Quadratic stability and stabilization of bimodal piecewise linear systems

This paper deals with quadratic stability and feedback stabilization problems for continuous bimodal piecewise linear systems. First, we provide necessary and sufficient conditions in terms of linear matrix inequalities for quadratic stability and stabilization of this class of systems. Later, these conditions are investigated from a geometric control point of view and a set of sufficient conditions (in terms of the zero dynamics of one of the two linear subsystems) for feedback stabilization are obtained.     

Null controllability of planar bimodal piecewise linear systems

This article investigates the null controllability of planar bimodal piecewise linear systems, which consist of two second order LTI systems separated by a line crossing through the origin. It is interesting to note that even when both subsystems are controllable in the classical sense, the whole piecewise linear system may be not null controllable. On the other hand, a piecewise linear system could be null controllable even when it has uncontrollable subsystems. First, the evolution directions from any non-origin state are studied from the geometric point of view, and it turns out that the directions usually span an open half space. Then, we derive an explicit and easily verifiable necessary and sufficient condition for a planar bimodal piecewise linear system to be null controllable. Finally, the article concludes with several numerical examples and discussions on the results and future work.     

一类非线性组合大系统的稳定性

本文研究一类比较广泛的非线性组合大系统的稳定性问题.基于矩阵代数Riccati 方程存在对称正定解的新判据(引理2),给出了检验这类非线性组合大系统稳定性的新准 则,所给出的稳定性准则只涉及子系统的稳定性以及子系统之间的互联项,因此,当应用该准 则检验这类非线性组合大系统的稳定性时,避免了构造子系统以及大系统的Lyapunov函数 的困难.     

Graph‐Based Dwell Time Computation Methods for Discrete‐Time Switched Linear Systems

Analytical computation methods are proposed for evaluating the minimum dwell time and the average dwell time guaranteeing the asymptotic stability of a discrete‐time switched linear system whose switchings are assumed to respect a given directed graph. The minimum and average dwell time can be found using the graph that governs the switchings, and the associated weights. This approach, which is used in a previous work for continuous‐time systems having non‐defective subsystems, has been adapted to discrete‐time switched systems and generalized to allow defective subsystems. Moreover, we present a novel method to improve the dwell time estimation in the case of bimodal switched systems. In this method, scaling algorithms to minimize the condition number have been used to give better minimum dwell time and average dwell time estimates.     

Robust stability analysis of singular linear system with delay and parameter uncertainty

1Introduction It is well known that the existence of a delay in adynamical system may induce instability or poorperformances in various systems suchas electric,pneumatic,and hydraulic networks,chemical processes,longtransmission lines,etc.For a survey of time_delay systems,the reader can refer to a recent overview paper[1].Control of singular systems has been extensively studied inthe past years due to the fact that singular systems betterdescribe physical systems than regular ones.Agreat numb…     

Stability bound of discrete multiple time-delay singularly perturbed systems

In this paper, the exact stability bound for discrete multiple time-delay singularly perturbed systems is examined. The obtained results, based on state space approach for the critical stability criterion of discrete systems, are sufficient conditions for the original system and its reduced subsystems about stability problems. Besides, we show that the subsystems are, indeed, zero-order approximations to the original systems. An example is given to illustrate the proposed schemes.     

Passivity-based decentralized stabilization of multi-agent systems with uncertainty and mixed delays

In this paper, we investigate a decentralized stabilization problem of uncertain multi-agent systems with mixed delays including discrete and distributed time-varying delays based on passivity stability. We design a decentralized state-feedback stabilization scheme such that the family of closed-loop feedback subsystems enjoys the delay-dependent passivity stability for each subsystem. Then, by employing a new Lyapunov-Krasovskii function, a linear matrix inequality （LMI） approach is developed to establish the delay-dependent criteria for the passivity stability of multi-agent systems. The sufficient condition is given for checking the passivity stability. The proposed LMI result is computationally efficient. An example is given to show the effectiveness of the method.     

Robust stability analysis of singular linear system with delay and parameter uncertainty

This paper deals with th e problem of robust stability for continuous-time sin gular systems with state delay and parameter uncerta inty.The uncertain singular systems with delay consi dered in this paper are assumed to be regular and impulse free.By decomposing the systems into slow and fast subsystems,a robust delay-dependent asymptotic stability criteria based on linear matrix inequality is proposed,which is derived by using Lyapunov-Krasovskii functionals,neither model transformatio n nor bounding for cross terms is required in the derivation of our delay-dependent result.The r obust delay-dependent stability criterion proposed in th is paper is a sufficient condition.Finally,numerical examples and Matlab simulation are provided to illustrate the effectiveness of the proposed method.     

On the stabilization of a class of bilinear systems

In this paper the problem of the stabilization of a class of continuous time bilinear systems, containing the class of the homogeneous ones, is studied, and simple stabiliza-bility conditions are derived, expressed in terms of the stability or stabilizability of some linear or homogeneous bilinear subsystems of the given system. On this basis, it is shown that this problem can be decomposed into subproblems and reduced to the stabilization of homogeneous subsystems of the system, obtained through successive decompositions of it. A sufficient condition for the stabilization of a homogeneous bilinear system is also given.     

On the Kamke-Müller conditions, monotonicity and continuity for bi-modal piecewise-smooth systems

We show that the Kamke-Müller conditions for bimodal piecewise-smooth systems are equivalent to simple conditions on the vector fields defining the system. As a consequence, we show that for a specific class of such systems, monotonicity is equivalent to continuity. Furthermore, we apply our results to derive a stability condition for piecewise positive linear systems.     

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.     

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.     

时滞半Markov切换随机系统输入–状态稳定性分析的时变驻留时间条件方法

本文针对时滞半Markov切换随机系统,建立了一种基于时变驻留时间条件的不定多重Lyapunov-Razumikhin函数方法,给出了系统输入–状态稳定性和积分输入–状态稳定性判别条件.一方面,构造了一种不定多重Lyapunov函数,不要求每个子系统对应的Lyapunov-Razumikhin函数导数总是保持负定,从而放宽了对于子系统稳定性要求,甚至允许了不稳定子系统的存在;另一方面,提出了一种时变驻留时间条件来表示半Markov切换信号的切换次数与子系统驻留时间之间的关系,利用时变函数对切换次数进行估计,间接放松了不定多重Lyapunov-Razumikhin函数选取的限制.最后,数值算例验证了所提方法的有效性.     

一类具有对称内联子系统的不确定大系统的二次稳定性

本文研究了一类由若干个相似子系统和一个外部系统通过对称地内联组成的不确定大系统的二次稳定性问题用二个低阶子系统以及相应的Ｈ∞范数的术语给出了这样一个系统是二次稳定的充分条件，并且证明了这个条件对于这个系统的过界系统的二次稳定性也是必要的。     

Stability results for nonlinear feedback systems

This paper presents an approach towards deriving sufficient conditions for the stability of nonlinear feedback systems. The central features of the approach are twofold. Firstly, useful stability tests are obtained for the case when the subsystems have nonlinear dynamics; secondly, a unifying set of general stability criteria are given, from which known situations can be treated as special cases and new ones are handled with equal ease. The results are obtained by use of a recently developed theory of dissipative systems.     

A geometry-based algorithm for the stability of planar switching systems

This paper describes a geometric procedure, whose aim is to provide necessary and sufficient conditions for the stability of a class of planar switching systems against arbitrary switching. The main idea is to build the worst switching sequence and to exploit it to get information about the stability for every switching sequence. The analysis is conveyed initially for LTI subsystems, then it is extended to homogeneous subsystems of the same degree having the origin as the unique equilibrium point. Finally, a simple method to build suitable Lyapunov functions to verify the stability of such systems is provided.     

On the stability issues of switched singular time‐delay systems with slow switching based on average dwell‐time

The issue of exponential stability analysis of continuous‐time switched singular systems consisting of a family of stable and unstable subsystems with time‐varying delay is investigated in this paper. It is very difficult to analyze the stability of such systems because of the existence of time‐delay and unstable subsystems. In this regard, on the basis of the free‐weighting matrix approach, by constructing the new Lyapunov‐like Krasovskii functional, and using the average dwell‐time approach, delay‐dependent sufficient conditions are derived and formulated in terms of LMIs to check the exponential stability of such systems. This paper also highlights the relationship between the average dwell‐time of the switched singular time‐delay system, its stability, exponential convergence rate of differential states, and algebraic states. Finally, a numerical example is given to confirm the analytical results and illustrate the effectiveness of the proposed strategy. 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.     

On the status of stability of interconnected systems

In the present paper we report on the status of stability of complex dynamical systems. We consider those systems which may be viewed as an interconnection of lower order subsystems. We emphasize those types of results which are phrased in terms of the stability properties of the subsystems and in terms of the qualitative properties of the system interconnecting structure. We consider both Lyapunov stability and input-output stability results. In our presentation, we concentrate on selected sample results and we show how some of these results are related. We then use these sample results to identify modifications, extensions, generalizations, and applications. In this way, we are able to provide a reasonably good picture of the entire subject on hand.