Stability and stabilization of continuous‐time switched systems: A multiple discontinuous convex Lyapunov function approach

In this paper, the problems of stability and stabilization are considered for a class of switched linear systems with slow switching and fast switching. A multiple convex Lyapunov function and a multiple discontinuous convex Lyapunov function are first introduced, under which the extended stability and stabilization results are derived with a mode‐dependent average dwell time switching strategy, where slow switching and fast switching are exerted on stable and unstable subsystems, respectively. These two types of Lyapunov functions are established in a constructive manner by virtue of a set of time‐varying functions. By using our proposed approaches, larger stability regions of system parameters are identified, and tighter bounds can be obtained for the mode‐dependent average dwell time. New mode‐dependent and time‐varying controllers are constructed for a class of switched control systems with stabilizable and unstabilizable subsystems as well. All the stability and stabilization conditions can be given in terms of strict linear matrix inequalities (LMIs), which can be checked easily by using recently developed algorithms in solving LMIs. Finally, two numerical examples are provided to show the effectiveness of the obtained results compared with the existing results.     

Passivity Analysis and Synthesis for a Class of Discrete‐Time Switched Stochastic Systems with Time‐Varying Delay

This paper deals with the problems of passivity and passification for a class of discrete‐time switched stochastic systems with time‐varying delay. Based on the average dwell time approach, the piecewise Lyapunov function technique, and the free‐weighting matrix method, a new Lyapunov functional is proposed and sufficient conditions for mean‐square exponential stability and stochastic passivity are developed under average dwell time switching. Moreover, an estimate of state decay can be calculated in terms of linear matrix inequalities (LMIs). Then, the solvability condition for passification is established and the corresponding controller is designed. Two numerical examples are given to show the effectiveness of the proposed methods.     

Stability and l2‐gain of discrete‐time switched systems with unstable modes

This paper addresses stability and l₂‐gain for discrete‐time switched systems with unstable modes based on slow/fast mode‐dependent average dwell time (MDADT) switching strategies. Firstly, by employing a class of multiple discontinuous Lyapunov functions (MDLFs) and developing a kind of alternative switching signals, the sufficient conditions on stability are established for the system without external disturbances under a slow/fast MDADT switching scheme with a tighter bounds on the dwell time. Furthermore, by defining indicator functions and exploring the features of slow/fast MDADT switching, the weighted l₂‐gain conditions are achieved for the system with external disturbances. Particularly, the criteria of stability and l₂‐gain are also established for the corresponding discrete‐time switched linear systems with unstable modes via the MDLFs method and the slow/fast MDADT switching strategy. Finally, two numerical examples are presented to illustrate the advantages of the proposed methods.     

Exponential stability of nonlinear impulsive switched systems with stable and unstable subsystems

Exponential stability and robust exponential stability relating to switched systems consisting of stable and unstable nonlinear subsystems are considered in this study. At each switching time instant, the impulsive increments which are nonlinear functions of the states are extended from switched linear systems to switched nonlinear systems. Using the average dwell time method and piecewise Lyapunov function approach, when the total active time of unstable subsystems compared to the total active time of stable subsystems is less than a certain proportion, the exponential stability of the switched system is guaranteed. The switching law is designed which includes the average dwell time of the switched system. Switched systems with uncertainties are also studied. Sufficient conditions of the exponential stability and robust exponential stability are provided for switched nonlinear systems. Finally, simulations show the effectiveness of the result.     

Robust state‐feedback control of stochastic state‐multiplicative discrete‐time linear switched systems with dwell time

Linear discrete‐time switched stochastic systems are considered, where the problems of mean square stability, stochastic l₂‐gain and state‐feedback control design are treated and solved. Solutions are obtained for both nominal and polytopic‐type uncertain systems. In all these problems, the switching obeys a dwell time constraint. In our solution, to each subsystem of the switched system, a Lyapunov function is assigned that is nonincreasing at the switching instants. The latter function is allowed to vary piecewise linearly, starting at the end of the previous switch instant, and it becomes time invariant after the dwell. In order to guarantee asymptotic stability, we require the Lyapunov function to be negative between consecutive switchings. We thus obtain Linear Matrix Inequalities conditions. Based on the solution of the stochastic l₂‐gain problem, we derive a solution to the state‐feedback control design, where we treat a variety of special cases. Being affine in the system matrices, all the aforementioned solutions are extended to the uncertain polytopic case. The proposed theory is demonstrated by a practical example taken from the field of flight control. Copyright © 2015 John Wiley & Sons, Ltd.     

On the stochastic finite-time stability for switched stochastic nonlinear systems

In this paper, the stochastic finite-time stability (SFTS) criteria for switched stochastic nonlinear systems (SSNS) with nonlocal Lipschitz coefficients are established. The main tools that we used here are mode-dependent limiting average dwell time (MD-LADT), stopping time technology and the stochastic analysis theory. The established criteria show that the diffusion coefficient, the drift coefficient and the switching law play crucial roles in obtaining the SFTS for the SSNS. Besides, our criteria provide some approaches on how to design the switching law to achieve the SFTS if a switched stochastic system contains SFTS subsystems and stochastic finite-time instable (SFTI) subsystems simultaneously. Finally, an example is given to illustrate the importance and usefulness of the theoretical results.     

Finite‐time stability of switched positive linear systems

This brief paper addresses the finite‐time stability problem of switched positive linear systems. First, the concept of finite‐time stability is extended to positive linear systems and switched positive linear systems. Then, by using the state transition matrix of the system and copositive Lyapunov function, we present a necessary and sufficient condition and a sufficient condition for finite‐time stability of positive linear systems. Furthermore, two sufficient conditions for finite‐time stability of switched positive linear systems are given by using the common copositive Lyapunov function and multiple copositive Lyapunov functions, a class of switching signals with average dwell time is designed to stabilize the system, and a computational method for vector functions used to construct the Lyapunov function of systems is proposed. Finally, a concrete application is provided to demonstrate the effectiveness of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd.     

Tracking control for switched linear systems with time‐delay: a state‐dependent switching method

Tracking control for switched linear systems with time‐delay is investigated in this paper. Based on the state‐dependent switching method, sufficient conditions for the solvability of the tracking control problem are given. We use single Lyapunov function technique and a typical hysteresis switching law to design a tracking control law such that the H_∞ model reference tracking performance is satisfied. The controller design problem can be solved efficiently by using linear matrices inequalities. Since convex combination techniques are used to derive the delay independent criteria, some subsystems are allowed to be unstable. It is highly desirable that a non‐switched time‐delay system can not earn such property. Simulation example shows the feasibility and validity of the switching control law. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

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.     

Exponential Stability and Asynchronous Stabilization of Switched Systems With Stable and Unstable Subsystems

This paper deals with the exponential stability and asynchronous stabilization of continuous‐time switched systems. By delicately constructed piecewise Lyapunov‐like functions and the minimum dwell time switching method, exponential stability of the switched systems with stable or unstable subsystems is obtained. Based on the result of the stability, the problem of controller design of the switched systems under asynchronous switching is also solved, and the delay that causes asynchronous phenomena can be unbounded. The stability results and control laws of the switched systems are formulated in the form of linear matrix inequalities that are numerically feasible. Finally, two illustrative numerical examples are presented to show the effectiveness of the obtained theoretical results.     

Stability analysis of switched singular stochastic linear systems

ABSTRACT

This paper is devoted to study the stability of switched singular stochastic linear systems with both stable and unstable subsystems. By using the method of multiple Lyapunov functions and the notion of average dwell time, we provide sufficient conditions for the exponential mean-square stability of switched singular stochastic systems in terms of a proper switching rule and the linear matrix inequalities. An example is given to illustrate the effectiveness of the obtained results.     

Robust stability analysis of uncertain switched linear systems with unstable subsystems

The problem of robust stability for switched linear systems with all the subsystems being unstable is investigated. Unlike the most existing results in which each switching mode in the system is asymptotically stable, the subsystems may be unstable in this paper. A necessary condition of stability for switched linear systems is first obtained with certain hypothesis. Then, under two assumptions, sufficient conditions of exponential stability for both deterministic and uncertain switched linear systems are presented by using the invariant subspace theory and average dwell time method. Moreover, we further develop multiple Lyapunov functions and propose a method for constructing multiple Lyapunov functions for the considered switched linear systems with certain switching law. Several examples are included to show the effectiveness of the theoretical findings.     

Exponential Stabilization of Switched Discrete‐Time Systems with All Unstable Modes

This paper studies the exponential stabilization of switched discrete‐time systems whose subsystems are unstable. A new sufficient condition for the exponential stability of the class of systems is proposed. The result obtained is based on the determination of a lower bound of the maximum dwell time by virtue of the multiple Lyapunov functions method. The key feature is that the given stability condition does not need the value of the Lyapunov function to uniformly decrease at every switching instant. An example is provided to illustrate the effectiveness of the proposed result.     

Finite‐time stability of switched nonlinear time‐varying systems via indefinite Lyapunov functions

This note considers the problem of finite‐time stability (FTS) for switched nonlinear time‐varying systems. First, a relaxed condition is proposed to verify the FTS of nonlinear time‐varying systems by using an indefinite Lyapunov function. Then, the result obtained is extended to study the FTS of switched nonlinear time‐varying systems. Several relaxed conditions are given by using a common indefinite Lyapunov function and multiple indefinite Lyapunov functions. Moreover, the corresponding estimates on convergence regions and times of systems are also given. Comparing with the existing results, the conditions obtained allow the time derivative of Lyapunov functions of subsystems (or systems) to be indefinite and all subsystems to be not finite‐time stable or even unstable. Finally, a numerical example is given to illustrate the theoretical results.     

Stability analysis of switched positive linear systems with stable and unstable subsystems

This paper addresses the stability problem of switched positive linear systems with stable and unstable subsystems. Based on a multiple linear copositive Lyapunov function, and by using the average dwell time approach, some sufficient stability criteria of global uniform exponential stability are established in both the continuous-time and the discrete-time cases, respectively. Finally, some numerical examples are given to show the effectiveness of the proposed results.     

Stability of discrete-time switched systems with admissible edge-dependent switching signals

The problem of stability is studied in this paper for a class of discrete-time switched systems with unstable subsystems. Two new definitions of slow switching and fast switching on the basis of admissible edge-dependent average dwell time are proposed, respectively. Some conditions are established by using multiple Lyapunov function method to guarantee the global uniform exponential stability of discrete-time switched systems. Finally, a numerical example is presented to demonstrate the effectiveness of the proposed results.     

Stability analysis and synthesis for linear impulsive stochastic systems

This paper presents a general framework for analyzing stability of linear impulsive stochastic systems (LISSs). Some simple mean square stability criteria for the three types of LISSs are firstly derived by analyzing an equivalent system. By exploring the hybrid characteristics of impulsive systems, the novel quasi‐periodic composite polynomial Lyapunov function and the time‐varying discretized Lyapunov function are developed, which leads to unified dwell‐time–based criteria for mean square stability and almost sure stability of LISSs without imposing the stability condition on continuous‐ and discrete‐time dynamics. Next, based on the established stability criteria, the synthesis problem of state‐feedback controller is solved. The computational complexity and the comparison with existing results on the deterministic systems are discussed. Finally, numerical examples are provided to illustrate the usefulness of the proposed results.     

Stabilization of switched continuous-time systems with all modes unstable via dwell time switching

Stabilization of switched systems composed fully of unstable subsystems is one of the most challenging problems in the field of switched systems. In this brief paper, a sufficient condition ensuring the asymptotic stability of switched continuous-time systems with all modes unstable is proposed. The main idea is to exploit the stabilization property of switching behaviors to compensate the state divergence made by unstable modes. Then, by using a discretized Lyapunov function approach, a computable sufficient condition for switched linear systems is proposed in the framework of dwell time; it is shown that the time intervals between two successive switching instants are required to be confined by a pair of upper and lower bounds to guarantee the asymptotic stability. Based on derived results, an algorithm is proposed to compute the stability region of admissible dwell time. A numerical example is proposed to illustrate our approach.     

Unified mode‐dependent average dwell time stability criteria for discrete‐time switched systems

In this article, a unified mode‐dependent average dwell time (MDADT) stability result is investigated, which could be applied to switched systems with an arbitrary combination of stable and unstable subsystems. Combined with MDADT analysis method, we classified subsystems into two categories: switching stable subsystems and switching unstable subsystems. State divergence caused by switching unstable subsystems could be compensated by activating switching stable subsystems for a sufficiently long time. Based on the above considerations, a new globally exponentially stability condition was proposed for discrete‐time switched linear systems. Under the premise of not resolving the LMIs, the MDADT boundary of the new stability condition is allowed to be readjusted according to the actual switching signal. Furthermore, the new stability result is a generalization of the previous one, which is more suitable for the case of more unstable subsystems. Some simulation results are given to show the advantages of the theoretic results obtained.     

Stability and stabilization of switched stochastic systems under asynchronous switching

This paper studies the stability and stabilization problems for a class of switched stochastic systems under asynchronous switching. The asynchronous switching refers to that the switching of the candidate controllers does not coincide with the switching of system modes. Two situations are considered: (1) time-delayed switching situation, that is, the switching of the candidate controllers has a lag to the switching of the system modes; (2) mismatched switching situation, the switching of the candidate controllers does not match the switching of the system modes. Using average dwell time and Lyapunov-like function, sufficient conditions are established for stochastic input-to-state stability of the whole system. Also, the stabilizing controller design approach is proposed for switched stochastic linear systems. The minimal average dwell time and the controller gain are achieved. Finally, a numerical example is used to demonstrate the validity of the developed results.