Robust Stability Analysis for Systems with Time‐Varying Delays: A Delay Function Approach

This paper is concerned with the robust stability of time‐varying delay systems with structured uncertainties. Stability conditions are provided through a Lyapunov‐Krasovskii functional (LKF) method. The proposed method introduces a linear function of the time‐varying delay to construct the LKF. With this function, two‐dimensional partition is conducted on the integral domain in the derivative of LKF. Quadratic convex combination then is employed to present stability criteria in the form of linear matrix inequalities (LMIs). The method not only exploits the information of delay at different time instants, but also enables the handling of its derivative to reduce conservatism. Numerical examples are given to show the effectiveness of our method.     

Stability of Linear Systems With Time‐Varying Delay: a Generalized Discretized Lyapunov Functional Approach

This paper investigates the stability of linear uncertain systems with time‐varying delay. Stability criteria are derived based on a generalized discretized Lyapunov functional approach. The kernel of the functional, which is a function of two variables, is chosen as piecewise linear. The stability conditions are written in the form of linear matrix inequalities. Numerical examples indicate significant improvements over the existing results.     

Delay‐Dependent Stability Criterion for Discrete‐Time Systems with Time‐Varying Delays

The stability analysis problem is considered for linear discrete‐time systems with time‐varying delays. A novel summation inequality is proposed, which takes the double summation information of the system state into consideration. The inequality relaxes the recently proposed discrete Wirtinger inequality and its improved version. Based on construction of a suitable Lyapunov‐Krasovskii functional and the novel summation inequality, an improved delay‐dependent stability criterion for asymptotic stability of the systems is derived in terms of linear matrix inequalities. Numerical examples are given to demonstrate the advantages of the proposed method.     

Synchronization of Uncertain Complex Networks with Time‐Varying Node Delay and Multiple Time‐Varying Coupling Delays

This paper investigates the synchronization problem of a class of complex dynamical networks via an adaptive control method. It differs from existing works in considering intrinsic delay and multiple different time‐varying coupling delays, and uncertain couplings. A simple approach is used to linearize the uncertainties with the norm‐bounded condition. Simple but suitable adaptive controllers are designed to drive all nodes of the complex network locally and globally synchronize to a desired state. In addition, several synchronization protocols are deduced in detail by virtue of Lyapunov stability theory and a Cauchy matrix inequality. Finally, a simulation example is presented, in which the dynamics of each node are time‐varying delayed Chua chaotic systems, to demonstrate the effectiveness of the proposed adaptive method.     

Improved Stabilization for Continuous Dynamical Systems with Two Additive Time‐Varying Delays

This paper studies the problem of stabilization criteria for systems with two additive time‐varying delays. First, the delay‐dependent stability condition for the systems is established through computing the more general Lyapunov functional. The Lyapunov functional is constructed by making full use of the property and the information of the systems, and the condition has advantages over the existing ones in the skillful combination of the delay decomposition and the reciprocal convex approach. Second, considered to be more flexible for the controller design with the introduced positive scalar, a new controller method is presented. Finally, two examples are provided to demonstrate the advantage of the results in this paper.     

New Augmented Lyapunov‐Krasovskii Functional for Stability Analysis of Systems with Additive Time‐Varying Delays

This paper is concerned with stability analysis for continuous‐time systems with additive time‐varying delays in the Lyapunov‐Krasovskii(L‐K) framework. Firstly, in view of the relationships between the upper bounds of the two time‐varying delays, a new augmented L‐K functional is constructed by using the information of the two upper bounds. Secondly, the free‐matrix‐based integral inequality is used to estimate the derivative of the constructed L‐K functional. Thirdly, a less conservative criterion is derived to assess stability. Finally, a numerical example is presented to demonstrate the effectiveness of the criterion.     

Guaranteed Cost Stabilization of Cellular Neural Networks with Time‐Varying Delay

Guaranteed cost stabilization of cellular neural networks with time‐varying delay (DCNNs) is considered in this paper. Via applying the zoned discussion and maximum synthesis (ZDMS) in DCNNs and Lyapunov–Krasovskii functional, a less conservative feedback control law in the form of quadratic matrix inequality (QMI) is derived to achieve globally asymptotic stability of the system.     

具有时变时滞的递归神经网络的渐近稳定性分析

当神经网络应用于最优化计算时,理想的情形是只有一个全局渐近稳定的平衡点,并且以指数速度趋近于平衡点,从而减少神经网络所需计算时间.研究了带时变时滞的递归神经网络的全局渐近稳定性.首先将要研究的模型转化为描述系统模型,然后利用Lyapunov-Krasovskii稳定性定理、线性矩阵不等式(LMI)技术、S过程和代数不等式方法,得到了确保时变时滞递归神经网络渐近稳定性的新的充分条件,并将它应用于常时滞神经网络和时滞细胞神经网络模型,分别得到了相应的全局渐近稳定性条件.理论分析和数值模拟显示,所得结果为时滞递归神经网络提供了新的稳定性判定准则.     

Synchronization of Switched Neural Networks with Time‐varying Delays via Sampled‐data Control

This paper investigates sampled‐data synchronization control of switched neural networks with time‐varying delays under average dwell time. Based on the delay system method, the sampled‐data synchronization system is proposed with time‐varying delays and input delays in the unified framework for switched neural networks. By constructing a suitable Lyapunov‐Krasovskii functional and free‐weighting matrix, the relationship between the average dwell time and the maximum sampling interval is revealed to form delay‐dependent exponentially synchronization criteria. The desired mode‐dependent controller under the maximum sampling interval and decay rate is designed. Finally, two numerical examples are provided to demonstrate the effectiveness and feasibility of the proposed techniques.     

带两个不同时延神经网络的稳定性研究

讨论了带两个不同时延且有两个神经元系统的局部稳定性,得到了判定神经网络稳定性的一些准则,这些准则有的是与时延有关,而有的是与时延无关（这种情形也称为“无害时延”）;研究方法对于带不同时延且多个神经元网络的稳定性的研究有重要的指导意义。     

New Results on the Robust Stability of Cohen–Grossberg Neural Networks with Delays

In this letter, a generalized type of Cohen–Grossberg neural networks with time delays are discussed and their global robust stability of the equilibrium point is investigated. By introducing a set of Lyapunov functionals, several new sufficient conditions guaranteeing the global robust convergence are derived. The results show that the amplification function a _i (x) is harmless to the robust stability of Cohen–Grossberg neural networks. Two examples are given to demonstrate the applicability of the proposed results.     

一类具有变时滞的Hopfield型神经网络的全局指数稳定性

通过构造适当的Lyapunov函数,利用Halanay不等式和Young不等式,讨论一类具有变时滞的Hopfield型神经网络的全局指数稳定性.在对网络施加两个不同的神经元激励函数的条件下,导出网络全局指数稳定的一个充分条件,得到的充分条件在实际应用中易于验证,且有较小的保守性,因而对网络的应用和设计具有重要意义.最后,一个数值实例进一步验证结果的正确性.     

多时滞Hopfield神经网络的鲁棒稳定性分析及吸引域的估计

The robust stability of a class of Hopfield neural networks with multiple delays and parameter perturbations is analyzed. The sufficient conditions for the global robust stability of equilibrium point are given by way of constructing a suitable Lyapunov functional. The conditions take the form of linear matrix inequality (LMI), so they are computable and verifiable efficiently. Furthermore, all the results are obtained without assuming the differentiability and monotonicity of activation functions. From the viewpoint of system analysis, our results provide sufficient conditions for the global robust stability in a manner that they specify the size of perturbation that Hopfield neural networks can endure when the structure of the network is given. On the other hand, from the viewpoint of system synthesis, our results can answer how to choose the parameters of neural networks to endure a given perturbation.     

Delay‐Dependent Exponential Stability for Discrete‐Time Singular Switched Systems with Time‐Varying Delay

In this paper, the exponential stability problem is investigated for a class of discrete‐time singular switched systems with time‐varying delay. By using a new Lyapunov functional and average dwell time scheme, a delay‐dependent sufficient condition is established in terms of linear matrix inequalities for the considered system to be regular, causal, and exponentially stable. Different from the existing results, in the considered systems the corresponding singular matrices do not need to have the same rank. A numerical example is given to demonstrate the effectiveness of the proposed result.     

Adaptive Nonlinear Actuator Fault Diagnosis for Uncertain Nonlinear Systems with Time‐Varying Delays

An actuator fault diagnosis method is presented in this paper for a class of time‐delayed nonlinear systems via the use of adaptive updating rules. The considered system is represented by a dynamic state space model where the time delays are embedded into the state vector. Model parameters are not perfectly known, which lead us to consider norm bounded uncertainties. An adaptive fault diagnosis observer is designed where the Lyapunov stability theory is used to obtain the required adaptive tuning rule for estimation of the nonlinear actuator fault. A simulated numerical example is included to demonstrate the effectiveness of the proposed approach.     

时变时滞神经网络系统的时滞依赖的鲁棒稳定性准则

基于Lyapunov稳定性理论和线性矩阵不等式技术,针对一类带有范数有界不确定性的时变时滞神经网络系统,给出时滞依赖的鲁棒稳定性准则.系统稳定的充分条件是在激励函数满足一类更为通用的条件下得到的,即激励函数不必是单调可微的,并且消除对时变时滞导数的限制.所给的准则可用Matlab中的线性矩阵不等式控制工具箱进行验证.仿真结果进一步证明结论的有效性.     

Finite‐Time Stabilization of Coupled Systems on Networks with Time‐Varying Delays via Periodically Intermittent Control

In this paper, finite‐time stabilization of coupled systems on networks with time‐varying delays (CSNTDs) via periodically intermittent control is studied. Both delayed subsystems and delayed couplings are considered; the self‐delays of different subsystems in delayed couplings are not identical. A periodically intermittent controller is designed to stabilize CSNTDs within finite time, and the stabilization duration is closely related to the topological structures of networks. Furthermore, two sufficient criteria are developed to ensure CSNTDs under periodically intermittent control can be stabilized within finite time by using an approach that combines the Lyapunov method with Kirchhoff's Matrix Tree Theorem. Then finite‐time stabilization of coupled oscillators with time‐varying delays is given as a practical application and sufficient criteria is obtained. Finally, a numerical simulation is proposed to support our results and show the effectiveness of the controller.     

Delay‐Dependent Robust Stability Criteria For Uncertain Neutral Systems With Mixed Time‐Varying Discrete And Neutral Delays

In this paper, a new class of augmented quasi full size Lypunov‐Krasovskii functional is introduced for the robust stability of uncertain neutral systems with mixed time‐varying discrete and neutral delays. The nonlinear parameter perturbations and norm‐bounded uncertainties are taken into consideration separately. Delay‐dependent robust stability criteria are derived in the form of linear matrix inequalities. Numerical examples are presented to illustrate the significant improvement on the conservativeness of the delay bound over some reported results in the literature.     

Robust Stability of Time‐Varying Delay Systems: The Quadratic Separation Approach

In this article, we are interested in analysing the stability of systems that incorporate time‐varying delays in their dynamic. The Lyapunov‐Krasovskii approach is definitely the most popular method to address this issue and many results have proposed new functionals and enhanced techniques for deriving less conservative stability conditions. In the present work, we propose an original approach: the quadratic separation. To this end, the delay operator properties are exploited to provide delay range stability conditions. In particular, L₂‐norm of delay‐dependent operators are computed so as to reduce the conservatism of the approach. Moreover, the main result is able to assess the stability of non‐small delay systems, i.e, it can detect a stability interval for systems that are unstable without any delay. Several examples illustrate the benefit of our methodology.