Stability of linear switched differential algebraic equations with stable and unstable subsystems

This article studies linear switched differential algebraic equations (DAEs), which contains stable and unstable subsystems. We prove sufficient conditions for stability of switched DAEs based on the existence of suitable Lyapunov functions. The result shows that stability is preserved under switching with an average dwell time and an additional condition involving consistency projectors holds. Furthermore, we also give an example to illustrate the result.     

一类多输入级联非线性切换系统的全局镇定

研究一类带有部分线性系统的多输入级联非线性切换系统的全局镇定问题. 首先, 给出保证线性部分有一致规范型的充分条件. 其次, 利用一致规范型及其零动态的共同二次Lyapunov函数设计状态反馈使得线性部分在任意切换律下镇定. 最后, 通过构造共同Lyapunov函数能实现闭环系统在任意切换律下的全局渐近稳定性.     

Stability and Stabilizability of Switched Linear Systems: A Survey of Recent Results

During the past several years, there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems. This paper aims to briefly survey recent results in this field. First, the stability analysis for switched systems is reviewed. We focus on the stability analysis for switched linear systems under arbitrary switching, and we highlight necessary and sufficient conditions for asymptotic stability. After a brief review of the stability analysis under restricted switching and the multiple Lyapunov function theory, the switching stabilization problem is studied, and a variety of switching stabilization methods found in the literature are outlined. Then the switching stabilizability problem is investigated, that is under what condition it is possible to stabilize a switched system by properly designing switching control laws. Note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature. A necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here.      

Stability of switched systems: a Lie-algebraic condition

We present a sufficient condition for asymptotic stability of a switched linear system in terms of the Lie algebra generated by the individual matrices. Namely, if this Lie algebra is solvable, then the switched system is exponentially stable for arbitrary switching. In fact, we show that any family of linear systems satisfying this condition possesses a quadratic common Lyapunov function. We also discuss the implications of this result for switched nonlinear systems.     

H-infinity control for cascade minimum-phase switched nonlinear systems

1Introduction H_∞control theory has become a powerful tool to solverobust stabilization or disturbance attenuation problems.Many results about linear H∞control have appeared,andlinear H∞theory has been generalized to nonlinear systems[1~5].Two major approaches have been used to providesolutions to nonlinear H∞control problems.One is basedon the dissipativity theory and differential games theory[2,6].The other is based on the nonlinear versionofclassical bounded real lemma[3~5].Both of th…     

H-infinity control for cascade minimum-phase switched nonlinear systems

This paper is concerned wi th the H-infinity control problem for a class of cascade switched nonlinear systems.Each switched syste m in this class is composed of a zero-input asymptotically stable nonlinear part,which is also a switched system,and a linearizable part which i s controllable.Conditions under which the H-infinity con trol problem is solvable under arbitrary switching l aw and under some designed switching law are der ived respectively.The nonlinear state feedback and s witching law are designed.We exploit the structural characteristics of the switched nonlinear systems t o construct common Lyapunov functions for arbitrary switching and to find a single Lyapunov function for designed switching law.The proposed methods do not rely on the solutions of Hamilton-Jacobi in equalities.     

Uniform stability of switched linear systems: extensions of LaSalle's Invariance Principle

This paper addresses the uniform stability of switched linear systems, where uniformity refers to the convergence rate of the multiple solutions that one obtains as the switching signal ranges over a given set. We provide a collection of results that can be viewed as extensions of LaSalle's Invariance Principle to certain classes of switched linear systems. Using these results one can deduce asymptotic stability using multiple Lyapunov functions whose Lie derivatives are only negative semidefinite. Depending on the regularity assumptions placed on the switching signals, one may be able to conclude just asymptotic stability or (uniform) exponential stability. We show by counter-example that the results obtained are tight.     

On hybrid impulsive and switching systems and application to nonlinear control

In this note, a new class of hybrid impulsive and switching models is introduced and their asymptotic stability properties are investigated. Using switched Lyapunov functions, some new general criteria for exponential stability and asymptotic stability with arbitrary and conditioned impulsive switching are established. In addition, a new hybrid impulsive and switching control strategy for nonlinear systems is developed. A typical example, the unified chaotic system, is given to illustrate the theoretical results.     

Convergence rate of nonlinear switched systems

This paper is concerned with the convergence rate of the solutions of nonlinear switched systems.We first consider a switched system which is asymptotically stable for a class of switching signals but not for all switching signals. We show that solutions corresponding to that class of switching signals converge arbitrarily slowly to the origin.Then we consider analytic switched systems for which a common weak quadratic Lyapunov function exists. Under two different sets of assumptions we provide explicit exponential convergence rates for switching signals with a fixed dwell-time.     

On the Asymptotic Stability Problem for Solutions of Difference Switched Systems

We consider difference systems obtained by discretizing certain classes of differential systems. It is assumed that the system under consideration can operate in several modes. The problem is to establish conditions that guarantee the asymptotic stability of a given equilibrium position when switching regimes. We use the method of Lyapunov functions. We study the case when solutions of the system under various operating modes can have features of both linear and nonlinear behavior.     

Stability analysis of some classes of nonlinear switched systems with time delay

Stability of certain classes of nonlinear time-delay switched systems is studied. For the corresponding families of subsystems, conditions of the existence of common Lyapunov–Razumikhin functions are found. The fulfilment of these conditions provides the asymptotic stability of the zero solutions of the considered hybrid systems for any switching law and for any nonnegative delay. Some examples are presented to demonstrate the effectiveness of the obtained results.     

Robust Output Feedback Stabilization of Switched Nonlinear Systems with Average Dwell Time

For some switched nonlinear systems, stabilization can be achieved under arbitrary switching with state feedback control. Due to switching zero dynamics, output feedback stabilization for some switched nonlinear systems needs dwell time between switching to guarantee system stability. In this paper, we consider a class of switched nonlinear systems with unknown parameters and unknown switching signals. We design a robust output feedback controller that stabilizes the system under a class of switching signals with average dwell time (ADT) where the value of ADT can be reduced by adjusting the control gain. For some special cases, common quadratic Lyapunov functions of the closed‐loop systems can be found and the value of ADT is further relaxed. Some examples and simulations are provided to validate the results.     

Uniform Asymptotic Stability of Nonlinear Switched Systems With an Application to Mobile Robots

This paper is concerned with the study of, both local and global, uniform asymptotic stability for general nonlinear and time-varying switched systems. Two concepts of Lyapunov functions are introduced and used to establish uniform Lyapunov stability and uniform global stability. With the help of output functions, an almost bounded output energy condition and an output persistent excitation condition are then proposed and employed to guarantee uniform local and global asymptotic stability. Based on this result, a generalized version of Krasovskii-LaSalle theorem in time-varying switched systems is proposed. For switched systems with persistent dwell-time, the output persistent excitation condition is guaranteed to hold under a zero-state observability condition. It is shown that several existing results in past literature can be covered as special cases using the proposed criteria. Interestingly, as opposed to previous work, the main results of this paper are applicable to the situation where some switching systems are not asymptotically stable at the origin. The robust practical regulation problem of nonholonomic mobile robots is studied as a way of demonstrating the power of the proposed new criteria. A novel switching controller is proposed with guaranteed robustness to orientation error and unknown parameters in mobile robots.      

Stability of switched nonlinear systems via extensions of LaSalle’s invariance principle

This paper studies the extension of LaSalle’s invariance principle for switched nonlinear systems. Un-like most existing results in which each switching mode in the system needs to be asymptotically stable, this paper allows the switching modes to be only stable. Under certain ergodicity assumptions of the switching signals, two extensions of LaSalle’s invariance principle for global asymptotic stability of switched nonlinear systems are obtained using the method of common joint Lyapunov function.     

Exponentially incremental dissipativity for nonlinear stochastic switched systems

ABSTRACT

In this paper, we investigate the exponentially incremental dissipativity for nonlinear stochastic switched systems by using the designed state-dependent switching law and multiple Lyapunov functions approach. Specifically, using incremental supply rate as well as a state dissipation inequality in expectation, a stochastic version of exponentially incremental dissipativity is presented. The sufficient conditions for nonlinear stochastic switched systems to be exponentially incrementally dissipative are given by the designed state-dependent switching law. Furthermore, the extended Kalman–Yakubovich–Popov conditions are derived by using two times continuously differentiable storage functions. Moreover, the incremental stability conditions in probability for nonlinear stochastic switched systems are derived based on exponentially incremental dissipativity. The exponentially incremental dissipativity is preserved for the feedback-interconnected nonlinear stochastic switched systems with the composite state-dependent switching law; meanwhile, the incremental stability in probability is preserved under some certain conditions. A numerical example is given to illustrate the validity of our results.     

Average dwell time based stability analysis for nonautonomous continuous‐time switched systems

Inspired by the idea of multiple Lyapunov functions and the average dwell time, we address the stability analysis of nonautonomous continuous‐time switched systems. First, we investigate nonautonomous continuous‐time switched nonlinear systems and successively propose sufficient conditions for their (uniform) stability, global (uniform) asymptotic stability, and global (uniform) exponential stability, in which an indefinite scalar function is utilized to release the nonincreasing requirements of the classical multiple Lyapunov functions. Afterwards, by using multiple Lyapunov functions of quadratic form, we obtain the corresponding sufficient conditions for (uniform) stability, global (uniform) asymptotic stability, and global exponential stability of nonautonomous switched linear systems. Finally, we consider the computation issue of our current results for a special class of nonautonomous switched systems (ie, rational nonautonomous switched systems), associated with two illustrative examples.     

Stability Properties of Switched Nonlinear Delay Systems with Synchronous or Asynchronous Switching

This paper is concerned with the problem of input‐to‐state stability (ISS) for a class of switched nonlinear delay systems. The cases where the switching signal of the system and the switching signal of the corresponding controller are synchronous and asynchronous are both considered. To study two asynchronous switching signals in a unified framework, we adopt the technique of the merging switching signal. Based on a piecewise Lyapunov–Krasovskii functional method, some sufficient conditions are explicitly given to guarantee the ISS of the switched nonlinear delay system under the average dwell time scheme. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed theory.     

离散时滞切换系统稳定性分析

对于一类子系统为离散时滞系统的切换系统,研究渐近稳定性条件和切换信号的选取方法．根据李亚普诺夫稳定性理论，推出以线性矩阵不等式表示的在任意切换信号作用下系统渐近稳定的两个充分性条件，在此基础上进一步给出了系统渐近稳定的凸组合条件和切换信号的选取方法．仿真实例验证了所设计的切换方案的有效性．     

Synchronization of complex dynamical networks with switching topology: A switched system point of view

In this paper, we study the synchronization problem for complex dynamical networks with switching topology from a switched system point of view. The synchronization problem is transformed into the stability problem for time-varying switched systems. We address two basic problems: synchronization under arbitrary switching topology, and synchronization via design of switching within a pre-given collection of topologies when synchronization cannot be achieved by using any topology alone in this collection. For the both problems, we first establish synchronization criteria for general connection topology. Then, under the condition of simultaneous triangularization of the connection matrices, a common Lyapunov function (for the first problem) and a single Lyapunov and multiple Lyapunov functions (for the second problem) are systematically constructed respectively by those of several lower-dimensional dynamic systems. In order to achieve synchronization using multiple Lyapunov functions, a stability condition and switching law design method for time-varying switched systems are also presented, which avoid the usual non-increasing condition.     

切换非线性正系统的有限时间稳定性

研究模型依赖平均驻留时间(MDADT)切换信号下一类齐次度为1的切换非线性正系统的有限时间稳定问题.首先,通过构造恰当的切换最大分离Lyapunov函数,借助于Dini导数,基于MDADT切换信号,给出切换非线性正系统有限时间稳定的充分条件.与已有的指数稳定性结果相比,进一步说明有限时间稳定与指数稳定的区别.其次,将所得结论应用于切换线性正系统,得到切换线性正系统在MDADT或平均驻留时间(ADT)切换信号下有限时间稳定的充分条件.最后,通过仿真算例验证所得结论的有效性.