不确定时滞系统的鲁棒稳定性分析
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摘要
时滞广泛存在于众多的实际系统中,而时滞的存在是导致系统不稳定的主要原因之一。另一方面,系统的稳定性不可避免地受到系统误差、线性化、外部随机扰动、系统参数振动和系统信息不全等诸多不确定性因素的影响。鉴于此,研究不确定时滞系统的鲁棒稳定性具有重要的理论和实际意义。近年来,不确定时滞系统的稳定性研究吸引了国内外众多学者的的广泛关注。本文基于Lyapunov稳定性理论和随机理论,以自由权矩阵方法、积分等式方法(有限和等式)为手段处理交叉乘积项的定界问题,研究了中立时滞系统、离散时间系统、随机Hopfield神经网络及随机双向联想记忆(BAM)神经网络等几类时滞不确定动力系统的鲁棒稳定性,获得了如下成果：
(1)针对具有区间时变时滞的不确定中立系统和具有非线性扰动的中立系统,给出了两类系统基于Lyapunov-Krasovskii方法和线性不等式技术的鲁棒稳定性新准则。分别采用了改进的自由权矩阵方法和积分等式方法,有效地克服了现有文献在处理交叉项的定界时的保守估计。通过比较与仿真研究,可以看出本文所得到的稳定性条件较以往文献是具有先进性的。
(2)研究了一类具有分布已知的时滞离散时间系统的鲁棒稳定性。由于环境噪声的影响,时滞往往具有随机性。而现有文献所讨论的离散时间系统的时滞都为时变的形式,得到的条件仅与时滞的上下界或时滞变化范围有关。针对这一情况,本文提出一种具有随机时滞的离散时间系统模型,该模型的时滞分布已知,包含了现有文献的模型。基于Lyapunov稳定性理论和随机理论提出一种有限和等式方法,得到了时滞及时滞分布相关的鲁棒稳定性条件。仿真示例说明了,该方法对扩大时滞的上界是有效的。
(3)讨论了一类具有混合时滞的不确定随机Hopfield神经网络的平衡点的时滞区间相关的稳定性,通过构造一个新的Lyapunov-Krasovskii泛函,并利用积分等式方法获得了若干时滞区间相关的神经网络平衡点全局鲁棒渐近稳定的判定准则,刻画了随机扰动对时滞Hopfield神经网络稳定性的影响；研究了具有马尔科夫跳变和混合时滞的不确定Hopfield神经网络全局鲁棒指数稳定性,基于Lyapunov稳定性定理、随机理论和积分等式,得到了与时滞区间相关的判定准则,且可以任意选取衰减指数率。
(4)分别研究了一类具有模式相关的连续型Markov跳跃参数时滞BAM神经网络和一类具有Markov跳跃参数的随机离散型BAM神经网络的全局稳定性。连续模型中考虑了两种不同形式的随机扰动,激励函数为泛化的形式,较Lipschitz条件更一般。基于积分等式和Lyapunov稳定性定理,给出了系统的线性矩阵不等式形式的稳定性条件。对于离散时间模型,给出了基于Lyapunov稳定性定理及有限和等式的LMI稳定性判据。仿真结果表明判据是有效的。
Time delays widely occur in many practical systems, and may cause undesirable dynamic behaviors such as oscillation and instability. On the other hand, uncertainties are unavoidable due to modeling errors, measurement errors, and linearization approxi-mations, external perturbations, environmental noises and incomplete information of the system. The characteristics of dynamic systems are significantly affected by the pres-ence of the uncertainty, even to the extend of instability in extreme situation. In view of this, robust stability analysis of dynamic systems with time delays and uncertainties has received considerable attention by many researchers, a great deal of results have been reported in the literature. This dissertation focuses on the robust stability for several delayed dynamical systems with uncertainties such as neutral systems, stochastic Hop-field neural networks, discrete-time systems and bidirectional associative memory (BAM) neural networks based on Lyapunov stability theory, stochastic theory and linear matrix inequality (LMI) technique. The main contributions of this dissertation are as follows:
(1)The problem of robust stability for the uncertain neutral system with interval time varying discrete delay and the neutral system with nonlinear perturbations is investigated. Based on Lyapunov-Krasovskii functional method and LMI technique, several new criteria are derived. The developed free weight matrices technique and two integral equalities are employed to overcome the conservative estimation when dealing with the cross terms. Numerical examples are given to demonstrate the effectiveness and the improvement of the proposed method.
(2) Impacted on the surrounding environment noise, time delays tend to be ran-dom. It is worth noting that only the deterministic time delay case was concerned and the stability criteria were derived based only on the information of variation range or bounds of the time delay in the existing references. To this end, a new type of discrete system with random delay is proposed. The problem of robustly globally exponentially stable in the mean square sense for the proposed system is investigated. By defining a Lyapunov-Krasovskii functional by utilizing some new finite sum equalities for the bound-ing of cross term, a delay-distribution-dependent criterion is obtained. Different from the existing ones, the proposed criterion depends on not only the size of the delay but also the probability distribution of it. Numerical examples suggest that the results are effective.
(3)The problem of robustly stable in the mean square for a class of stochastic Hop-field neural networks with mixed time-varying delays and parameter uncertainties is in-vestigated. The time-varying discrete delay is assumed to belong to an interval. By defining a new Lyapunov-Krasovskii functional and employing the stochastic stability theory, several delay-range-dependent stability criteria are established in terms of linear matrix inequalities (LMIs), which describe the effect of the stochastic noise on the system stability. Two bounding integral equalities are utilized to reduce the conservativeness. Then, the problem of robustly exponentially stable in the mean square for delayed un-certain Hopfield neural networks with Markovian jumping parameters is studied. Based on Lyapunov-Krasovskii functional method and the integral equality, some delay-range-dependent exponential stability criteria are presented in terms of LMIs. The decay rate can be any finite positive value without any other constraints, which is more general than the conventional assumptions in some relative lectures.
(4)The stochastic stability problem for a class of Markovian jumping continuous time BAM neural networks with mode-dependent time delays and a class of Markovian jumping discrete time BAM neural networks with time delays are studied. Stochastic perturbation in two forms and generalized excitation function which is more general then Lipschitz conditions are considered in the former. Based on Lyapunov method and stochastic analysis techniques, mode-dependent criteria that ensure the mean-square stability of the system are deduced. The criteria are established in terms of LMIs. In view of the discrete model, we utilize the Lyapunov stability theorem and two finite sum equalities and obtain some stability conditions in terms of LMIs. Simulation results show the effectiveness of the proposed stability conditions.
(1)针对具有区间时变时滞的不确定中立系统和具有非线性扰动的中立系统,给出了两类系统基于Lyapunov-Krasovskii方法和线性不等式技术的鲁棒稳定性新准则。分别采用了改进的自由权矩阵方法和积分等式方法,有效地克服了现有文献在处理交叉项的定界时的保守估计。通过比较与仿真研究,可以看出本文所得到的稳定性条件较以往文献是具有先进性的。
(2)研究了一类具有分布已知的时滞离散时间系统的鲁棒稳定性。由于环境噪声的影响,时滞往往具有随机性。而现有文献所讨论的离散时间系统的时滞都为时变的形式,得到的条件仅与时滞的上下界或时滞变化范围有关。针对这一情况,本文提出一种具有随机时滞的离散时间系统模型,该模型的时滞分布已知,包含了现有文献的模型。基于Lyapunov稳定性理论和随机理论提出一种有限和等式方法,得到了时滞及时滞分布相关的鲁棒稳定性条件。仿真示例说明了,该方法对扩大时滞的上界是有效的。
(3)讨论了一类具有混合时滞的不确定随机Hopfield神经网络的平衡点的时滞区间相关的稳定性,通过构造一个新的Lyapunov-Krasovskii泛函,并利用积分等式方法获得了若干时滞区间相关的神经网络平衡点全局鲁棒渐近稳定的判定准则,刻画了随机扰动对时滞Hopfield神经网络稳定性的影响；研究了具有马尔科夫跳变和混合时滞的不确定Hopfield神经网络全局鲁棒指数稳定性,基于Lyapunov稳定性定理、随机理论和积分等式,得到了与时滞区间相关的判定准则,且可以任意选取衰减指数率。
(4)分别研究了一类具有模式相关的连续型Markov跳跃参数时滞BAM神经网络和一类具有Markov跳跃参数的随机离散型BAM神经网络的全局稳定性。连续模型中考虑了两种不同形式的随机扰动,激励函数为泛化的形式,较Lipschitz条件更一般。基于积分等式和Lyapunov稳定性定理,给出了系统的线性矩阵不等式形式的稳定性条件。对于离散时间模型,给出了基于Lyapunov稳定性定理及有限和等式的LMI稳定性判据。仿真结果表明判据是有效的。
Time delays widely occur in many practical systems, and may cause undesirable dynamic behaviors such as oscillation and instability. On the other hand, uncertainties are unavoidable due to modeling errors, measurement errors, and linearization approxi-mations, external perturbations, environmental noises and incomplete information of the system. The characteristics of dynamic systems are significantly affected by the pres-ence of the uncertainty, even to the extend of instability in extreme situation. In view of this, robust stability analysis of dynamic systems with time delays and uncertainties has received considerable attention by many researchers, a great deal of results have been reported in the literature. This dissertation focuses on the robust stability for several delayed dynamical systems with uncertainties such as neutral systems, stochastic Hop-field neural networks, discrete-time systems and bidirectional associative memory (BAM) neural networks based on Lyapunov stability theory, stochastic theory and linear matrix inequality (LMI) technique. The main contributions of this dissertation are as follows:
(1)The problem of robust stability for the uncertain neutral system with interval time varying discrete delay and the neutral system with nonlinear perturbations is investigated. Based on Lyapunov-Krasovskii functional method and LMI technique, several new criteria are derived. The developed free weight matrices technique and two integral equalities are employed to overcome the conservative estimation when dealing with the cross terms. Numerical examples are given to demonstrate the effectiveness and the improvement of the proposed method.
(2) Impacted on the surrounding environment noise, time delays tend to be ran-dom. It is worth noting that only the deterministic time delay case was concerned and the stability criteria were derived based only on the information of variation range or bounds of the time delay in the existing references. To this end, a new type of discrete system with random delay is proposed. The problem of robustly globally exponentially stable in the mean square sense for the proposed system is investigated. By defining a Lyapunov-Krasovskii functional by utilizing some new finite sum equalities for the bound-ing of cross term, a delay-distribution-dependent criterion is obtained. Different from the existing ones, the proposed criterion depends on not only the size of the delay but also the probability distribution of it. Numerical examples suggest that the results are effective.
(3)The problem of robustly stable in the mean square for a class of stochastic Hop-field neural networks with mixed time-varying delays and parameter uncertainties is in-vestigated. The time-varying discrete delay is assumed to belong to an interval. By defining a new Lyapunov-Krasovskii functional and employing the stochastic stability theory, several delay-range-dependent stability criteria are established in terms of linear matrix inequalities (LMIs), which describe the effect of the stochastic noise on the system stability. Two bounding integral equalities are utilized to reduce the conservativeness. Then, the problem of robustly exponentially stable in the mean square for delayed un-certain Hopfield neural networks with Markovian jumping parameters is studied. Based on Lyapunov-Krasovskii functional method and the integral equality, some delay-range-dependent exponential stability criteria are presented in terms of LMIs. The decay rate can be any finite positive value without any other constraints, which is more general than the conventional assumptions in some relative lectures.
(4)The stochastic stability problem for a class of Markovian jumping continuous time BAM neural networks with mode-dependent time delays and a class of Markovian jumping discrete time BAM neural networks with time delays are studied. Stochastic perturbation in two forms and generalized excitation function which is more general then Lipschitz conditions are considered in the former. Based on Lyapunov method and stochastic analysis techniques, mode-dependent criteria that ensure the mean-square stability of the system are deduced. The criteria are established in terms of LMIs. In view of the discrete model, we utilize the Lyapunov stability theorem and two finite sum equalities and obtain some stability conditions in terms of LMIs. Simulation results show the effectiveness of the proposed stability conditions.
引文
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