Delay‐dependent dissipativity analysis and synthesis of switched delay systems

In this paper, we study the problem of dissipative analysis for a class of switched systems with time‐varying delays. Sufficient conditions for dissipativity are developed for a class of switching signals with average dwell time. These conditions express delay‐dependent exponential stability and are provided in terms of linear matrix inequalities (LMIs). It is shown that the derived results encompass some available results on ??_∞ approach and arbitrary switching case. Numerical examples are given to illustrate the developed results. Copyright © 2010 John Wiley & Sons, Ltd.     

Exponential quasi‐(Q,S,R)‐dissipativity and practical stability for switched nonlinear systems

In this paper, the problems of exponential quasi‐(Q,S,R)‐dissipativity and practical stability analysis for a switched nonlinear system are addressed. First, the concept of exponential quasi‐(Q,S,R)‐dissipativity for switched nonlinear systems without requiring the exponential quasi‐(Q,S,R)‐dissipativity property of each subsystem is proposed. Then, we show that an exponentially quasi‐(Q,S,R)‐dissipative switched nonlinear system is practically stable. Second, this exponential quasi‐(Q,S,R)‐ dissipativity property for a switched nonlinear system is obtained by the design of a state‐dependent switching law. Third, a composite state‐dependent switching law is designed to render the feedback interconnection of switched nonlinear systems exponentially quasi‐(Q,S,R)‐dissipative. This switching law allows interconnected switched nonlinear systems to switch asynchronously. Finally, the effectiveness of the results is verified by a numerical example.     

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.     

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.     

切换对称组合系统的稳定性

首次提出了切换对称组合系统的概念, 研究了此类系统在任意切换下渐近稳定的条件, 同时分别利用多李雅普诺夫函数方法和单李雅普诺夫函数方法, 给出使切换对称组合系统渐近稳定的切换律, 利用切换对称组合系统的结构特点, 使其切换律的设计条件得到简化.     

A Lyapunov-based small-gain theorem for interconnected switched systems

Stability of an interconnected system consisting of two switched systems is investigated in the scenario where in both switched systems there may exist some subsystems that are not input-to-state stable (non-ISS). We show that, providing the switching signals neither switch too frequently nor activate non-ISS subsystems for too long, a small-gain theorem can be used to conclude global asymptotic stability (GAS) of the interconnected system. For each switched system, with the constraints on the switching signal being modeled by an auxiliary timer, a correspondent hybrid system is defined to enable the construction of a hybrid ISS Lyapunov function. Apart from justifying the ISS property of their corresponding switched systems, these hybrid ISS Lyapunov functions are then combined to establish a Lyapunov-type small-gain condition which guarantees that the interconnected system is globally asymptotically stable.     

String stability of infinite-dimension stochastic interconnected large-scale systems with time-varying delay

The exponential string stability for a class of nonlinear interconnected large-scale systems with time-varying delay is analysed by using the box theory and constructing a vector Lyapunov function. Under the assumption that the time delay is bounded and continuous, a criterion for exponential string stability of the systems is obtained by analysing the stability of differential inequalities with time-varying delay. The large-scale system is exponential string stable when the conditions associating with the coefficient matrices of the system and the solutions of the Lyapunov equations, interconnected with the system, are satisfied. Since it is independent of the delays and simplifies the calculation, the criterion is easy to apply.     

Links between different stabilities of switched homogeneous systems with delays and uncertainties

This paper addresses the links between three stabilities (attractivity, asymptotic stability, and exponential stability) of switched homogeneous systems with delays and uncertainties. A system has a certain property over a given set of switching signals if the property holds for all switching signals in . It is shown that a switched homogeneous system of degree one is exponentially stable over a given set of switching signals if it is attractive or asymptotically stable over the same set. The result is then applied to switched linear systems with delays and uncertainties. Finally, an example follows to show that ‘being over a given set of switching signals’ is necessary to guarantee the equivalence between different stabilities. Copyright © 2015 John Wiley & Sons, Ltd.     

Stability analysis and control synthesis of switched impulsive systems

In this paper, we investigate the stability analysis problem of switched impulsive nonlinear systems and several stabilization problems of switched discrete‐time linear systems are studied. First, sufficient conditions ensuring globally uniformly asymptotically stability of switched nonlinear impulsive system under arbitrary and DDT (dynamical dwell time which defines the length of the time interval between two successive switchings) switching are derived, respectively. In the DDT switching case, we first consider the switched system composed by stable subsystems, then we extend the results to the case where not all subsystems are stable. The stabilizations of switched discrete‐time linear system under arbitrary switching, DDT switching and asynchronous switching are investigated respectively. Based on the stability analysis results, the control synthesis consists of controller design for each subsystem and state impulsive jumping generators design at switching instant. With the aid of the state impulsive jumping generators at switching instant, the ‘energy’ produced by switching can be minimized, which leads to less conservative results. Several numerical examples are given to illustrate the proposed results within this paper. Copyright © 2011 John Wiley & Sons, Ltd.     

Stability of linear discrete switched systems with delays based on average dwell time method

This paper deals with the problem of exponential stability for a class of linear discrete switched systems with constant delays.The switched systems consist of stable and unstable subsystems.Based on the average dwell time method, some switching signals will be found to guarantee exponential stability of these systems.The explicit state decay estimation is also given in the form of the solutions of linear matrix inequalities(LMIs).An example relating to networked control systems(NCSs) illustrates the effect...     

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.     

ROBUST FAULT‐TOLERANT CONTROL FOR A CLASS OF SWITCHED NONLINEAR SYSTEMS IN LOWER TRIANGULAR FORM

This paper investigates the problem of robust fault‐tolerant control for a class of uncertain switched nonlinear systems in lower triangular form. A system of this class involves parameter uncertainties and unknown nonlinear disturbances. A sufficient condition for the problem to be solvable under arbitrary switching is given in terms of linear matrix inequalities (LMIs). State feedback controllers of subsystems are designed by using the solutions to the matrix inequalities to guarantee global asymptotic stability of the closed‐loop systems in presence of actuator failures and under arbitrary switching. A practical system of hybrid haptic display is analyzed to demonstrate the proposed design method.     

Stability analysis of 2-D switched systems with multiplicative noise under arbitrary and restricted switching signals

This paper introduces a new class of discrete-time two-dimensional (2-D) switched systems with multiplicative noise. Firstly, we extend the definition of the asymptotic stability and establish a sufficient stochastically stability condition for this new model under arbitrary switching signal. Then, by introducing the average dwell time into this new model and combining with the Lyapunov function, we investigate the extended stochastic exponential stability of the 2-D switched systems with multiplicative noise for the restricted switching case. Moreover, some remarks and discussions are given to illustrate the significance of obtained results, which generalise and comprise some previous results the literature. Finally, two examples are provided to show the effectiveness of the theoretical results.     

Design proportional‐integral‐derivative/proportional‐derivative controls for second‐order time‐varying switched nonlinear systems

Based on proportional‐integral‐derivative (PID)/PD controls, we in the article investigate the tracking problem of a class of second‐order time‐varying switched nonlinear systems. To start with, for tracking a given point under arbitrary switching signals, we propose a sufficient condition about PID controller parameters, which can be implicitly described as semialgebraic sets. Successively, we consider the tracking problem under average dwell time (ADT)‐based switching signals and propose an alternative sufficient condition about PID controller parameters. Especially, for tracking an equilibrium point of the system without controls, we can further simply utilize the proportional‐derivative control and similarly construct corresponding semialgebraic conditions about proportional‐derivative controller parameters under arbitrary switching signals and ADT‐based switching signals. Finally, two examples are given to show the applicability of our theoretical results.     

Stability analysis of discrete-time switched systems: a switched homogeneous Lyapunov function method

This paper addresses the stability issue of discrete-time switched systems with guaranteed dwell-time. The approach of switched homogeneous Lyapunov function of higher order is formally proposed. By means of this approach, a necessary and sufficient condition is established to check the exponential stability of the considered system. With the observation that switching signal is actually arbitrary if the dwell time is one sample time, a necessary and sufficient condition is also presented to verify the exponential stability of switched systems under arbitrary switching signals. Using the augmented argument, a necessary and sufficient exponential stability criterion is given for discrete-time switched systems with delays. A numerical example is provided to show the advantages of the theoretical 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.     

任意相对阶下非线性切换系统的事件触发漏斗控制

针对一类具有任意相对阶且带有部分非输入到状态稳定逆动态的非线性切换系统, 提出一种动态事件触 发漏斗跟踪控制方案. 首先, 引入一个虚拟输出将任意相对阶的非线性切换系统转换为相对阶为一的非线性切换系 统. 其次, 设计各子系统的事件触发漏斗控制器和切换的动态事件触发机制, 解决候选事件触发漏斗控制器和子系 统之间的异步切换问题, 所提方案消除已有文献中为所有子系统设计共同控制器带来的保守性. 在一类具有平均驻 留时间切换信号的作用下, 保证切换闭环系统的所有信号都是有界的, 且跟踪误差一直在预设的漏斗内演化, 并排 除采样中的奇诺现象. 最后, 一个仿真例子验证方案的实用性和有效性.     

Superstability of linear switched systems

This paper applies the concept of superstability to switched linear systems as a particular case of linear time-varying systems. A generalised concept of superstability, applied to complex matrices, and extended superstability, is introduced in order to obtain a new result for guaranteeing the asymptotic stability of a switched system under arbitrary switching. The relation between extended superstable and stable simultaneously triangularizable sets of matrices is also discussed. It is shown that stable triangularizable matrices are a proper subset of extended superstable ones, pointing out that the presented stability result is a generalisation of the previous well-known stability theorems to a broader class of switched dynamical systems.     

Stability of a class of switched stochastic nonlinear systems under asynchronous switching

The stability of a class of switched stochastic nonlinear retarded systems with asynchronous switching controller is investigated. By constructing a virtual switching signal and using the average dwell time approach incorporated with Razumikhin-type theorem, the sufficient criteria for pth moment exponential stability and global asymptotic stability in probability are given. It is shown that the stability of the asynchronous stochastic systems can be guaranteed provided that the average dwell time is sufficiently large and the mismatched time between the controller and the systems is sufficiently small. This result is then applied to a class of switched stochastic nonlinear delay systems where the controller is designed with both state and switching delays. A numerical example illustrates the effectiveness of the obtained results.     

Finite‐time stabilization of a class of switched stochastic nonlinear systems under arbitrary switching

This paper investigates the finite‐time stabilization of a class of switched stochastic nonlinear systems under arbitrary switching, where each subsystem has a chained integrator with the power r (0 < r < 1). By using the technique of adding a power integrator, a continuous state‐feedback controller is constructed, and it is proved that the solution of the closed‐loop system is finite‐time stable in probability. Two simulation examples are provided to show the effectiveness of the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.