Mean square stability of linear stochastic neutral‐type time‐delay systems with multiple delays

This paper studies mean square exponential stability of linear stochastic neutral‐type time‐delay systems with multiple point delays by using an augmented Lyapunov‐Krasovskii functional (LKF) approach. To build a suitable augmented LKF, a method is proposed to find an augmented state vector whose elements are linearly independent. With the help of the linearly independent augmented state vector, the constructed LKF, and properties of the stochastic integral, sufficient delay‐dependent stability conditions expressed by linear matrix inequalities are established to guarantee the mean square exponential stability of the system. Differently from previous results where the difference operator associated with the system needs to satisfy a condition in terms of matrix norms, in the current paper, the difference operator only needs to satisfy a less restrictive condition in terms of matrix spectral radius. The effectiveness of the proposed approach is illustrated by two numerical examples.     

Robust model predictive control of discrete nonlinear systems with time delays and disturbances via T–S fuzzy approach

In this paper, robust fuzzy model predictive control of a class of nonlinear discrete systems subjected to time delays and persistent disturbances is investigated. Based on the modeling method of delay difference inclusions, nonlinear discrete time-delay systems can be represented by T–S fuzzy systems comprised of piecewise linear delay difference equations. Moreover, Lyapunov–Razumikhin function (LRF), instead of Lyapunov–Krasovskii functional (LKF), is employed for time-delay systems due to its ability to reflect system original state space and its advantages in controller synthesis and computation. The robust positive invariance and input-to-state stability with respect to disturbance under such circumstances are investigated. A robust constraint set is adopted that the system state is converged to this set round the desired point. In addition, the controller synthesis conditions are derived by solving a set of matrix inequalities. Simulation results show that the proposed approach can be successfully applied to the well-known continuous stirred tank reactor (CSTR) systems subjected to time delay.     

Robust stabilizers for an implicit dynamical delay system of the neutral type

A feedback stabilization problem is investigated for a class of imperfectly known implicit systems with discrete and distributed delays. The imperfections acting on the systems, which may be time‐, state‐, delayed state‐, and/or input‐dependent, are modelled as additive nonlinear perturbations influencing a known set of nonlinear functional differential equations of the neutral type. Sufficient conditions, which include a delay‐dependent matrix inequality and a delay‐dependent stability criterion involving some bounding parameters for the uncertainty in the system, are presented and a class of robust feedback stabilizers, including both memoryless and those with memory, are designed to guarantee a prescribed stability property for the class of implicit systems. Copyright © 2005 John Wiley & Sons, Ltd.     

Small gain problem in coupled differential‐difference equations,time‐varying delays,and direct Lyapunov method

This article presents a Lyapunov–Krasovskii formulation of scaled small gain problem for systems described by coupled differential‐difference equations. This problem includes H_∞ problem with block‐diagonal uncertainty as a special case. A discretization may be applied to reduce the conditions into linear matrix inequalities. As an application, the stability problem of systems with time‐varying delays is transformed into the scaled small gain problem through a process of either one‐term approximation or two‐term approximation. The cases of time‐varying delays with and without derivative upper‐bound are compared. Finally, it is shown that similar conditions can also be obtained by a direct Lyapunov–Krasovskii functional method for coupled differential‐functional equations. Numerical examples are presented to illustrate the effectiveness of the method in tackling systems with time‐varying delays. Copyright © 2010 John Wiley & Sons, Ltd.     

Exponential stability of linear continuous time difference systems with multiple delays

Some recent results on exponential stability of linear continuous time difference systems with discrete and distributed delay terms are extended to the case of multiple delays. New sufficient conditions for the exponential stability and exponential estimates for the solutions by using Lyapunov–Krasovskii functionals are derived. Special attention is paid to the case of systems with commensurate discrete and distributed delays.     

On the Liapunov-Krasovskii methodology for the ISS of systems described by coupled delay differential and difference equations

The input-to-state stability of time-invariant systems described by coupled differential and difference equations with multiple noncommensurate and distributed time delays is investigated in this paper. Such equations include neutral functional differential equations in Hale’s form (which model, for instance, partial element equivalent circuits) and describe lossless propagation phenomena occurring in thermal, hydraulic and electrical engineering. A general methodology for systematically studying the input-to-state stability, by means of Liapunov-Krasovskii functionals, with respect to measurable and locally essentially bounded inputs, is provided. The technical problem concerning the absolute continuity of the functional evaluated at the solution has been studied and solved by introducing the hypothesis that the functional is locally Lipschitz. Computationally checkable LMI conditions are provided for the linear case. It is proved that a linear neutral system in Hale’s form with stable difference operator is input-to-state stable if and only if the trivial solution in the unforced case is asymptotically stable. A nonlinear example taken from the literature, concerning an electrical device, is reported, showing the effectiveness of the proposed methodology.     

Improved stability conditions for discrete‐time systems under dynamic network protocols

This paper deals with the stability of discrete‐time networked systems with multiple sensor nodes under dynamic scheduling protocols. Access to the communication medium is orchestrated by a weighted try‐once‐discard or by an independent and identically‐distributed stochastic protocol that determines which sensor node can access the network at each sampling instant and transmit its corresponding data. Through a time‐delay approach, a unified discrete‐time hybrid system with time‐varying delays in the dynamics and in the reset conditions is formulated under both scheduling protocols. Then, a new stability criterion for discrete‐time systems with time‐varying delays is proposed by the discrete counterpart of the second‐order Bessel‐Legendre integral inequality. The developed approach is applied to guarantee the stability of the resulting discrete‐time hybrid system model with respect to the full state under try‐once‐discard or independent and identically‐distributed scheduling protocol. The communication delays can be larger than the sampling intervals. Finally, the efficiency of the presented approach is illustrated by a cart‐pendulum system.     

具多时滞的不确定多变量反馈系统的鲁棒稳定化

用矩阵测度研究了具有时滞的不确定系统,给出了在多变量反馈控制律作用下,系统指数渐近稳定的判别准则,推广和改进了文[1～3]的工作.     

具多时滞的不确定多变量反馈系统的鲁棒稳定化

用矩阵测度研究了具有时滞的不确定系统, 给出了在多变量反馈控制律作用下, 系统指数渐近稳定的判别准则, 推广和改进了文 [1～ 3]的工作.     

New results on stability and L∞‐gain analysis for positive linear differential‐algebraic equations with unbounded time‐varying delays

This article addresses the problems of stability and L_∞‐gain analysis for positive linear differential‐algebraic equations with unbounded time‐varying delays for the first time. First, we consider the stability problem of a class of positive linear differential‐algebraic equations with unbounded time‐varying delays. A new method, which is based on the upper bounding of the state vector by a decreasing function, is presented to analyze the stability of the system. Then, by investigating the monotonicity of state trajectory, the L_∞‐gain for differential‐algebraic systems with unbounded time‐varying delay is characterized. It is shown that the L_∞‐gain for differential‐algebraic systems with unbounded time‐varying delay is also independent of the delays and fully determined by the system matrices. Two numerical examples are given to illustrate the obtained results.     

Exponential stability of impulsive positive switched systems with discrete and distributed time‐varying delays

This paper studies the exponential stability problems of discrete‐time and continuous‐time impulsive positive switched systems with mixed (discrete and distributed) time‐varying delays, respectively. By constructing novel copositive Lyapunov‐Krasovskii functionals and using the average dwell time technique, delay‐dependent sufficient conditions for the solvability of considered problems are given in terms of fairly simple linear matrix inequalities. Compared with the most existing results, by introducing an extra real vector, restrictive conditions on derivative of the time‐varying delays (less than 1) are relaxed, thus the obtained improved stability criteria can deal with a wider class of continuous‐time positive switched systems with time‐varying delays. Finally, two simple examples are provided to verify the validity of theoretical results.     

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.     

Effect of a distributed delay on relative stability of diffusely coupled systems,with application to synchronized equilibria

We perform a linear stability analysis of synchronized equilibria in networks of diffusely coupled oscillators, affected by distributed delays in the coupling, and we characterize the structure of the emanating solutions in the bifurcations, under the assumption that the delay kernels are equal to each other. The motivation comes from the fact that valuable quantitative and qualitative information about the occurrence and type of synchronous or partially synchronous solutions can be obtained from this linear analysis. We analyze stability of synchronized equilibria as a function of the parameters of a so‐called shifted gamma‐distributed delay, which allows to represent or approximate a large class of distributed delay kernels. We also present an asymptotic analysis method, which is particularly suitable for studying the effect of a distribution of delays on stability. We apply the methods to networks of coupled Lorenz systems, where we highlight that stability is favored by a distributed delay compared with a discrete delay with the same average value. Among others, we show that if the coupling is diffusive, the synchronized equilibria become asymptotically stable for large values of the coupling gain, that is, the systems locally asymptotically synchronize at an equilibrium, independently of the network topology. Copyright © 2015 John Wiley & Sons, Ltd.     

Asymptotic Stability of Cohen–Grossberg BAM Neutral Type Neural Networks with Distributed Time Varying Delays

This paper is concerned with the problem of asymptotic stability of neutral type Cohen–Grossberg BAM neural networks with discrete and distributed time-varying delays. By constructing a suitable Lyapunov–Krasovskii functional (LKF), reciprocal convex technique and Jensen’s inequality are used to delay-dependent conditions are established to analysis the asymptotic stability of Cohen–Grossberg BAM neural networks with discrete and distributed time-varying delays. These stability conditions are formulated as linear matrix inequalities (LMIs) which can be easily solved by various convex optimization algorithms. Finally numerical examples are given to illustrate the usefulness of our proposed method.     

Exponential estimates for neutral time delay systems with multiple delays

Exponential estimates and sufficient conditions for the exponential stability of linear neutral time delay for systems with multiple delays are given. The case of systems with uncertainties, including uncertainties in the difference operator, is considered. The proofs follows from new results on non‐homogeneous difference equations evolving in continuous time combined with the Lyapunov–Krasovskii functionals approach. The conditions are expressed in terms of linear matrix inequalities. The particular case of neutral time delay systems with commensurate delays, which leads to less restrictive exponential estimates, is also addressed. Copyright © 2005 John Wiley & Sons, Ltd.     

Filtering in nonlinear time delay systems

Linear and nonlinear (extended Kalman-Bucy) filters are derived for systems governed by coupled partial and integro-differential equations. The framework used is sufficiently general that filters for 1) lumped parameter systems having multiple time varying or constant time delays, 2) coupled lumped and hyperbolic distributed parameter systems, and 3) lumped parameter systems with functional time delays, evolve as special cases. Although the filtering equations are the final result, the corresponding smoothing equations are developed as well. The performance of the filter is illustrated through application to a well stirred chemical reactor with external heat exchange.     

Stochastic stability of Markovian jumping Hopfield neural networks with constant and distributed delays

This paper investigates the problem of stability analysis for Markovian jumping Hopfield neural networks (MJHNNs) with constant and distributed delays. Some new delay-dependent stochastic stability criteria are derived based on a novel Lyapunov–Krasovskii functional (LKF) approach. These new criteria based on the delay partitioning idea prove to be less conservative, since the conservatism could be notably reduced by thinning the delay partitioning. Numerical examples are provided to show the effectiveness and advantage of the proposed techniques.     

New delay‐dependent stability analysis and synthesis of T–S fuzzy systems with time‐varying delay

In this paper, a new method is proposed for stability analysis and synthesis of Takagi–Sugeno (T–S) fuzzy systems with time‐varying delay. Based on a new Lyapunov–Krasovskii functional (LKF), some less conservative delay‐dependent stability criteria are established. In the derivation process, some additional useful terms, ignored in previous methods, are considered and new free‐weighting matrices are introduced to estimate the upper bound of the derivative of LKF for T–S fuzzy systems with time‐varying delay. The proposed stability criterion and stabilization condition are represented in terms of linear matrix inequalities. Numerical examples are given to demonstrate the effectiveness and the benefits of the proposed method. Copyright © 2009 John Wiley & Sons, Ltd.