Stability analysis of impulsive switched systems with time delays

In this paper, exponential stability criteria of impulsive switched systems with variable delays are introduced. Based on some impulsive delay differential inequalities, some general criteria for the exponential stability are obtained. Finally, an example is given to illustrate the effectiveness of the theory.     

A LMI approach to stability analysis and synthesis of impulsive switched systems with time delays

This paper studies the asymptotic stability problem for a class of impulsive switched systems with time invariant delays based on linear matrix inequality (LMI) approach. Some sufficient conditions, which are independent of time delays and impulsive switching intervals, for ensuring asymptotical stability of these systems are derived by using a Lyapunov–Krasovskii technique. Moreover, some appropriate feedback controllers, which can stabilize the closed-loop systems, are constructed. Illustrative examples are presented to show the effectiveness of the results obtained.     

Stability of singularly perturbed switched systems with time delay and impulsive effects

This paper studies the stability properties of singularly perturbed switched systems with time delay and impulsive effects. Such systems are assumed to consist of both unstable and stable subsystems. By using the multiple Lyapunov functions technique and the dwell time approach, some stability criteria are established. Our results show that impulses do contribute in order to obtain stability properties even when the system consists of only unstable subsystems. Numerical examples are presented to verify our theoretical results.     

Stability analysis of impulsive delayed switched systems and applications

In this paper, a general class of impulsive delayed switched systems is considered. By employing the Lyapunov–Razumikhin method and some analysis techniques, we established several global asymptotic stability and global exponential stability criteria for the considered impulsive delayed switched systems, which improve and extend some recent works. As an application, the result of global exponential stability are used to study a class of uncertain linear switched systems with time‐varying delays. Several LMI‐based conditions are proposed to guarantee the global robust stability and global exponential stabilization. The designed memoryless state feedback controller can be easily checked by the LMI toolbox in Matlab. Moreover, the dwell time constraint is imposed for the switching law. Finally, two numerical examples and their simulations are given to show the effectiveness of our proposed results. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Stability analysis of a class of nonlinear positive switched systems with delays

This paper addresses the stability problem of delayed nonlinear positive switched systems whose subsystems are all positive. Both discrete-time systems and continuous-time systems are studied. In our analysis, the delays in systems can be unbounded. Two conditions are established to test the local asymptotic stability of the considered systems. The method to compute the domains of attraction is also proposed provided that the system is locally asymptotically stable. When reduced to general nonlinear positive systems, that is, the considered switched system consists of only one mode, an interesting conclusion follows that the proposed nonlinear positive system is locally asymptotically stable for all admissible delays and nonnegative nonlinearities which satisfy an extra condition at the origin, if and only if the system represented by the linear part is asymptotically stable for all admissible delays. Finally, a numerical example is presented to illustrate the obtained results.     

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.     

Exponential stability of impulsive discrete systems with time delays

The purpose of this paper is to investigate the global exponential stability of impulsive discrete systems with time delays. By using Lyapunov functionals, a number of new global exponential stability criteria are provided. It is shown that a discrete system with time delays can be globally exponentially stabilized by impulses even if it may be unstable itself. Some examples are also presented to illustrate the effectiveness and the superiority 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.     

Controllability analysis of linear time-varying systems with multiple time delays and impulsive effects

In this paper, the issue of controllability for linear time-varying systems with multiple time delays in the control and impulsive effects is addressed. The solution of such systems based on the variation of parameters is derived. Several sufficient and necessary algebraic conditions for two kinds of controllability, i.e., controllability to the origin and controllability, are derived. The relation among these conditions are established. A numerical example is provided to illustrate the effectiveness of the proposed methods.     

Stability analysis of controlled multiple-link robotic manipulator systems with time delays

Controlled robotic manipulator systems, normally operating in an acceptable manner, may become unstable, when time delays exist in the feedback loop. This instability may cause oscillations and thus deteriorate the system performance. In this paper we introduce a method to analyze the performance of robotic manipulators with small time delays in the feedback loop. Specifically, we may predict the limits of a stable system Operation, i.e., the robustness and expected system error bounds, as a function of the time delay and system parameters. The analysis method uses nonlinear control theory concepts, such as Lyapunov's second method, and takes advantage of norms to establish operational bounds. Previous research of time-delay systems analysis is also reviewed, in general, and pertaining to robotic controls, in particular. The results of the analysis can be displayed graphically and may be used to adjust the controller gains and trajectories to ensure satisfactory operation. Predictably, the system response is slowed as the time delay increases. The validity of the analysis is demonstrated with a numerical example of a two-link manipulator using a computed-torque controller. The time responses of the example, run on a computer-generated model, supports the analysis results and predicted performance.     

Robust exponential stability of impulsive switched systems with switching delays: A Razumikhin approach

In this paper, we focus on the robust exponential stability of a class of uncertain nonlinear impulsive switched systems with switching delays. We introduce a novel type of piecewise Lyapunov-Razumikhin functions. Such functions can efficiently eliminate the impulsive and switching jump of adjacent Lyapunov functions at impulsive switching instants. By Razumikhin technique, the delay-independent criteria of exponential stability are established on the minimum dwell time. Finally, an illustrative numerical example is presented to show the effectiveness of the obtained theoretical results.     

时滞脉冲切换系统的实用稳定性-带Razumikhin条件的Lyapunov函数方法

用带Razumikhin条件的Lyapunov函数方法研究了一般形式的时滞脉冲切换系统的实用稳定性,得到了时滞脉冲切换系统实用稳定、一致实用稳定的充分条件.最后,给出了具体的例子及其数值模拟.     

Stability of interconnected impulsive switched systems subject to state dimension variation

The contribution of this article is twofold. First we deal with the stability of continuous-time interconnected impulsive switched systems for which the dimension of the system’s state vector may change at different modes. A dwell time condition is derived to ensure the exponential convergence of the state trajectories, and when the system is subject to a non-vanishing perturbation, a superior bound of the state trajectories is provided as well. The main advantage of the solution we propose is its direct application to practical fields such as metallurgy or traffic control through the use of semi-definite programming solvers. Second, we introduce in the framework a platoon of vehicles in automated highway systems where vehicles may join or leave the platoon. Then, we detail the longitudinal stability problem of such a system and illustrate the features of the proposed stability conditions through numerical simulations.     

Stability properties of nonlinear stochastic impulsive systems with time delay

This article establishes mean square stability and stabilization for stochastic delay systems with impulses. Using Razumikhin methodology, two approaches, classical Lyapunov-based method and comparison principle, are proposed to develop sufficient conditions that guarantee the stability and stabilization properties. It is shown that if the continuous system is stable and the impulses are destabilizing, the impulses should not be applied frequently. On the other hand, if the continuous system is unstable, but the impulses are stabilizing, the impulses should occur frequently to compensate the continuous state growth. Numerical examples are also presented to clarify the proposed theoretical results.     

Stability analysis of impulsive control systems

This paper studies impulsive control systems. Several stability criteria are established by employing the method of Lyapunov functions. These criteria may be used for impulsive feedback control design. As an application, impulsive control of the Lorenz chaotic system is discussed. Numerical experiments are carried out for the control of the Lorenz system. It is shown that small and frequent impulses need to be used in order to stabilize the Lorenz system.     

Synchronization of switched neural networks with mixed delays via impulsive control

This paper concerns the problem of global exponential synchronization for a class of switched neural networks with time-varying delays and unbounded distributed delays via impulsive control method. By using Lyapunov stability theory, new synchronization criterion is derived. In our synchronization criterion, the switching law can be arbitrary and the concept of average impulsive interval is utilized such that the obtained synchronization criterion is less conservative than those based on maximum of impulsive intervals. Numerical simulations are given to show the effectiveness and less conservativeness of the theoretical results.     

Stability criteria for impulsive systems on time scales

In this paper we study stability of impulsive system on time scales. By using comparison method, Lyapunov function and analysis method, the asymptotic stability criteria for system with impulses at fixed times and impulses at variable times on time scales are obtained, respectively. An example is presented to illustrate the efficiency of proposed result.     

Stability analysis of impulsive fractional differential systems with delay

In this paper, a class of impulsive fractional differential systems with finite delay is considered. Some sufficient conditions for the finite-time stability of above systems are obtained by using generalized Bellman–Gronwall’s inequality, which extend some known results.