大系统关于部分变元的指数稳定性

本文利用Lyapunov函数法讨论了大系统关于部分变元的指数稳定性,得到了一些定理.通过把高阶系统看作低阶关联子系统的复合,从而使独立子系统的部分变元指数稳定性反映了整个系统同样的特性.     

时变离散大系统的稳定性

本文首先给出了线性时变离散系统稳定性的一个充分条件.然后研究当孤立子系统满足上述条件时的线性及非线性时变离散大系统的稳定性.利用向量李雅普诺夫函数法结合时变离散系统的比较原理,得到了时变离散大系统在稳定性中的集结模型.直接由集结系统的稳定性得到大系统稳定性的条件.     

一类变时滞神经网络的全局指数稳定性

本文研究一类变时滞神经网络平衡点的全局指数稳定性.在不要求激活函数全局Lipschitz条件下,利用Lyapunov函数方法,并结合Young不等式和Halanay时滞微分不等式,得到了系统全局指数稳定的充分条件.文末,一个数值例子用以说明本文结果的有效性.     

Exponential stability and applications of switched positive linear impulsive systems with time-varying delays and all unstable subsystems

The global uniform exponential stability of switched positive linear impulsive systems with time-varying delays and all unstable subsystems is studied in this paper, which includes two types of distributed time-varying delays and discrete time-varying delays. Switching behaviors dominating the switched systems can be either stabilizing and destabilizing in the new designed switching sequence. We design new linear programming algorithm process to find the feasible ratio of stabilizing switching behaviors, which can be compensated by unstable subsystems, destabilizing switching behaviors, and impulses. Specically, we add a kind of nonnegative impulses which is consistent with the switching behaviors for the systems. Employing a multiple co-positive Lyapunov-Krasovskii functional, we present several new sufficient stability criteria and design new switching sequence. Then, we apply the obtained stability criteria to the exponential consensus of linear delayed multi-agent systems, and obtain the new exponential consensus criteria. Three simulations are provided to demonstrate the proposed stability criteria.     

A novel approach to exponential stability of nonlinear systems with time-varying delays

In this paper, the stability of nonlinear systems with time-varying delays is investigated by means of the concepts of generalized Dahlquist constant, generalized relative Dahlquist constant and relative minimal Lipschitz constant. In detail, two sufficient conditions are derived for the exponential stability of nonlinear systems with time-varying delays and the exponential decay of the solutions is also estimated. Compared with some existing results, our stability conditions are less conservative. Some examples are given to illustrate the effectiveness of the obtained results.     

Stability analysis of time-varying switched systems via indefinite difference Lyapunov functions

Asymptotic stability of time-varying switched systems is investigated in this paper. The less conservative sufficient criteria for asymptotic stability of time-varying discrete-time switched systems are proposed via common indefinite difference Lyapunov functions and multiple indefinite difference Lyapunov functions introduced in this note, respectively. Common indefinite difference Lyapunov functions can be used to analyze stability of a switched system with asymptotic stable subsystems and arbitrary switching signal. Multiple indefinite difference Lyapunov functions can be used to investigate stability of a switched system with unstable subsystems and a given switching signal. The difference of the proposed Lyapunov function may be positive at some instants for an asymptotically stable subsystem. We compare these main results and illustrate the effectiveness of the obtained theorems by three numerical examples.     

On the Robust Practical Global Stability of Nonlinear Time-varying Systems

The goal of this paper is twofold. The first part presents a Lyapunov statement and proof of the concept of robust global practical exponential stability (RpGES) for nonlinear time varying systems. A RpGES-Lyapunov function for the overall system is exhibited. Its proof is a consequence of some results on converse Lyapunov theorems obtained by Tsinias in 19. The second part provides sufficient conditions for the robust practical globally uniformly asymptotically stability (RpGUAS).     

Stability of perturbed switched nonlinear systems with delays

This paper addresses the stability problems of perturbed switched nonlinear systems with time-varying delays. It is assumed that the nominal switched nonlinear system (perturbation-free system) is uniformly exponentially stable and that the perturbations satisfy a linear growth bound condition. It is revealed that there exists an upper bound of perturbation guaranteeing that the perturbed system preserves the stability property of the nominal system, locally or globally, depending on both perturbations and the nominal system itself. An example is provided to illustrate the proposed theoretical results.     

Robust Exponential Stability of Stochastically Nonlinear Jump Systems with Mixed Time Delays

In this paper, the problem of robust exponential stability is investigated for a class of stochastically nonlinear jump systems with mixed time delays. By applying the Lyapunov–Krasovskii functional and stochastic analysis theory as well as matrix inequality technique, some novel sufficient conditions are derived to ensure the exponential stability of the trivial solution in the mean square. Time delays proposed in this paper comprise both time-varying and distributed delays. Moreover, the derivatives of time-varying delays are not necessarily less than 1. The results obtained in this paper extend and improve those given in the literature. Finally, two numerical examples and their simulations are provided to show the effectiveness of the obtained results.     

Stability of switched systems with time delay

In this paper, we study the qualitative properties of linear and nonlinear delay switched systems which have stable and unstable subsystems. First, we prove some inequalities which lead to the switching laws that guarantee: (a) the global exponential stability to linear switched delay systems with stable and unstable subsystems; (b) the local exponential stability of nonlinear switched delay systems with stable and unstable subsystems. In addition, these switching laws indicate that if the total activation time ratio among the stable subsystems, unstable subsystems and time delay is larger than a certain number, the switched systems are exponentially stable for any switching signals under these laws. Some examples are given to illustrate the main results.     

On exponential stability of nonlinear differential systems with time-varying delay

General nonlinear differential systems with time-varying delays are considered. Several explicit criteria for exponential stability are presented. An example is given to illustrate the obtained results. To the best of our knowledge, the results of this note are new.     

一类周期线性时变切换系统的稳定性

讨论一类包含一组线性时变子系统和一个周期切换信号的切换系统的稳定性.在系统满足一定条件的情况下,利用Floquet定理,给出了周期线性时变切换系统指数稳定的充分必要条件.     

Robust exponential stability of switched delay interconnected systems under arbitrary switching

The problem of robust exponential stability for a class of switched nonlinear dynamical systems with uncertainties and unbounded delay is addressed. On the assumption that the interconnected functions of the studied systems satisfy the Lipschitz condition, by resorting to vector Lyapunov approach and M-matrix theory, the sufficient conditions to ensure the robust exponential stability of the switched interconnected systems under arbitrary switching are obtained. The proposed method, which neither require the individual subsystems to share a Common Lyapunov Function (CLF), nor need to involve the values of individual Lyapunov functions at each switching time, provide a new way of thinking to study the stability of arbitrary switching. In addition, the proposed criteria are explicit, and it is convenient for practical applications. Finally, two numerical examples are given to illustrate the correctness and effectiveness of the proposed theories.     

Stability of interconnected impulsive systems with and without time delays,using Lyapunov methods

In this paper, we consider the input-to-state stability (ISS) of impulsive control systems with and without time delays. We prove that, if the time-delay system possesses an exponential Lyapunov–Razumikhin function or an exponential Lyapunov–Krasovskii functional, then the system is uniformly ISS provided that the average dwell-time condition is satisfied. Then, we consider large-scale networks of impulsive systems with and without time delays and prove that the whole network is uniformly ISS under the small-gain and the average dwell-time condition. Moreover, these theorems provide us with tools to construct a Lyapunov function (for time-delay systems, a Lyapunov–Krasovskii functional or a Lyapunov–Razumikhin function) and the corresponding gains of the whole system, using the Lyapunov functions of the subsystems and the internal gains, which are linear and satisfy the small-gain condition. We illustrate the application of the main results on examples.     

Decentralized Stabilizing State Feedback Controllers for a Class of Large-Scale Systems Including State Delays in the Interconnections

The problem of the decentralized stabilization of a class of large-scale nonlinear and linear systems including time-varying delays in the interconnections is considered. By combining the Razumikhin-type theorem with the Lyapunov stability theory, we propose a class of decentralized state feedback controllers which can guarantee always some type of stability of large-scale time-delay systems. In this paper, we assume that the time-varying delays are continuous and bounded nonnegative functions. Furthermore, since the proposed decentralized state feedback controllers are independent of the delays, the results obtained in this paper are applicable to systems without exact knowledge of the delays, i.e., systems with perturbed delays. Finally, a numerical example is given to demonstrate the validity of the results.     

Uniform stability of switched nonlinear systems

In this paper, we investigate the stability properties of a general class of nonautonomous switched nonlinear systems. Sufficient conditions for uniform stability, uniform asymptotic stability and uniform exponential stability are derived via multiple Lyapunov functions. Our results provide stability criteria for switched systems with both stable and unstable subsystems. Particularly, our results include some existing results as special cases or improve those in the literature. Several numerical examples are worked out to illustrate our results.     

Note on exponential stability of certain nonlinear differential systems with time-varying delays

One of the main results in a recent paper [P.H.A. Ngoc, On exponential stability of nonlinear differential systems with time-varying delay, Applied Mathematics Letters 25 (2012) 1208–1213] is extended to a more general nonlinear differential system with time-varying delays. A restrictive condition for global exponential stability in the above paper is also removed.     

Exponential quasi-dissipativity and boundedness property for switched nonlinear systems

This paper is concerned with exponential quasi-dissipativity for switched nonlinear systems using multiple storage functions and multiple supply rates. First, a new concept of exponential quasi-dissipativity for a switched nonlinear system is proposed. In contrast with dissipative system which does not produce energy in some abstract sense, the supply rate of quasi-dissipativity is the sum of the conventional dissipativity supply rate and the constant supply rate. This means quasi-dissipative system can produce energy by itself. Each active subsystem is only required to be exponentially quasi-dissipative, while the energy changing of each inactive subsystem characterized by cross-supply rates is considered as the transformation of “energy” from the active subsystem. By weakening the conventional dissipation inequality, a broader class of open-loop systems and controllers are admissible, leading to broader application. Second, the ultimate boundedness property is obtained under some constraints on the energy changing of inactive subsystems. Third, the exponential quasi-dissipativity conditions for switched nonlinear systems are obtained by the design of a state-dependent switching law. In particular, the sufficient conditions of exponential quasi-passivity and exponential finite power gain for switched nonlinear systems are given thereby providing generalizations of KYP conditions and HJI inequality. Finally, the effectiveness of the obtained results is verified by two examples.     

Global exponential stability of certain switched systems with time-varying delays

This paper is focused on global exponential stability of certain switched systems with time-varying delays. By using an average dwell time (ADT) approach that is different from the method in [P.H.A. Ngoc, On exponential stability of nonlinear differential systems with time-varying delay, Applied Mathematics Letters 25 (2012) 1208–1213], we establish a new global exponential stability criterion for the switched linear time-delay system under the ADT switching. We also apply this method to a general switched nonlinear time-delay system. A numerical example is given to show the effectiveness of our results.     

参数不确定非线性组合大系统的鲁棒稳定性及参数鲁棒域的估计(英)

本文研究了参数不确定非线性组合大系统．首先利用现代微分几何理论将系统化为由线性子系统互联而成的组合大系统，然后给出了参数不确定非线性组合大系统鲁棒稳定性的若干判据．最后通过一个例子说明了本文的结论及参数鲁棒域的估计方法．