切换线性系统稳定性研究进展

切换线性系统的稳定性分析和稳定化设计是近年来研究的热点.为此,对近期关于这一问题研究所得到的主要成果进行综述.首先给出任意切换以及带约束切换两种情况下系统稳定性分析的主要结论;然后给出切换线性系统稳定化中的主要方法;最后简要概述目前切换系统的实用稳定性和有限时间稳定性问题,并就这一领域今后的发展方向进行了展望.     

On linear co-positive Lyapunov functions for sets of linear positive systems

In this paper we derive necessary and sufficient conditions for the existence of a common linear co-positive Lyapunov function for a finite set of linear positive systems. Both the state dependent and arbitrary switching cases are considered. Our results reveal an interesting characterisation of “linear” stability for the arbitrary switching case; namely, the existence of such a linear Lyapunov function can be related to the requirement that a number of extreme systems are Metzler and Hurwitz stable. Examples are given to illustrate the implications of our results.     

Stabilization of continuous-time switched nonlinear systems

The paper considers three problems for continuous-time nonlinear switched systems. The first result of this paper is a open-loop stabilization strategy based on dwell time computation. The second considers a state switching strategy for global stabilization. The strategy is of closed loop nature (trajectory dependent) and is designed from the solution of what we call nonlinear Lyapunov–Metzler inequalities from which the stability condition is expressed. Finally, results on the stabilization of nonlinear time varying polytopic systems are provided.     

Realization of continuous-time positive linear systems

Positive linear systems are used in biomathematics, economics, and other research areas. For discrete-time positive linear systems, part of the realization problem has been solved. In this paper the solution of the corresponding problem for continuous-time positive linear systems will be presented, which can be deduced from that of the discrete-time case by a transformation. Sufficient and necessary conditions for the existence of a positive realization are presented. To solve the problem of minimality, the solution of the factorization of positive matrices is needed.     

Stability of switched positive linear delay systems with mixed impulses

ABSTRACT

This paper investigates the global exponential stability of both continuous-time and discrete-time switched positive linear delay systems with the coexistence of destabilising and stabilising impulses. Based on the comparison principle, with the help of discretised multiple linear copositive Lyapunov functions, some delay-independent stability results are obtained. It should be noted that it is the first time that discrete-time switched delay systems with asynchronous switching and impulses are considered, and the switching effect can be stabilising in the obtained results of this paper. Some numerical examples are provided to demonstrate the effectiveness and superiority of the derived results.     

Dual approach to stability and stabilisation of uncertain switched positive systems

This paper addresses two kinds of dual approaches to stability and stabilisation of uncertain switched positive systems under arbitrary switching and average dwell-time switching, respectively. The uncertainties in systems refer to polytopic ones. A new parameter-dependent switched linear copositive Lyapunov function is first proposed for uncertain switched positive systems. By using the new Lyapunov function associated with arbitrary switching and average dwell-time switching, respectively, sufficient conditions for the stability of the systems are established. Two alternative stability criteria based on two kinds of dual approaches are addressed. It is shown that the alternative criteria hold for not only the primal switched positive system but also its dual system. Then, the stabilisation of primal and dual switched positive systems under arbitrary switching and average dwell-time switching is solved, respectively. All present conditions are solvable in terms of linear programming. By some comparisons with existing results, the less conservativeness of the obtained results is verified. Finally, a practical example is provided to illustrate the effectiveness of the theoretical findings.     

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.     

A Lagrange duality characterisation for stability under arbitrary switching in switched positive linear systems

The present communication is concerned with uniform exponential stability, under arbitrary switching, in discrete-time switched positive linear systems. Lagrange duality is used in order to obtain a new characterisation for uniform exponential stability which is in terms of sets of inequalities involving each of the matrices that represent the modes of the system. These sets of inequalities are shown to generalise the classical linear Lyapunov inequality that characterises, in positive matrices, the property of being Schur. Each solution to these sets of inequalities is shown to provide a representation, in terms of a number of linear functionals, for a common Lyapunov function for the switched positive linear system. A result is further presented which conveys to, a conservative upper bound on the minimum required number of linear functionals (in the above mentioned representation), and also to a method for computing them. Our proof for the aforementioned characterisation is based on another (equivalent) characterisation, in terms of the solvability of a dynamic programming equation associated to the switched positive linear system, which is also reported in the paper. In particular, it is shown that the associated dynamic programming equation has at most one solution. And this solution is shown to be convex, monotonic, positively homogeneous, and it yields a common Lyapunov function for the switched positive linear system.     

Necessary conditions for positive realizability of continuous-time linear systems

This paper deals with the positive realization problem. The problem is to find, from a given transfer function, a state equation in which state variables and the output take nonnegative values whenever initial states and inputs are nonnegative. Necessary conditions are investigated and a new one is given, together with some related results.     

Improved results on stability and stabilization criteria for uncertain linear systems with time-varying delays

In this paper, the problems of stability and stabilization for linear systems with time-varying delays and norm-bounded parameter uncertainties are considered. By constructing augmented Lyapunov functionals and utilizing auxiliary function-based integral inequalities, improved delay-dependent stability and stabilization criteria for guaranteeing the asymptotic stability of the system are proposed with the framework of linear matrix inequalities. Four numerical examples are included to show that the proposed results can reduce the conservatism of stability and stabilization criteria by comparing maximum delay bounds.     

On the reachability in any fixed time for positive continuous-time linear systems

This paper deals with the reachability of continuous-time linear positive systems. The reachability of such systems, which we will call here the strong reachability, amounts to the possibility of steering the state in any fixed time to any point of the positive orthant by using nonnegative control functions. The main result of this paper essentially says that the only strongly reachable positive systems are those made of decoupled scalar subsystems. Moreover, the strongly reachable set is also characterized.     

On the quadratic stability of switched linear systems associated with symmetric transfer function matrices

In this paper we give necessary and sufficient conditions for weak and strong quadratic stability of a class of switched linear systems consisting of two subsystems, associated with symmetric transfer function matrices. These conditions can simply be tested by checking the eigenvalues of the product of two subsystem matrices. This result is an extension of the result by Shorten and Narendra for strong quadratic stability, and the result by Shorten et al. on weak quadratic stability for switched linear systems. Examples are given to illustrate the usefulness of our results.     

On the asymptotic stability of switched homogeneous systems

The stability of switched systems generated by the family of autonomous subsystems with homogeneous right-hand sides is investigated. It is assumed that for each subsystem the proper homogeneous Lyapunov function is constructed. The sufficient conditions of the existence of the common Lyapunov function providing global asymptotic stability of the zero solution for any admissible switching law are obtained. In the case where we can not guarantee the existence of a common Lyapunov function, the classes of switching signals are determined under which the zero solution is locally or globally asymptotically stable. It is proved that, for any given neighborhood of the origin, one can choose a number L>0 (dwell time) such that if intervals between consecutive switching times are not smaller than L then any solution of the considered system enters this neighborhood in finite time and remains within it thereafter.     

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.     

线性切换容错控制系统稳定性的新判据*

研究了线性切换容错控制系统的稳定性问题。利用分段李雅普诺夫函数方法,结合梅茨勒矩阵的性质和矩阵不等式的分析技巧,得到了基于李雅普诺夫—梅兹勒线性矩阵不等式判定系统稳定的新结果。设计依赖于状态的切换规则便于计算、易于检验。最后利用MATLAB工具箱得到的仿真实例验证了本结果的可行性。     

Reachability of linear switched systems: Differential geometric approach

The paper investigates the structure of the reachable set of linear switched systems. The structure of the reachable set is determined using techniques from classical nonlinear systems theory, namely, the theory of orbits developed by H. Sussman and the realization theory for nonlinear systems developed by B. Jakubczyk.     

Stability analysis and antiwindup design of uncertain discrete-time switched linear systems subject to actuator saturation

This paper investigates the stability analysis and antiwindup design problem for a class of discrete-time switched linear systems with time-varying norm-bounded uncertainties and saturating actuators by using the switched Lyapunov function approach.Supposing that a set of linear dynamic output controllers have been designed to stabilize the switched system without considering its input saturation,we design antiwindup compensation gains in order to enlarge the domain of attraction of the closed-loop system in the presence of saturation.Then,in terms of a sector condition,the antiwindup compensation gains which aim to maximize the estimation of domain of attraction of the closed-loop system are presented by solving a convex optimization problem with linear matrix inequality(LMI)constraints.A numerical example is given to demonstrate the effectiveness of the proposed design method.     

Robust stability of positive switched systems with dwell time

This paper studies robust stability of positive switched systems (PSSs) with polytopic uncertainties in both discrete-time and continuous-time contexts. By using multiple linear copositive Lyapunov functions, a sufficient condition for stability of PSSs with dwell time is addressed. Being different from time-invariant multiple linear copositive Lyapunov functions, the Lyapunov functions constructed in this paper are time-varying during the dwell time and time-invariant afterwards. Then, robust stability of PSSs with polytopic uncertainties is solved. All conditions are solvable via linear programming. Finally, illustrative examples are given to demonstrate the validity of the proposed results.     

H-infinity filtering for discrete-time switched linear systems under arbitrary switching

This paper is concerned with the problem of H-infinity filtering for discrete-time switched linear systems under arbitrary switching laws.New sufficient conditions for the solvability of the problem are given via switched quadratic Lyapunov functions.Based on Finsler’s lemma,two sets of slack variables with special structure are introduced to provide extra degrees of freedom in optimizing the guaranteed H-infinity performance.Compared to the existing methods,the proposed one has better performances and less conservatism.An example is given to illustrate its effectiveness.