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.      

Switching Stabilizability for Continuous-Time Uncertain Switched Linear Systems

This paper investigates the switching stabilizability problem for a class of continuous-time switched linear systems with time-variant parametric uncertainties. First, a necessary and sufficient condition for the asymptotic stabilizability of such uncertain switched linear system is derived, under the assumption that the closed-loop switched system does not generate sliding motions. Then, an additional condition is introduced to exclude the possibility of unstable sliding motions. Finally, a necessary and sufficient for the asymptotic stabilizability of such continuous-time uncertain switched linear systems is presented. This result improves upon conditions found in the literature which are either sufficient only or necessary only     

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.     

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.     

一类非线性离散切换系统基于观测器的指数镇定

对一类离散非线性切换系统, 考虑了基于观测器的指数镇定问题. 借助微分中值定理(DMVT), 将非线性切换系统转化为线性参数(LPV)切换系统. 当状态变量不完全可获得时, 基于多Lyapunov函数方法, 给出系统在基于观测器的输出反馈控制器下指数镇定的充分条件. 所设计的滞后切换规则能够避免产生滑动模态. 并且将结果推广到系统方程含有不确定性的情况. 最后, 仿真例子说明了设计方法的有效性.     

具有松弛条件的一类切换模糊系统的稳定性分析

使用切换技术以及多Lyapuriov函数方法．研究一类切换模糊系统的稳定性问题．给出了有连续控制输入时该切换模糊系统的一种松弛稳定性条件,避免了并行分配补偿法中固模糊规则数较多而求解公共矩阵P的困难,同时给出了实现系统全局渐近稳定的切换策略．主要条件以LMI的形式给出,具有较强的可解性．空气调节系统的设计实例表明了所提出设计方法的可行性和有效性．     

Asymptotic stabilization of driftless systems

Issues of asymptotic stabilization of a class of non-linear driftless systems are presented. In addition to the necessary and sufficient condition for the existence of a smooth time-invariant asymptotic stabilizer, sufficient condition for the existence of a quadratic-type Lyapunov function candidate is also proposed herein to alleviate the construction of stabilizing control laws. Following the deduction of the equivalence of the sufficient condition and the determination of the local definiteness of a defined scalar function, the stabilizability checking conditions are then derived in terms of system dynamics and its derivatives at the origin only. These are achieved by taking Taylor's series expansion on system dynamics. The derived conditions are shown to be consistent with those obtained by Brockett. Comparative results of Liaw and Liang are also included. Finally, examples are given to demonstrate the use of the main results.     

Decentralized measurement feedback stabilization of large-scale systems via control vector Lyapunov functions

This paper studies the problem of decentralized measurement feedback stabilization of nonlinear interconnected systems. As a natural extension of the recent development on control vector Lyapunov functions, the notion of output control vector Lyapunov function (OCVLF) is introduced for investigating decentralized measurement feedback stabilization problems. Sufficient conditions on (local) stabilizability are discussed which are based on the proposed notion of OCVLF. It is shown that a decentralized controller for a nonlinear interconnected system can be constructed using these conditions under an additional vector dissipation-like condition. To illustrate the proposed method, two examples are given.     

Global stabilization of semilinear systems using switching controls

This paper considers feedback stabilization of distributed semilinear systems using switching controls. The semilinear processes, together with the switched controllers, constitute a type of switched nonlinear system. Three kinds of stabilizabilities, namely weak, strong and exponential stabilizabilities, are investigated one by one. Sets of sufficient conditions are obtained for each case. A necessary condition for strong stabilizability is given. The stabilizing control is characterized using a minimization problem. Applications to parabolic and hyperbolic like equations are considered.     

Sufficient and necessary conditions for discrete-time nonlinear switched systems with uniform local exponential stability

In this paper, we investigate sufficient and necessary conditions of uniform local exponential stability (ULES) for the discrete-time nonlinear switched system (DTNSS). We start with the definition of T-step common Lyapunov functions (CLFs), which is a relaxation of traditional CLFs. Then, for a time-varying DTNSS, by constructing such a T-step CLF, a necessary and sufficient condition for its ULES is provided. Afterwards, we strengthen it based on a T-step Lipschitz continuous CLF. Especially, when the system is time-invariant, by the smooth approximation theorem, the Lipschitz continuity condition of T-step CLFs can further be replaced by continuous differentiability; and when the system is time-invariant and homogeneous, due to the extension of Weierstrass approximation theorem, T-step continuously differentiable CLFs can even be strengthened to be T-step polynomial CLFs. Furthermore, three illustrative examples are additionally used to explain our main contribution. In the end, an equivalence between time-varying DTNSSs and their corresponding linearisations is discussed.     

Controllability and stabilizability of switched linear-systems

In this paper, we study the conjecture made in (IEEE Trans. Automat. Control 47 (8) (2002) 1401) concerning the existence of basic switching sequence for switched systems, and give it an affirmative answer and a construction method. This paper proves that for switched linear system, there exists a switching sequence such that the controllable state set of this switching sequence is equal to the controllable state set of the system. Hence, the controllability can be realized by using only one switching sequence. Then, the result is extended to the systems with time-delay in controls. Furthermore, we define the stabilizability of switched linear systems, and based on certain transformation of state coordinate, we give a necessary and sufficient condition for the stabilizability. Two examples are presented to illustrate the obtained results.     

Necessary and sufficient stability condition for second-order switched systems: a phase function approach

To find a unified approach for the stability analysis of second-order switched system, the concept of phase function is proposed in this paper. First, the basic properties of phase function are explored. Following this concept and its properties, the phase-based stability criterion is investigated based on the Lyapunov theory, and a necessary and sufficient stability condition is obtained in the phase function approach. Moreover, the connection between phase-based stability conditions and algebraic condition of system matrices is also discussed. Finally, numerical examples are provided to exemplify the main result and make necessary comparisons with the existing methods.     

切换奇异布尔网络的稳定性分析

基因调控网络的稳定性分析是系统生物学的研究热点问题之一.本文利用矩阵半张量积方法研究了切换奇异布尔网络的稳定性问题.首先给出了切换奇异布尔网络的代数表示,基于该代数表示,建立了系统解存在唯一的充要条件.然后通过将切换奇异布尔网络转化为等价的切换布尔网络,分别得到了系统在任意切换下稳定以及切换可稳的充要条件.最后给出例子验证所得结果的有效性.     

Non-conservative matrix inequality conditions for stability/stabilizability of linear differential inclusions

This paper shows that the matrix inequality conditions for stability/stabilizability of linear differential inclusions derived from two classes of composite quadratic functions are not conservative. It is established that the existing stability/stabilizability conditions by means of polyhedral functions and based on matrix equalities are equivalent to the matrix inequality conditions. This implies that the composite quadratic functions are universal for robust, possibly constrained, stabilization problems of linear differential inclusions. In particular, a linear differential inclusion is stable (stabilizable with/without constraints) iff it admits a Lyapunov (control Lyapunov) function in these classes. Examples demonstrate that the polyhedral functions can be much more complex than the composite quadratic functions, to confirm the stability/stabilizability of the same system.     

On Linear Copositive Lyapunov Functions and the Stability of Switched Positive Linear Systems

We consider the problem of common linear copositive Lyapunov function existence for positive switched linear systems. In particular, we present a necessary and sufficient condition for the existence of such a function for switched systems with two constituent linear time-invariant systems. Several applications of this result are also given.     

Stability analysis for a class of switched nonlinear systems

This paper addresses the stability analysis of a class of switched nonlinear systems. The switched systems have uncertain nonlinear functions constrained in a sector set, which are called admissible sector nonlinearities. A sufficient condition in terms of linear inequalities is presented to guarantee the existence of a common Lyapunov function, and thereby to ensure that the switched system is stable for an arbitrary switching signal and any admissible sector nonlinearities. A constructive algorithm based on the modified Gaussian elimination procedure is given to find the solutions of the linear inequalities. The obtained results are applied to a population model with switchings of parameter values and the conditions of ultimate boundedness of its solutions are investigated. Another example of an automatic control system is considered to demonstrate the effectiveness of the proposed approaches.     

Frequency-domain stability conditions for hybrid systems

Consideration was given to a special class of the hybrid systems with switchings of time-invariant linear right-hand sides. A narrower subclass of such systems, that of connected switched linear systems, was specified among them. The necessary and sufficient frequencydomain conditions (criteria) for the existence of a common quadratic Lyapunov function providing stability of the switched systems were proposed for them. The specified subclass includes control systems with several nonstationary nonlinearities from the finite sectors that are the matter at issue of the theory of absolute stability. For the connected switched linear systems of a special kind (triangular type systems), the separate necessary and separate sufficient existence conditions were obtained for such Lyapunov functions. The interrelations between these conditions were discussed in the example.     

Input-to-state stability of switched nonlinear systems

The input-to-state stability （ISS） problem is studied for switched systems with infinite subsystems. By using multiple Lyapunov function method, a sufficient ISS condition is given based on a quantitative relation of the control and the values of the Lyapunov functions of the subsystems before and after the switching instants. In terms of the average dwell-time of the switching laws, some sufficient ISS conditions are obtained for switched nonlinear systems and switched linear systems, respectively.     

Stabilization of Finite Automata with Application to Hybrid Systems Control

This paper discusses the state feedback stabilization problem of a deterministic finite automaton (DFA), and its application to stabilizing model predictive control (MPC) of hybrid systems. In the modeling of a DFA, a linear state equation representation recently proposed by the authors is used. First, this representation is briefly explained. Next, after the notion of equilibrium points and stabilizability of the DFA are defined, a necessary and sufficient condition for the DFA to be stabilizable is derived. Then a characterization of all stabilizing state feedback controllers is presented. Third, a simple example is given to show how to follow the proposed procedure. Finally, control Lyapunov functions for hybrid systems are introduced based on the above results, and the MPC law is proposed. The effectiveness of this method is shown by a numerical example.     

Homogeneous Polynomial Forms for Simultaneous Stabilizability of Families of Linear Control Systems: A Tensor Product Approach

This note uses the formalism of tensor products in order to deal with the problem of simultaneous stabilizability of a family of linear control systems by means of Lyapunov functions which are homogeneous polynomial forms. While the feedback synthesis seems to be nonconvex, the simultaneous stability by means of homogeneous polynomial forms of the uncontrollable modes yields (convex) necessary but not sufficient conditions for simultaneous stabilizability.