几类动力系统的稳定性研究
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摘要
众所周知,动力系统中不可避免地存在时间滞后现象。时滞是影响系统稳定的重要因素之一,甚至带来振荡、分叉以及混沌等动力学行为。此外,动力系统的稳定性容易受到不可避免的系统误差,外部扰动,系统参数振动,系统信息不全等诸多不确定性因素的影响。因此,研究时滞以及不确定性对动力系统稳定性的影响就显得非常重要。在很多实际的系统中,如在物理电路、生物系统、化学反应过程中,随机因素的干扰在动力系统中起着非常重要的作用。因此,动力系统稳定性还需考虑随机因素的影响。近年来,动力系统的稳定性研究吸引了大量的研究人员的浓厚兴趣,并取得了丰富的结果。
本论文主要致力于几类动力系统的渐近稳定性和鲁棒稳定性的分析,获取了一些有意义的成果。其主要内容和创新之处可概述如下:
①具有两个累加时变时滞的不确定系统的鲁棒稳定性研究
在诸如网络控制系统等实际系统中,信号从系统的一个结点向另一个结点的传输过程中,要经历网络的几个组成部分。由于网络传输条件的变化,可能产生几个连续的、具有不同属性的时滞。本节基于一个新的具有几个连续累加时滞的系统模型,研究了不确定时滞系统的稳定性。我们仔细考虑了系统状态向量带有两个累加时滞的情况,得到了带两个连续时滞的不确定系统稳定的一些新的充分条件。其思想可以很容易的推广到带多个连续时滞的线性系统中。
②时变时滞神经网络与时滞区间相关的稳定性分析
对于许多具有实际意义的系统,时滞的下界并不一定为0,即时滞包含在一个有界的区间[τ,τ]内,其中τ> 0是区间的下界。由此,我们研究了一类时变时滞神经网络平衡点的时滞区间相关的稳定性。得到了几个与时滞区间相关和与时滞导数无关/相关的神经网络平衡点全局渐近稳定和鲁棒稳定的判定准则。
③基于时滞分段方法的静态递归神经网络的稳定性分析
利用时滞分段方法,研究了一类静态递归神经网络的全局渐近稳定性问题。得到了几个与时滞相关的静态时滞递归神经网络渐近稳定性的充分条件,该条件与已有结论相比不仅形式简单,而且具有更少的保守性。实验结果也表明,时滞分段技术对扩大时滞的上界是有效的。
④基于LMI方法的带区间变时滞基因调控网络的稳定性分析
研究了带区间变时滞的参数不确定基因调控网络的全局渐近稳定性和鲁棒稳定性问题。利用自由权值矩阵和LMI方法,首先得到了几个时滞区间相关和时滞导数相关/无关的时滞基因调控网络的全局渐近稳定判定条件。然后研究了基因调控网络的鲁棒稳定性问题。所得到稳定性条件克服了时变时滞导数必须小于1的限制,使得其适用范围更宽。由于采用了LMI方法,使得这些结果更易于验证。
⑤随机噪声对时滞基因调控网络的稳定性影响
由于细胞中的分子事件受到热力学波动和噪声过程的支配,基因表达可视作一个随机过程。特别是在分子数目较少或反应速率较慢时,这种影响的作用将更加显著。因此,基因调控网络较精确的模型应该包括随机噪声。本节研究带随机噪声干扰和区间时滞的不确定基因调控网络的全局渐近稳定性和鲁棒稳定性。得到了几个判断基因调控网络在均方意义下渐近稳定和鲁棒稳定的充分条件,这些条件刻画了随机噪声和时滞对基因调控网络稳定性的影响。
It is well known that time delay is unavoidable in dynamical systems. Time delays may affect the stability of the system, even lead to instability, oscillation or chaos phenomena. Furthermore, in the applications and designs of networks, some unavoidable uncertainties which result from using an approximate system model for simplicity, external perturbations, parameter fluctuations, and data errors, etc, must be integrated into the system model. Such time delays, parametric uncertainties may significantly influence on the overall behavior of a dynamical system. Hence, it is significant and of prime importance to consider the effect of time delay and parametric uncertainties on the stability property of dynamical systems. In many practical systems, such as in the physical circuits, biological systems, chemical reaction process, stochastic disturbances in dynamical systems play a very important role. Therefore, the stability of dynamical system must take into account their effect. Recently, the stability analysis of dynamical systems has attracted a large number of researchers, and a series of significant results have been established.
This dissertation focuses on the asymptotical and robust stability for several dynamical systems. The main contributions and originality contained in this dissertation are as follows:
①Robust stability analysis of uncertain systems with two additive time-varying delay components
Sometimes in practical situations, for example, in networked control system, however, signals transmitted from one point to another may experience a few segments of networks, which can possibly induce successive delays with different properties due to the variable network transmission conditions. The problem of stability analysis for uncertain systems is concerned. The systems are based on a new time-delay model proposed recently, which contains multiple successive delay components in the state. We consider the case where only two successive delay components appear in the state. As a result, some less conservative stability criteria are established for systems with two successive delay components and parameter uncertainties. And the idea behind the proposed results can be easily extended to systems with multiple successive delay components.
②Delay-range dependent stability of uncertain neural networks with interval time-varying delays
In practical engineering systems, time-varying delay is a time delay that varies in an intervalτ≤τ(t )≤τ, in which the lower boundτis not restricted to be 0. In this thesis, the stability analysis for neural networks with interval time-varying delays and parameter uncertainties is addressed and some delay-derivative-independent stability criteria are established in term of linear matrix inequality (LMI).
③Asymptotical stability analysis for static recurrent neural networks with time delay: delay fractioning approach
The asymptotical stability analysis for static recurrent neural networks with time delay is studied by means of a delay fractioning approach. Some delay-dependent asymptotical stability criteria for static recurrent neural networks with time delay are established. The obtained criteria not only have the advantage of simple form but also are less conservative than some existing ones in the literature. Experimental results also show that the delay fractioning approach is effective to expand the upper bound of time delay.
④Stability analysis for genetic regulatory networks with interval time-varying delays: an LMI approach
The asymptotical and robust stability of genetic regulatory networks with interval time-varying delays and parameter uncertainties is investigated. First, by employing some free-weighting matrices and linear matrix inequalities, new delay-range-dependent and delay-derivative-dependent/independent stability criteria are derived. Then, the robust stability of genetic regulatory networks with interval time-varying delays and parameter uncertainties is addressed. Furthermore, the rigorous requirement of other literatures that the time derivatives of time-varying delays must be smaller than one is abandoned in the proposed scheme. As a result, the new criteria have wider range of applications and are applicable to both fast and slow time-varying delays. Since the criteria are obtained by LMIs, the results are easily verified
⑤Stabilizing effects of stochastic noises in genetic regulatory networks with interval time-varying delays
As molecular events in cells dominated by the thermal fluctuations and noise, gene expression can be regarded as a random process. Especially in low copy number of molecules, or a slower reaction rate, the role of this effect will be more prominent. Therefore, gene regulatory network should be described by more accurate models which include random noise. The stabilizing effects of stochastic noises in genetic regulatory networks with interval time-varying delays are concerned amd some new stability criteria are established to guarantee the delayed genetic regulatory networks to be robustly asymptotically stable in the mean square. The obtained criteria characterize the aggregated effects of the stochastic noises and time-varying delays on the stability of the considered genetic regulatory networks.
本论文主要致力于几类动力系统的渐近稳定性和鲁棒稳定性的分析,获取了一些有意义的成果。其主要内容和创新之处可概述如下:
①具有两个累加时变时滞的不确定系统的鲁棒稳定性研究
在诸如网络控制系统等实际系统中,信号从系统的一个结点向另一个结点的传输过程中,要经历网络的几个组成部分。由于网络传输条件的变化,可能产生几个连续的、具有不同属性的时滞。本节基于一个新的具有几个连续累加时滞的系统模型,研究了不确定时滞系统的稳定性。我们仔细考虑了系统状态向量带有两个累加时滞的情况,得到了带两个连续时滞的不确定系统稳定的一些新的充分条件。其思想可以很容易的推广到带多个连续时滞的线性系统中。
②时变时滞神经网络与时滞区间相关的稳定性分析
对于许多具有实际意义的系统,时滞的下界并不一定为0,即时滞包含在一个有界的区间[τ,τ]内,其中τ> 0是区间的下界。由此,我们研究了一类时变时滞神经网络平衡点的时滞区间相关的稳定性。得到了几个与时滞区间相关和与时滞导数无关/相关的神经网络平衡点全局渐近稳定和鲁棒稳定的判定准则。
③基于时滞分段方法的静态递归神经网络的稳定性分析
利用时滞分段方法,研究了一类静态递归神经网络的全局渐近稳定性问题。得到了几个与时滞相关的静态时滞递归神经网络渐近稳定性的充分条件,该条件与已有结论相比不仅形式简单,而且具有更少的保守性。实验结果也表明,时滞分段技术对扩大时滞的上界是有效的。
④基于LMI方法的带区间变时滞基因调控网络的稳定性分析
研究了带区间变时滞的参数不确定基因调控网络的全局渐近稳定性和鲁棒稳定性问题。利用自由权值矩阵和LMI方法,首先得到了几个时滞区间相关和时滞导数相关/无关的时滞基因调控网络的全局渐近稳定判定条件。然后研究了基因调控网络的鲁棒稳定性问题。所得到稳定性条件克服了时变时滞导数必须小于1的限制,使得其适用范围更宽。由于采用了LMI方法,使得这些结果更易于验证。
⑤随机噪声对时滞基因调控网络的稳定性影响
由于细胞中的分子事件受到热力学波动和噪声过程的支配,基因表达可视作一个随机过程。特别是在分子数目较少或反应速率较慢时,这种影响的作用将更加显著。因此,基因调控网络较精确的模型应该包括随机噪声。本节研究带随机噪声干扰和区间时滞的不确定基因调控网络的全局渐近稳定性和鲁棒稳定性。得到了几个判断基因调控网络在均方意义下渐近稳定和鲁棒稳定的充分条件,这些条件刻画了随机噪声和时滞对基因调控网络稳定性的影响。
It is well known that time delay is unavoidable in dynamical systems. Time delays may affect the stability of the system, even lead to instability, oscillation or chaos phenomena. Furthermore, in the applications and designs of networks, some unavoidable uncertainties which result from using an approximate system model for simplicity, external perturbations, parameter fluctuations, and data errors, etc, must be integrated into the system model. Such time delays, parametric uncertainties may significantly influence on the overall behavior of a dynamical system. Hence, it is significant and of prime importance to consider the effect of time delay and parametric uncertainties on the stability property of dynamical systems. In many practical systems, such as in the physical circuits, biological systems, chemical reaction process, stochastic disturbances in dynamical systems play a very important role. Therefore, the stability of dynamical system must take into account their effect. Recently, the stability analysis of dynamical systems has attracted a large number of researchers, and a series of significant results have been established.
This dissertation focuses on the asymptotical and robust stability for several dynamical systems. The main contributions and originality contained in this dissertation are as follows:
①Robust stability analysis of uncertain systems with two additive time-varying delay components
Sometimes in practical situations, for example, in networked control system, however, signals transmitted from one point to another may experience a few segments of networks, which can possibly induce successive delays with different properties due to the variable network transmission conditions. The problem of stability analysis for uncertain systems is concerned. The systems are based on a new time-delay model proposed recently, which contains multiple successive delay components in the state. We consider the case where only two successive delay components appear in the state. As a result, some less conservative stability criteria are established for systems with two successive delay components and parameter uncertainties. And the idea behind the proposed results can be easily extended to systems with multiple successive delay components.
②Delay-range dependent stability of uncertain neural networks with interval time-varying delays
In practical engineering systems, time-varying delay is a time delay that varies in an intervalτ≤τ(t )≤τ, in which the lower boundτis not restricted to be 0. In this thesis, the stability analysis for neural networks with interval time-varying delays and parameter uncertainties is addressed and some delay-derivative-independent stability criteria are established in term of linear matrix inequality (LMI).
③Asymptotical stability analysis for static recurrent neural networks with time delay: delay fractioning approach
The asymptotical stability analysis for static recurrent neural networks with time delay is studied by means of a delay fractioning approach. Some delay-dependent asymptotical stability criteria for static recurrent neural networks with time delay are established. The obtained criteria not only have the advantage of simple form but also are less conservative than some existing ones in the literature. Experimental results also show that the delay fractioning approach is effective to expand the upper bound of time delay.
④Stability analysis for genetic regulatory networks with interval time-varying delays: an LMI approach
The asymptotical and robust stability of genetic regulatory networks with interval time-varying delays and parameter uncertainties is investigated. First, by employing some free-weighting matrices and linear matrix inequalities, new delay-range-dependent and delay-derivative-dependent/independent stability criteria are derived. Then, the robust stability of genetic regulatory networks with interval time-varying delays and parameter uncertainties is addressed. Furthermore, the rigorous requirement of other literatures that the time derivatives of time-varying delays must be smaller than one is abandoned in the proposed scheme. As a result, the new criteria have wider range of applications and are applicable to both fast and slow time-varying delays. Since the criteria are obtained by LMIs, the results are easily verified
⑤Stabilizing effects of stochastic noises in genetic regulatory networks with interval time-varying delays
As molecular events in cells dominated by the thermal fluctuations and noise, gene expression can be regarded as a random process. Especially in low copy number of molecules, or a slower reaction rate, the role of this effect will be more prominent. Therefore, gene regulatory network should be described by more accurate models which include random noise. The stabilizing effects of stochastic noises in genetic regulatory networks with interval time-varying delays are concerned amd some new stability criteria are established to guarantee the delayed genetic regulatory networks to be robustly asymptotically stable in the mean square. The obtained criteria characterize the aggregated effects of the stochastic noises and time-varying delays on the stability of the considered genetic regulatory networks.
引文
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