Box invariance in biologically-inspired dynamical systems

A dynamical system is box invariant if there exists a box-shaped positively invariant region. We show that box invariance can be checked in cubic time for linear and affine systems, and that it remains decidable for classes of nonlinear systems of interest (with polynomial structure). We present results on the robustness of box invariance for linear systems using spectral properties of Metzler matrices. We also present sufficient conditions for establishing box invariance of switched and hybrid systems. In general, we argue that box invariance is a characteristic of many biologically-inspired dynamical models.     

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.     

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.     

线性切换系统的积分不变性原理

将积分不变性原理进行推广, 用来讨论线性切换系统的稳定性. 作为LaSalle不变性原理的推广, 文中推广的积分不变性原理无需构造Lyapunov函数, 而是利用系统输出的可积性, 讨论线性切换系统输出为零的状态集合的稳定性. 另外, 讨论了切换系统状态集合稳定性与切换信号之间的关系. 利用线性切换系统的可观测性, 讨论了系统平衡点的渐近稳定性. 数值例子说明了文中方法的有效性.     

Stability of switched nonlinear systems via extensions of LaSalle’s invariance principle

This paper studies the extension of LaSalle’s invariance principle for switched nonlinear systems. Un-like most existing results in which each switching mode in the system needs to be asymptotically stable, this paper allows the switching modes to be only stable. Under certain ergodicity assumptions of the switching signals, two extensions of LaSalle’s invariance principle for global asymptotic stability of switched nonlinear systems are obtained using the method of common joint Lyapunov function.     

Observability analysis of conewise linear systems via directional derivative and positive invariance techniques

Belonging to the broad framework of hybrid systems, conewise linear systems (CLSs) form a class of Lipschitz piecewise linear systems subject to state triggered mode switchings. Motivated by state estimation of nonsmooth switched systems, this paper exploits directional derivative and positive invariance techniques to characterize finite-time and long-time local observability of a general CLS. For the former observability notion, directional derivative results are developed from the simple switching property, and they yield improved observability conditions. For the latter notion, we focus on the case where a nominal trajectory has finitely many switchings. In order to characterize long-time behaviors of the CLS, necessary and sufficient conditions are obtained for the interior of a positively invariant cone. By employing these conditions, we establish connections between finite-time and long-time local observability; underlying positive invariance properties are unveiled.     

Stability and stabilisation of planar switched linear systems via LaSalle's invariance principle

This paper investigates the problem of uniformly asymptotical stability (UAS) and stabilisation of planar switched linear systems using LaSalle's invariance principle of switched systems. First, we show that a common weak quadratic Lyapunov function (WQLF) is enough to assure the UAS of a switched linear system with stable modes. Then the necessary and sufficient conditions for the existence of common WQLF are obtained. Secondly, we consider the problem of uniformly asymptotical stabilisation (UASZ) of single-input planar switched linear systems. Necessary and sufficient conditions for the closed-loop system with proper feedback to share a common WQLF are presented. It is also proved that a common WQLF is enough to assure the UAS of the closed-loop system.     

On the necessity of looped-functionals arising in the analysis of pseudo-periodic,sampled-data and hybrid systems

Looped-functionals have been shown to be relevant for the analysis of a wide variety of systems. However, the conditions obtained in previous papers on the analysis of sampled-data, impulsive and switched systems have only been shown to be sufficient for the characterisation of their associated discrete-time stability conditions. We prove here that these conditions are also necessary. This result is derived for a wider class of linear systems, referred to as impulsive pseudo-periodic systems, that encompass periodic, impulsive, sampled-data and switched systems as special cases.     

Extended invariance principle for nonautonomous switched systems

Stability analysis is developed for nonlinear nonautonomous switched systems, trajectories of which admit, generally speaking, a nonunique continuation on the right. For these systems Krasovskii-LaSalle's invariance principle is extended in such a manner to remain true even in the nonautonomous case. In addition to a nonsmooth Lyapunov function with negative-semidefinite time derivatives along the system trajectories, the extended invariance principle involves a coupled indefinite function to guarantee asymptotic stability of the system in question. As an illustration of the capabilities of this principle, a switched regulator of a fully-actuated manipulator with frictional joints is constructed.