切换线性系统的若干动力学性质研究

Switched systems are one of the main branches of hybrid systems, which can be used to model many practical systems. Since switched systems have multiple subsystems and flexible switching patterns, they exhibit abundant dynamic characteristics. In general, switched systems have two main features: one is that their dynamics is much more complex than that of general dynamic systems, and another one is that switching signals have important impact on the dynamics of systems. Based on these two basic features, this proposal will address the following four aspects: (1) The dynamics of switched positive systems and switched definite systems will be studied, and the findings will be applied to study the dynamics of general switched systems. (2) The relationship between the dynamics of two mutually dual switched systems and the relationship between different dynamics (mainly including exponential stability, asymptotic stability, and attractivity) of a switched system will be explored. This will lead to a new approach to investigate the dynamics of switched systems. (3) For given sets of switching signals, the conditions that are able to judge whether the dynamics (including exponential stability, asymptotic stability, and convergence) of a switched system with time-delay is uniform or not will be derived. (4) The criteria will be established to determine different Pseudo-stability properties of switched systems. The project attempts to investigate the switched systems from a new point of view, finds some new methods that are more effective for studying dynamics of switched systems, and explores some novel ways or tools to deal with them. According to the state of the art in the area of study, this project is challenging and of theoretically significant. Due to the profound practical background of switched systems, the project will have broad applications in the future.

切换系统是混杂系统的一个主要分支, 可以建模大量的实际系统。由于具有多个子系统和灵活的切换方式，其动力学性质非常丰富。切换系统有两个重要特征：其动力学性质远比一般动力学系统复杂；切换信号对切换系统的动力学有着至关重要的影响。本项目从切换系统的基本特征出发，重点研究四个问题：一、揭示切换正系统和切换定号系统的动力学性质，并研究它们在一般切换系统中的应用方法。二、揭示互为对偶的切换系统间动力学性质的关系以及切换系统不同动力学性质间的关系，创新切换系统的研究方法。三、建立时滞切换系统在给定切换信号集上的动力学性质(指数稳定性, 渐近稳定性和收敛性)一致性的判定条件。四、建立切换系统各种Pseudo-稳定性的判定条件。本项目力图从新的角度展开研究，提出研究切换系统的新思路，联系该领域的研究前沿，从事具有挑战性的工作，具有重要的理论意义。因为切换系统有深刻的实际背景，本项目的结果具有广阔的应用前景。

(1).建立了正系统和切换正系统的系列结果，推动了(切换)正系统从线性系统到非线性系统的研究。改善了关于切换正系统、切换时滞正系统的稳定性判据。首次提出了定号系统并建立了其基础性质，为把一大类常规系统转化为正系统和定号系统并揭示其性质奠定了基础。.(2).揭示了互为对偶的切换系统间动力学性质的等价关系和齐次切换系统不同动力学性质间的等价关系，为把某些约束较强的稳定性(指数稳定性)转化为约束较弱的稳定性(渐近稳定性和吸引性)铺平了道路，提供了分析较强稳定性的新途径。.(3).进一步发展和提炼了包络法，提出了基于轨线比较的新型包络法，使得包络法可以有效克服传统Lyapunov方法的某些缺陷，如构造Lyapunov函数或泛函比较困难、Lyapunov函数或泛函对时滞的连续性、可微性、有界性的比较苛刻的约束。本质上讲，包络法是一种将复杂系统轨线利用理想简单系统轨线进行包络的方法。进一步发展包络法，将其提升为一种与Lyapunov方法并行、互补的分析方法，是接下来的重要工作。事实上，我们已经基于包络法做了大量的工作，建立了含时滞三角系统、非线性级联系统指数稳定的充要条件。.(4).从全新的角度较为全面地揭示了时滞非线性切换系统的动力学性质。包括时滞齐一次切换系统不同稳定性间的等价关系，非线性时滞系统在不同扰动下的动力学性质，解决了非线性级联系统稳定性的分解问题。揭示了动力学系统在一定扰动下具有“稳定性保守”的性质对时滞系统也同样成立。同时，级联系统稳定性的分解性质具有良好的应用前景。.(5).发展和完善了高次齐次Lyapunov函数方法。由于常用二次Lyapunov函数方法在很多情形下具有较高的保守性，发展高次齐次Lyapunov函数方法成为必然。本项目已经采用高次齐次Lyapunov函数方法建立了含时滞离散时间切换系统稳定的充要条件。进一步发展这项工作，可望提出高次齐次Lyapunov泛函方法，用以大幅降低复杂时滞系统稳定性分析的保守性，从而改变当前分析和设计时滞系统时采用二次Lyapunov泛函方法的现状。这对稳定性分析将产生重大的影响。