when estimating the upper bound of derivative of the Lyapunov functional.In this thesis, the convexity in the matrix inequality is picked up,and a new lemma is developed which is a supplement of the existing LMIs rule sets.Furthermore,this thesis improves the existing stability analysis approach of combining the Lyapunov method and free-weighting matrix technique.
On the other hand,the variation intervals of time delay are often assumed to be started from zero in the existing references.Actually,the lower bounds of time delay intervals are not zero in many important systems.This thesis analyzes the global asymptotical stability problem of continuous-time RNNs and discretetime RNNs with interval time delays,respectively.The delay-central-point method which is latest reported in the control system is reconsidered and a delay-centralpoint approach which is based on the piecewise analysis method is introduced to investigate the stability of RNNs with interval time delays.The existing robust global asymptotical stability criteria for continuous-time and for discrete-time RNNs, respectively,are both improved in this thesis.
(2)Robust global exponential stability of stochastic RNNs with time-varying delays
In real systems,the connection weights of the neurons depend on certain resistance and capacitance values which include uncertainties.On the other hand, the information storage and neurotransmission frequently suffer from the stochastic fluctuations.Therefore,when designing a neural network,both the parameter uncertainties and the stochastic fluctuations should be involved.By constructing a novel Lyapunov-Krasovskii functional and reconsidering the existing free-weighting matrix technique,some sufficient conditions of robust global exponential stability for a class of stochastic RNNs with norm-bounded parameter uncertainties are achieved,which can provide less conservatism than currently available stability criteria.
(3)Delay-distribution-dependent stability criteria for a class of RNNs
The existing stability criteria are often dependent on the variation range of time delays in term of the deterministic time delays.The stability problems for a class of RNNs in which the probability distribution of time delays can be obtained by statistical methods are studied.A new modeling method is firstly introduced. By utilizing a stochastic variable which satisfies Bernoulli random binary distribution, the probability distribution of the time delay is translated into the transformed NNs'parameter matrices.In terms of different kinds of RNNs,appropriative Lyapunov-Krasovskii functions are developed and distinguished analysis techniques are performed.Some delay-distribution-dependent stability criteria are provided for discrete-time RNNs,discrete-time stochastic RNNs with norm-bounded parameter uncertainties and continuous-time RNNs,respectively.Also,some comparisons are performed with the existing results.In addition,this modeling method can describe the time delays more exactly and can be extended to many other dynamic systems and control systems.
(4)Networked synchronization control of RNNs with time delays
The synchronization control methods of a class of RNNs with time delays are studied,in which the reference signals are transmitted through common communication networks.The networked synchronization models are firstly given in this thesis and the communication networks'Quality of Services(QoS)are indexed.The elementary principles of networked synchronization Control of stochastic RNNs are introduced.By utilizing the Lyapunov method and exploiting LMIs technique,the problem of mean-square exponential synchronization control of stochastic delayed RNNs is concerned about and the gained results are verified on the TrueTime platform. Then,according to the probability distribution of the indexed QoS of networks, the mathematic model of the network is improved and some new results based on the probability distribution of the communication networks'QoS are achieved.
从研究对象上看,本论文建立了两个新模型:(1)提出了一种基于时滞概率分布的递归神经网络模型并给出了相应的稳定性判别方法。在现有文献中,神经网络系统中所包含的时滞都是确定性的,所得稳定性判据可分为时滞依赖的(delay-dependent)和非时滞依赖的(delay-independent)两种。一般而言,时滞依赖的稳定性判据比非时滞依赖的能提供更小的保守性,特别是当时滞相对较小时。然而,现有的时滞依赖的稳定性判据只和时滞的变化范围有关,若对于一些神经网络中含有较大的时滞,但是发生较大时滞的概率却非常小的情况,这些确定性的方法显得比较保守。事实上,在许多实际应用系统中,时滞的存在往往具有一定的概率分布规律。本论文中充分利用这种时滞的分布规律对时滞神经网络系统进行建模,通过引入合适的随机变量将时滞的概率分布特性转化到新系统的系数矩阵中。进而利用Lyapunov函数法和线性矩阵不等式技术,得到了一些使时滞递归神经网络稳定的时滞分布依赖(delay-distribution-dependent)的判据。(2)提出了基于一般通信网络的时滞神经网络网络化同步控制(Networked Synchronization Control)模型,并给出了相应的控制设计策略。在同步控制系统的应用过程中,有时候信号的传输需要依赖于一般的通信网络。两个或多个神经网络之间通过公共网络平台协同工作,传递信息的同步系统被称为神经网络网络化同步控制系统。由于网络中信号的传递过程可能会经受网络诱导时滞、数据包丢失、外部随机干扰以及高误码率等不利因素的影响,从而使现有的神经网络同步控制策略不能直接应用到网络化同步控制系统中。针对这一问题,本论文提出了神经网络网络化同步控制的数学模型,并利用Lyapunov函数法以及线性矩阵不等式技术给出了相应的控制设计方法。
从研究方法上看,本论文所做的主要工作有:(1)提出了三个基于线性矩阵不等式的引理。这些引理是现有线性矩阵不等式规则集的补充,同时对改进现有的Lyapunov函数法与自由权值矩阵技术(Free-weighting Matrix Technique)相结合的稳定性分析方法起着重大的作用,因此具有一定的理论意义和实际应用价值.(2)提出了基于分段系统(Piecewise)分析方法的时滞区间中点法。通过引入时滞区间的中点,构造合适的Lyapunov函数,利用分段系统的研究方法探讨含有区间时滞的神经网络鲁棒全局渐近稳定性问题,改进了现有的判据。(3)通过构建新颖的Lyapunov-Krasovskii泛函并利用改进的自由权值技术,得到了使时滞随机神经网络全局指数稳定的新判据。
具体地说本论文研究的内容包括如下几方面:
(1)区间时滞递归神经网络鲁棒全局渐近稳定性分析
首先研究区间时滞神经网络的一个特例,即时滞的下界为零时的情况。通过构建合适的Lyapunov-Krasovskii泛函分别研究在慢变时滞和快变时滞情况下递归神经网络的鲁棒全局渐近稳定性。利用Lyapunov函数法进行稳定性研究,所得判据的性能不仅仅和Lyapunov函数的选取有关,而且和Lyapunov函数导数上界估计的方法有关。在现有的Lyapunov函数法和自由权值矩阵技术相结合的分析方法中,当进行Lyapunov函数导数上界估计时,有一些影响判据性能的项被丢弃掉。本论文通过深入发掘矩阵不等式中隐含的凸关系,提出了一个基于线性矩阵不等式的引理,该引理是现有线性矩阵不等式规则集的有力补充。通过这一引理,本文改进了现有时滞神经网络系统稳定性研究的方法,得到了具有较小保守性的判据。
另一方面,在现有的时滞神经网络稳定性分析的研究中,时滞的下界往往假设为零。现实生活中存在着许多重要的时滞系统,其时滞区间的下界是非零的。本论文针对含有一般区间时滞的神经网络,分别研究了连续型和离散型神经网络的鲁棒全局渐近稳定性。在分析过程中,通过引入时滞区间的中点,建立合适的Lyapunov-Krasovskii泛函。在分析方法上,根据时滞所取的不同区间将递归神经网络系统视为一个非线性分段时滞系统,从而提出了基于分段系统分析方法的时滞区间中点法,得到了使区间时滞神经网络鲁棒全局渐近稳定的新判据。
(2)随机时滞递归神经网络鲁棒全局指数稳定性分析
由于实现神经网络的电容、电阻、放大器等电子器件本身就存在一定的参数不确定性,以及神经网络运行过程中不可避免地会受到外部的随机干扰,因此建立的神经网络模型往往为含参数不确定项的随机模型。本论文通过构建合适的Lyapunov-Krasovskii泛函并利用改进的自由权值技术,分析一类含有参数不确定性的随机神经网络鲁棒全局指数稳定性,得到了使所述神经网络鲁棒全局指数稳定的新判据。
(3)时滞分布依赖的时滞递归神经网络稳定性判据
根据确定性时滞所得稳定性判据往往只和时滞区间的上界和下界有关。针对时滞出现概率分布规律的神经网络系统,探讨新颖的神经网络建模方法和相应的稳定性判别方法。首先给出了这类神经网络的数学建模。通过引入满足Bernoulli二项分布的随机变量,论文将时滞的概率分布转化到系统状态方程的系数矩阵中。根据神经网络类型的不同建立相应的Lyapunov函数,分别研究了离散型递归神经网络、带参数不确定项的离散型随机神经网络以及连续型递归神经网络的稳定性问题,得出了使相应的神经网络稳定的时滞分布依赖的判据,并将所得判据和现有结果进行了比较。这种建模方法能更细致地描述系统中时滞的存在对系统稳定性的影响,同时,也可以推广到其他动力系统或控制系统中。
(4)时滞递归神经网络的网络化同步控制
针对时滞递归神经网络的网络化同步问题进行了研究。首先建立了时滞递归神经网络网络化同步控制的系统模型。在该模型中,给出了同步控制系统中用于信号传输的网络的一般数学描述,并对网络的性能指标进行了定义。分析了随机神经网络网络化同步控制的基本思想,并利用Lyapunov函数法和线性矩阵不等式技术得到了网络化指数同步的充分条件以及相应的控制设计。论文给出了一个网络化混沌神经网络同步的数值例子,并通过TrueTime仿真平台验证了所得结论。进而,论文中根据网络性能指标的概率分布规律,改进了网络的描述方法。在前面章节所得研究成果的基础上,论文得到了使时滞递归神经网络网络化同步控制的网络QoS(Quality of Service)分布依赖的判据和相应的控制设计。
Neural networks(NNs)are a kind of nonlinear dynamic systems.They can exhibit a great of complex dynamic behaviors when the parameters are properly selected.Since a lot of successful applications have been witnessed in many fields, such as signal processing,pattern recognition,dynamic image processing,secure communication,global optimization,and so on,the study of dynamic behaviors of NNs have attracted considerable attention in recent years.NNs are adaptive and self-organizing systems and a lot of their engineering applications are strongly relative to their stability.On the other hand,time delays inevitably exist in NNs and are frequently important sources of oscillation or even chaos.Consequently,the studies of stability,including asymptotical stability,exponential stability,absolute stability,etc.,of NNs with time delays are of profound theoretical and practical significance.Also,a lot of research results on them have emerged in recent years. Meanwhile,many efforts have also been performed on the synchronization control of delayed NNs,especially on the synchronization of chaotic NNs.This thesis contains two parts in broad outline.In the former,it mainly focuses on the studies of the stability problems for recurrent neural networks(RNNs)with various time delays, including constant time delays,interval time delays and time delays with known probability distribution.By utilizing many useful tools,such as Lyapunov stability theory,S-procedure,linear matrix inequalities(LMIs)and so on,the corresponding stability criteria are achieved.In the latter,the networked synchronization control problems for RNNs with time delays are considered.
As far as the research subjects are concerned,we develop two new models in this thesis.(1)Establish a probability-distribution-dependent model for a class of delayed RNNs and provide corresponding stability criteria.In the existing references, the time delays in NNs are deterministic and the achieved stability criteria can be divided into two categories:delay-independent and delay-dependent.Generally speaking,the delay-dependent criteria can lead to less conservative results than these by the delay-independent criteria,especially when the time delays are small enough.However,these stability criteria were derived based only on the information of variation range of the time delays.When it comes to the case that some values of the time delays are very large but the probabilities of the delays taking such large values are very small,the existing methods may lead to more conservative results. Actually,the time delays in some NNs are often existent in a stochastic fashion and their probabilistic characteristics can often be measured by statistical methods. Considering these probability distributions,a new modeling method is introduced by translating the distribution probabilities of the time delays into parameter matrices of the transferred systems.Consequently,by combining the Lyapunov method and LMI technique,some delay-distribution-dependent stability criteria of the delayed RNNs are achieved.(2)Introduce a new model for networked synchronization control of delayed RNNs and provide the corresponding control design method.In real applications of many synchronization control systems,the signal transmissions sometimes have to relay on the common communication networks.Two or more NNs which are either chaotic or periodic sharing a common dynamic behavior by coupling or external forcing and transmitting signals mutually through common communication networks fall into the category of networked synchronization control of NNs. Since the communication will suffer from network-induced time delays,high rate of frame losses,high rate of bit errors,environment disturbance,and so on,the existing synchronization control schemes can not work well in the networked synchronization control systems.On this issue,a new mathematic model of networked synchronization control is introduced,in which the network performances are indexed and the stochastic fluctuations are described in term of a Brownian motion.By using Lyapunov method and some well known inequalities,the exponential mean-square networked synchronization control problems of delayed RNNs are concerned about and the corresponding control design approaches are provided.
As far as the analysis methods are concerned,this thesis contains the following: (1)Develop some LMI-based lemmas.These lemmas are important complements of the existing rule sets of LMI theory.Also,they play a great role in improving the existing stability criteria of delayed RNNs.(2)Introduce a new kind of delay-centralpoint method which is based on the piecewise analysis method.By introducing the center point of the time delay variation interval,developing appropriative Lyapunov functions and employing the piecewise analysis method,some sufficient conditions are achieved which ensure the stability of a class of RNNs with interval time delays.(3) By developing a new Lyapunov-Krasovskii functional and employing the improved LMI technique,the robust global exponential stability problem of a class of stochastic RNNs is investigated.
More specifically,the main contents of this thesis are as follows:
(1)Robust global asymptotical stability of RNNs with interval time delays
Firstly,a special case is considered,where the lower bound of the time delay interval is assumed to be zero.By constructing an appropriative Lyapunov-Krasovskii functional,this thesis studies the robust global asymptotical stability problem of a class of RNNs with slow time-varying delays and fast time-varying delays,respectively. It is well known that the performance of a gained criterion depends not only on the construction of Lyapunov functional,but also on the approach to estimate the upper bound of derivative of the Lyapunov functional.However,in the existing methods,some convexities in certain matrix equation are often ignored when estimating the upper bound of derivative of the Lyapunov functional.In this thesis, the convexity in the matrix inequality is picked up,and a new lemma is developed which is a supplement of the existing LMIs rule sets.Furthermore,this thesis improves the existing stability analysis approach of combining the Lyapunov method and free-weighting matrix technique.
On the other hand,the variation intervals of time delay are often assumed to be started from zero in the existing references.Actually,the lower bounds of time delay intervals are not zero in many important systems.This thesis analyzes the global asymptotical stability problem of continuous-time RNNs and discretetime RNNs with interval time delays,respectively.The delay-central-point method which is latest reported in the control system is reconsidered and a delay-centralpoint approach which is based on the piecewise analysis method is introduced to investigate the stability of RNNs with interval time delays.The existing robust global asymptotical stability criteria for continuous-time and for discrete-time RNNs, respectively,are both improved in this thesis.
(2)Robust global exponential stability of stochastic RNNs with time-varying delays
In real systems,the connection weights of the neurons depend on certain resistance and capacitance values which include uncertainties.On the other hand, the information storage and neurotransmission frequently suffer from the stochastic fluctuations.Therefore,when designing a neural network,both the parameter uncertainties and the stochastic fluctuations should be involved.By constructing a novel Lyapunov-Krasovskii functional and reconsidering the existing free-weighting matrix technique,some sufficient conditions of robust global exponential stability for a class of stochastic RNNs with norm-bounded parameter uncertainties are achieved,which can provide less conservatism than currently available stability criteria.
(3)Delay-distribution-dependent stability criteria for a class of RNNs
The existing stability criteria are often dependent on the variation range of time delays in term of the deterministic time delays.The stability problems for a class of RNNs in which the probability distribution of time delays can be obtained by statistical methods are studied.A new modeling method is firstly introduced. By utilizing a stochastic variable which satisfies Bernoulli random binary distribution, the probability distribution of the time delay is translated into the transformed NNs'parameter matrices.In terms of different kinds of RNNs,appropriative Lyapunov-Krasovskii functions are developed and distinguished analysis techniques are performed.Some delay-distribution-dependent stability criteria are provided for discrete-time RNNs,discrete-time stochastic RNNs with norm-bounded parameter uncertainties and continuous-time RNNs,respectively.Also,some comparisons are performed with the existing results.In addition,this modeling method can describe the time delays more exactly and can be extended to many other dynamic systems and control systems.
(4)Networked synchronization control of RNNs with time delays
The synchronization control methods of a class of RNNs with time delays are studied,in which the reference signals are transmitted through common communication networks.The networked synchronization models are firstly given in this thesis and the communication networks'Quality of Services(QoS)are indexed.The elementary principles of networked synchronization Control of stochastic RNNs are introduced.By utilizing the Lyapunov method and exploiting LMIs technique,the problem of mean-square exponential synchronization control of stochastic delayed RNNs is concerned about and the gained results are verified on the TrueTime platform. Then,according to the probability distribution of the indexed QoS of networks, the mathematic model of the network is improved and some new results based on the probability distribution of the communication networks'QoS are achieved.
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