切换广义线性系统的公共Lyapunov函数存在性问题研究

The existence of common Lyapunov functions is the first basic problem regarding stability of switched systems. This problem focuses on the stability for arbitrary switching sequences. For the existence of common Lyapunov functions of general switched linear systems, the research results are quite abundant. However, for switched descriptor systems, due to the system singularity, the study on this problem may be quite difficult. Based on the basic characteristics of system matrices, this proposal will address the following two aspects: (1) For general switched descriptor linear systems, the criteria of the existence of common quadric Lyapunov functions will be established. (2) For switched descriptor positive linear systems, the conditions will be derived to verify the existence of common linear copositive Lyapunov functions. This project is based on the research frontiers and is challengeable. The achievement of this project will enriched and perfected the theory of the existence of common Lyapunov functions, and thus has important theoretical values.

公共Lyapunov函数存在性问题是切换系统稳定性研究的首要基本问题，其存在性可以确保切换系统在任意切换信号下稳定。对于一般线性切换系统的公共Lyapunov函数存在性问题，研究成果已经相当丰富。然而对于切换广义系统，由于系统存在奇异性，使得该问题的研究具有相当大的难度。本项目从系统矩阵的基本特性出发，重点研究两个问题：一、针对一般切换广义线性系统，建立公共二次Lyapunov函数存在的判定条件。二、针对切换广义正线性系统，建立公共线性协正Lyapunov函数存在的判定准则。本项目立足于前沿课题，从事具有挑战性的工作。本项目的完成将丰富和完善切换系统公共Lyapunov函数存在性问题的理论体系，具有重要的理论价值。

切换系统可以模拟大量的实际问题。本项目主要研究广义切换系统和正切换系统在任意切换信号下的稳定性问题。取得的成果主要包含如下两个方面：（一）、针对正切换自治系统，研究其公共对角二次Lyapunov函数存在问题。建立了对角二次Lyapunov函数存在的代数等价条件。（二）针对广义切换时滞系统，建立了系统正则无脉冲和切换无脉冲的判定准则。基于此，通过构造一类新颖的分段Lyapunov泛函，利用最小驻留时间方法，建立了系统一致指数稳定的时滞无关条件。该项目所得成果将为该领域今后研究工作和大量的实际问题提供理论指导。