Superstable Linear Control Systems. I. Analysis

The notion of superstability of linear control systems was introduced. Superstability which is a sufficient condition for stability was formulated in terms of linear constraints on the entries of a matrix or the coefficients of a characteristic polynomial. In the first part of the paper, the properties of superstable systems were studied. The norms of solutions were proved to decrease exponentially monotonically in the absence of perturbations, and the solutions were proved to be uniformly bounded in the presence of bounded perturbations. A generalization to nonlinear and time varying systems was discussed. Spectral properties of superstable systems were studied. For interval matrices, a complete solution was given to the problem of robust superstability.     

A structural approach to robust control

We consider the problem of stabilizing linear stationary parametrically uncertain systems with a guaranteed stability margin. The methodology of our approach is based on the synthesis of superstable closed systems, done with the procedures derived from the block control principle and their modifications, the procedures consisting of sequential establishment of local connections in elementary blocks that provide for the superstability of each block and the closed system as a whole in the new coordinate basis. The fact that the notion of superstability is formulated in terms of the elements of the system matrix based on inequalities lets us provide for robust stability for all admissible values of indefinite parameters in such systems. The robust control algorithms that we have developed are applicable to a practically significant class of linear systems which, given that parameters change in known ranges, preserve structural controllability properties defined by the nominal system.     

Stabilization of discrete-time positive switched linear systems: A nonzero elements chain method

This article studies the stability problem of discrete-time positive switched linear systems by introducing a nonzero elements chain method. Compared to existing works, the highlight of this article is reflected in the inclusiveness of the subsystem, that is, each subsystem is allowed to be stable, marginally stable or even unstable. Some sufficient conditions depending on the characteristics of system matrices and the time-dependent switching signals are established, so as to guarantee the exponential stability of positive switched linear systems. In addition, the main result is extended to the case of time-varying delay. At last, the superiority of the obtained results is well verified by numerical examples.     

Design of stabilizing switching control laws for discrete- and continuous-time linear systems using piecewise-linear Lyapunov functions

In this paper, the stability of switched linear systems is investigated using piecewise linear Lyapunov functions. In particular, we identify classes of switching sequences that result in stable trajectories. Given a switched linear system, we present a systematic methodology for computing switching laws that guarantee stability based on the matrices of the system. In the proposed approach, we assume that each individual subsystem is stable and admits a piecewise linear Lyapunov function. Based on these Lyapunov functions, we compose 'global' Lyapunov functions that guarantee stability of the switched linear system. A large class of stabilizing switching sequences for switched linear systems is characterized by computing conic partitions of the state space. The approach is applied to both discrete-time and continuous-time switched linear systems.     

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.     

切换系统的不变性原理与不变集的状态反馈镇定

证明了一类切换系统的一个不变性原理,并将输入对状态稳定的概念推广到输入对系统某个非负能量函数稳定的情况.基于这个不变性原理以及输入对系统能量函数稳定的概念,利用多Lyapunov函数方法提出并证明了一类具有Lyapunov稳定子系统的切换系统的不变集可状态反馈镇定的条件.最后讨论了输入对系统能量函数稳定与输入对状态稳定的关系.仿真结果证明了该方法的可行性.     

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.     

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.     

切换线性超循环系统的稳定性分析

超循环---生物学中重要模型, 具有广泛的实际背景. 本文将超循环系统扩展为切换超循环系统. 循环矩阵的循环结构为研究切换超循环系统的稳定性提供了有效的方法, 给出切换线性时变超循环系统在任意切换律下渐近稳定的充要条件和切换线性定常超循环系统可切换镇定的充分条件.     

On the Stability of Switched Positive Linear Systems

It was recently conjectured that the Hurwitz stability of the convex hull of a set of Metzler matrices is a necessary and sufficient condition for the asymptotic stability of the associated switched linear system under arbitrary switching. In this note, we show that (1) this conjecture is true for systems constructed from a pair of second-order Metzler matrices; (2) the conjecture is true for systems constructed from an arbitrary finite number of second-order Metzler matrices; and (3) the conjecture is in general false for higher order systems. The implications of our results, both for the design of switched positive linear systems, and for research directions that arise as a result of our work, are discussed toward the end of the note.     

Superstable Linear Control Systems. II. Design

The notion of superstability introduced in [1] is applied to the design of stabilizing and optimal controllers. It is shown that a static output feedback controller which ensures superstability of the closed-loop system can be found (provided it exists) by means of linear programming (LP) techniques; finding a superstable matrix in the given affine family is a generalization of this problem. The ideology of superstability is also shown to be fruitful in optimal and robust control. This is exemplified by the problems of rejection of bounded disturbances, optimization of the integral performance index which involves absolute values (rather than squares) of the control and state, and by stabilization of an interval matrix family and simultaneous stabilization.     

Robust stability bounds on time-varying perturbations for state-space models of linear discrete-time systems

The stability robustness of linear discrete-time systems in the time domain is addressed using the Lyapunov approach. Bounds on linear time-varying perturbations that maintain the stability of an asymptotically stable linear time-invariant discrete-time nominal system are obtained for both structured and unstructured independent perturbations. Bounds are also derived assuming that various elements of the system matrix are perturbed dependently. The result for the structured perturbation case is extended to the stability analysis of interval matrices.     

PIECEWISE LYAPUNOV FUNCTIONS FOR SWITCHED SYSTEMS WITH AVERAGE DWELL TIME

The stability properties of linear switched systems consisting of both Hurwitz stable and unstable subsystems are investigated by using piecewise Lyapunov functions incorporated with an average dwell time approach. It is shown that if the average dwell time is chosen sufficiently large and the total activation time ratio between Hurwitz stable and unstable subsystems is not smaller than a specified constant, then exponential stability of a desired degree is guaranteed. The above result is also extended to the case where nonlinear norm‐bounded perturbations exist.     

Delay‐Dependent Exponential Stability for Discrete‐Time Singular Switched Systems with Time‐Varying Delay

In this paper, the exponential stability problem is investigated for a class of discrete‐time singular switched systems with time‐varying delay. By using a new Lyapunov functional and average dwell time scheme, a delay‐dependent sufficient condition is established in terms of linear matrix inequalities for the considered system to be regular, causal, and exponentially stable. Different from the existing results, in the considered systems the corresponding singular matrices do not need to have the same rank. A numerical example is given to demonstrate the effectiveness of the proposed result.     

一类切换线性系统的稳定性判别方法

研究了子系统中系统矩阵为对角标准型或者Jordan标准型的切换线性系统的稳定性．用状态方程的解来分析系统能量函数（状态向量的2范数）的单调性,得到系统在任意切换序列下渐近稳定的充分条件．另外,根据这些条件可以比较容易地设计出渐近稳定的切换序列．最后通过一个数值例子来说明所得到结论的效果．     

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 dwell time approach to the stability of switched linear systems based on the distance between eigenvector sets

In this article, a sufficient condition on the minimum dwell time that guarantees the stability of switched linear systems is given. The proposed method interprets the stability of switched linear systems through the distance between the eigenvector sets of subsystem matrices. Thus, an explicit relation in view of stability is obtained between the family of the involved subsytems and the set of admissible switching signals.     

On Padé approximations, quadratic stability and discretization of switched linear systems

In this note we consider the stability preserving properties of diagonal Padé approximations to the matrix exponential. We show that while diagonal Padé approximations preserve quadratic stability when going from continuous-time to discrete-time, the converse is not true. We discuss the implications of this result for discretizing switched linear systems. We also show that for continuous-time switched systems which are exponentially stable, but not quadratically stable, a Padé approximation may not preserve stability.     

Stability analysis of switched systems with stable and unstable subsystems: An average dwell time approach

We study the stability properties of switched systems consisting of both Hurwitz stable and unstable linear time-invariant subsystems using an average dwell time approach. We propose a class of switching laws so that the entire switched system is exponentially stable with a desired stability margin. In the switching laws, the average dwell time is required to be sufficiently large, and the total activation time ratio between Hurwitz stable subsystems and unstable subsystems is required to be no less than a specified constant. We also apply the result to perturbed switched systems where nonlinear vanishing or non-vanishing norm-bounded perturbations exist in the subsystems, and we show quantitatively that, when norms of the perturbations are small, the solutions of the switched systems converge to the origin exponentially under the same switching laws.     

Minimum Dwell Time for Global Exponential Stability of a Class of Switched Positive Nonlinear Systems

This paper will investigate global exponential stability analysis for a class of switched positive nonlinear systems under minimum dwell time switching, whose nonlinear functions for each subsystem are constrained in a sector field by two odd symmetric piecewise linear functions and whose system matrices for each subsystem are Metzler. A class of multiple time-varying Lyapunov functions is constructed to obtain the computable sufficient conditions on the stability of such switched nonlinear systems within the framework of minimum dwell time switching. All present conditions can be solved by linear/nonlinear programming techniques. An example is provided to demonstrate the effectiveness of the proposed result.