基于非Lyapunov函数法的切换非线性时滞系统的稳定性理论研究及应用

As a type of hybrid dynamical systems, switched systems have wide applications in electrical power, transportation, networking control, aeronautics and astronautics. In recent years, the topic of switched systems has become the research focus of complex system theory. Based on the existing problems and the deficiency in the stability of switched systems, this project utilizes integral inequality, positive system and matrix analysis to investigate stability theory of switched nonlinear time-delay systems with disturbances and its application in collaborative control of multiple agents. By using the estimation approach of integral inequality, several stability and stabilization criteria of switched nonlinear time-delay systems with non-Lipschitz internal disturbances are established; based on the typical non-Lyapunov functional method of positive systems, the stability of switched nonlinear time-delay positive systems with exogenous disturbances is investigated; the internal mechanism between the disturbances and the stability and convergence of the systems is demonstrated. As application, the consensus problem of multi-agent system is discussed from a new point of view by building a bridge between the stability of switched systems and the consensus of multi-agent systems. The study of this project can further enrich and perfect the stability theory of switched nonlinear systems, which can provide the new ideas and methods to distributed collaborative control of multi-agent systems.

切换系统作为一类特殊的混杂系统，在电力、交通、网络控制和航空航天等领域具有重要的应用。近年来，切换系统成为复杂系统研究的热点。针对切换系统稳定性理论研究中存在的问题和不足，本项目综合运用积分不等式、正系统和矩阵分析等理论工具，对一类具有扰动项的切换非线性时滞系统的稳定性理论及其在多智能体协作控制中的应用进行深入地研究。基于积分不等式的估计方法，建立具有非李普希兹内部扰动项的切换非线性时滞系统的稳定和镇定准则；基于正系统特有的非李雅普诺夫函数法，研究具有外部扰动项的切换非线性时滞正系统的稳定性问题；深刻揭示扰动项与系统稳定性和收敛性之间的内在机理。作为应用，搭建切换系统稳定性理论和多智能体系统一致性之间的桥梁，从新的角度探讨多智能体系统的一致性问题。本项目的研究能够进一步丰富和完善切换非线性系统稳定性理论体系，并且为多智能体系统的分布式协作控制提供新的思路和方法。

切换系统是一类特殊的混杂系统，在交通、网络控制和群体协同控制等领域具有广泛而重要的应用。近年来，切换系统的稳定性分析和控制器设计引起了国内外学者关注。本项目综合运用不等式技术、正系统和矩阵分析等理论工具，对一类切换非线性时滞系统的稳定性进行深入地研究。主要研究结果包括：基于不等式的估计方法，建立了具有扰动项的切换非线性时滞系统的稳定和镇定准则；基于正系统特有的非李雅普诺夫函数法和模态依赖平均驻留时间法，建立了切换齐次正系统的稳定性判据，借助比较原理，进一步将理论结果推广到时变切换非线性齐次时滞系统，深刻揭示了时滞与系统稳定性之间的影响机制；提出了对数压缩的平均驻留时间法，建立了一类具有不同齐次度的切换正系统的多项式稳定准则，给出了理论成果在多智能体系统一致性问题中的应用。所得结果丰富和完善了切换系统稳定性理论体系，并且为多智能体系统的分布式协作控制提供新的思路和方法。