【作者】 玉强；

【导师】 吴保卫；

【作者基本信息】 陕西师范大学 ， 基础数学， 2014， 博士

【摘要】 摘要切换系统作为最重要的一类混杂系统,常用来模型化各种控制问题及某些自然界的复杂过程.由于存在多个子系统和各种可能切换信号的多样性,导致切换系统虽然表现形式简单,却展现出非常复杂的动力学行为.近几十年来,关于切换系统稳定性的研究一直吸引着众多系统工程师,计算机和数学家们的广泛关注.本文主要以具有不稳定子系统的切换系统为研究对象,这类系统在实际中既具有理论上的挑战性又具有非常重要的应用背景.本文主要内容如下：第一章主要介绍了本文的研究意义和现状.第二章研究所有子系统均不稳定的切换线性系统的鲁棒稳定性性质.首先,在特定假设下,给出这类系统能稳的必要条件；其次,利用不变子空间理论及平均驻留时间技术,在给定的两个假设下,分别获得无不确定性和含不确定性的这类系统稳定性的充分条件.切换信号需要满足在两组不稳定子系统之间激活的时间比介于两个常数之间.其中这两个常数可以通过切换系统设计的稳定度计算得来.另外,本章也发展了多Lyapunov函数理论,并给出这种Lyapunov函数构造方法.第三章研究具有不稳定子系统的离散时间切换线性系统的稳定性问题.首先,给出这类离散切换系统能稳的必要条件；其次,在特定的假设下,利用平均驻留时间技术获得了这类系统稳定的充分条件.最后,给出两个仿真算例演示本章结论有效性.需要指出的是,已经有一些基于平均驻留时间技术研究离散时间切换系统的结论,这些结论多是运用相似于连续情形的手段进行处理.然而,对离散时间切换系统而言,由于系统的状态仅依赖于之前的采样时刻点而不是之前的任意时间点,这导致了可能出现尽管平均驻留时间大于所给出的最小平均驻留时间,切换仍然不能发生或没有意义.为解决这一矛盾,本章提出离散切换信号这一新概念,并对这类离散切换信号发展了平均驻留时间技术.第四章研究经典Lyapunov函数定理的扩展问题.特别的,本章结论放松了经典Lyapunov函数V需沿系统轨迹非增这一要求.从这一点来说,经典Lyapunov函数可以看做这里的广义Lyapunov函数的一种特例.特别的,非线性系统的稳定性定理可以按如下形式获得：用“V沿系统轨迹容许在某些适当的时间段上增加”替换“V沿系统轨迹非增”.不同于多Lyapunov函数方法,新理论仅需构造一个类Lyapunov函数而不是多个.本章也研究了理论发现在具有不稳定子系统的切换系统中的应用问题.最后两个算例仿真演示结论的有效性.第五章建立重置正系统与切换正系统的关系.首先,作为重置控制系统的一种扩展,这里提出了重置正系统这一新概念.其次,本章建立起重置正系统与切换正系统之间的某种等价关系.这种关系使我们可以将重置正系统稳定性问题转化为离散时间切换系统稳定性问题,而对于切换系统来说有多种处理方法.然后,借助这种联系,给出在任意重置下重置正系统渐进稳定的充要条件；同时也给出这类系统能稳的充要条件.最后,给出重置正系统的一些稳定判据.第六章考虑扩展平均驻留时间技术下的离散时间和连续时间切换系统稳定性问题.通过引入三个新的概念,即闭链,r—开链和拟循环切换信号,在这些新切换信号下可以获得平均驻留时间及模型依赖平均驻留时间切换相对应的稳定及能稳性结论.在特定的受限切换信号下,我们发展和丰富了切换系统的已有稳定性结论.另一方面,本章也提出一种获取平均驻留时间及模型依赖平均驻留时间更好下界的方案.更多还原

【Abstract】 Abstract As one of the most important hybrid systems, switched systems are often used for modeling various control problems and some complex processes in nature. In spite of their apparent simplicity, switched systems display a very complicated dynamical behaviors because of the multiple subsystems and various possible switching signals. In the last decade, the investigations of stability for switched systems have attracted consid-erable attention in Systems Engineering, Computer Sciences and Mathematics commu-nities. This dissertation is devoted to research the switched systems with unstable sub-systems, which is theoretically challenging and of fundamental importance to numerous applications. This paper is organized as follows.In Chapter1, we introduce some research background and status on the dissertation.Chapter2studies the problem of robust stability for switched linear systems with all the subsystems being unstable. A necessary condition of stability for switched linear systems is first obtained with certain hypothesis. Then, under two assumptions, suffi-cient conditions of exponential stability for both deterministic and uncertain switched lin-ear systems are presented by using the invariant subspace theory and average dwell time method. Under the given switching signals, the activation time ratio between two sets of unstable subsystems is required to be located between two constants which are computed using the desired stability degree of the switched system. 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.Chapter3discusses some stability properties of discrete-time switched linear sys-tems (DSLSs) with unstable subsystems. First, under certain hypothesis, the necessary condition of stability for DSLSs is obtained. Second, using the average dwell time (ADT) strategy, we discuss the sufficient condition of exponential stability for switched linear systems with two assumptions. Finally, two examples are presented to show the effec-tiveness of the proposed approach. It is worth pointing out that there are some results for DSLSs obtained by using ADT technique. Those results are based on the switching signal is arbitrary, which usually is used to deal with continuous-time switching law. However, the switching signal of DSLSs is not arbitrar}’because the state of those systems depends only on discrete-time points not on all time points. In order to solve this contradiction, this chapter proposes a novel concept of discrete-time switching signal (DTSS) and develops the ADT technique for DTSS.In Chapter4, the classical Lyapunov theorems for nonlinear autonomous systems are extended. In particular, our results relax the requirement of V along the system trajec-tories being non-increasing, and thus Lyapunov function is a special case of generalized Lyapunov function. In particular, stability theorems of nonlinear systems are present-ed by replacing "V along the system trajectories is non-increasing" with "V along the system trajectories may increase its value during some proper time intervals". Different from the multiple Lyapunov function method, it only needs to construct a Lyapunov-like function instead of a collection of Lyapunov-like functions. These results are applied to the research for stability of switched systems with unstable subsystems. Two numerical examples are presented to demonstrate the proposed approach.Chapter5investigates the relationship between reset positive systems and switched positive systems. Firstly, as an extension of reset control systems, a novel concept of reset positive systems (RPS) is proposed. Secondly, the relationship between RPS and switched positive systems (SPS) is established. These relationship provides us an effective approach to transfer the stability of reset control system to investigate the stability of the discrete-time switched system which may be solved with many tools. Thirdly, the sufficient and necessary condition of those RPS asymptotically stable for any reset is obtained, while the sufficient and necessary condition of those RPS stabilizable is also presented. Last but not least, some stability criteria for RPS are given.Chapter6concerns the problem of stability for switched systems with extended av-erage dwell time in both continuous-time and discrete-time cases. By introducing three novel concepts of closed-chain, r-open-chain and quasi-cyclic switching signals, the sta-bility and stabilization conditions of the switched systems with ADT or mode-dependent ADT (MDADT) switching are obtained, we develops and enriches the existing results of stability under constrained switching. On the other hand, the chapter provides a solution to the open problem of how to obtain a tighter bound on ADT and MDADT.更多还原