Dwell time and average dwell time methods based on the cycle ratio of the switching graph

A new method to find an upper bound on dwell time and average dwell time for switched linear systems is proposed. The method is based on computing the maximum cycle ratio and the maximum cycle mean of the directed graph that governs switchings. For planar switched systems, an upper bound for dwell time and average dwell time can be estimated by considering only the cycles of length two.     

An invariance principle for nonlinear switched systems

In this paper we address the problem of extending LaSalle Invariance Principle to switched system. We prove an extension of the invariance principle relative to dwell time switched solutions, and a second one relative to constrained switched systems.     

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.     

离散时滞切换系统的无记忆状态反馈镇定

针对一类子系统为离散时滞系统的切换系统,研究了稳定性与无记忆状态反馈镇定问题．采用多李雅普诺夫函数法,首先以线性矩阵不等式形式给出了在任意切换信号作用下离散时滞切换系统渐进稳定的一个充分性条件;然后给出了系统无记忆状态反馈镇定的控制器设计方案,并将结果推广到不确定离散时滞切换系统;最后用仿真算例验证了所提出设计方案的可行性。     

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.     

Switching control approach for stable navigation of mobile robots in unknown environments

This paper presents a stable switching control strategy for the parking problem of non-holonomic mobile robots. First, it is proposed a positioning-orientation switching controller for the parking problem. With this strategy robot backwards motions are avoided and the robot heading is always in the direction of the goal point facilitating the obstacle handling. Second, the avoidance of unexpected obstacles is considered in a reactive way by following the contour of the obstacles. Next, the stability of the switching parking/obstacle-avoider controller is analyzed showing stability under reasonable conditions. Finally, the good performance and the feasibility of this approach are shown through several experimental results.     

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．     

一类切换系统极限环的反馈镇定研究

利用多Lyapunov函数法讨论一类切换系统极限环的反馈镇定问题．首先给出切换系统极限环的有界状态反馈、有界输出反馈以及动态反馈镇定定理;然后利用Jurdjevic—Quinn方法给出反馈控制器的具体形式,并证明了在给定反馈控制器的作用下,切换系统的极限环渐近稳定．仿真例子验证了结论的正确性．     

H-infinity control for cascade minimum-phase switched nonlinear systems

1Introduction H_∞control theory has become a powerful tool to solverobust stabilization or disturbance attenuation problems.Many results about linear H∞control have appeared,andlinear H∞theory has been generalized to nonlinear systems[1~5].Two major approaches have been used to providesolutions to nonlinear H∞control problems.One is basedon the dissipativity theory and differential games theory[2,6].The other is based on the nonlinear versionofclassical bounded real lemma[3~5].Both of th…     

Probabilistic sorting and stabilization of switched systems

In this paper, we consider Lyapunov stability of switched linear systems whose switching signal is constrained to a subset of indices. We propose a switching rule that chooses the most stable subsystem among those belonging to the subset. This rule is based on an ordering of the subsystems using a common Lyapunov function. We develop randomized algorithms for finding the ordering as well as for finding a subset of systems for which a common Lyapunov function exists. It is shown that the class of randomized algorithms known as the Las Vegas type is useful in the design procedure. A third-order example illustrating the efficacy of the approach is presented.     

Improved results on stability of continuous-time switched positive linear systems

In this paper, the problems of stability for switched positive linear systems (SPLSs) under arbitrary switching are investigated in a continuous-time context. The so-called “copositive polynomial Lyapunov function” (CPLF) giving a generalization of copositive types of Lyapunov function is first proposed, which is formulated in a higher order form of the positive states of the underlying systems. It is illustrated in this paper that some classical types of Lyapunov functions can be seen as special cases of the proposed CPLF. Then, new stability conditions are developed by the new Lyapunov function approach. It is also proved that the conservativeness of the obtained criteria can be further reduced as the degree of the Lyapunov function increases. A numerical example is given to demonstrate the effectiveness and less conservativeness of the developed techniques.     

Modal and transition dwell time computation in switching systems: A set-theoretic approach

We consider a plant the dynamics of which switch among a family of systems. Each of these systems has a single stable equilibrium point. We assume that a constraint region for the state is assigned and we consider the problem of finding suitable limitations on the commutation speed in order to avoid constraints violations, even in the absence of state measurements. We introduce the concepts of modal and transition dwell times which lead to the definition of dwell time vector and dwell time graph (represented by a proper matrix), respectively. The former imposes a minimum permanence time on a discrete mode before commuting, the latter imposes the minimum permanence time on the current mode before switching to a specific new one. Both dwell time vector and dwell time graph can be computed via set-theoretic techniques. When the systems share a single equilibrium state, stability can be assured as a special case. Finally, under the assumption of affine dynamics, non-conservative values are achieved.