广义正切换系统的稳定性和镇定性研究及应用

The theory of switched systems has a position of great importance in the control of complex systems. In recent ten years, positive switched systems receive many researchers' attention due to its particularity. Based on some exising problems and deficiency in the theory of positive switched systems, and by using combined theory and methods of differential equation, matrix analysis, algebra and geometry, we will systematically study the stability and the stabilizability for several kinds of general positive switched systems, discover the algebraric and geometric structure of common (multiple) co-positive Lyapunov functions, and investigate the stability and the stabilizability of the involved systems under arbitrary or chosen switching signals. The effect of delays, impulses and disturbances on the stability and the stabilizability of generalized positive switched systems will be illustrated profoundly. What's more, the theoretical results will be applied to consensus of multi-agent systems. The research in this project is significant to reveal the leading role of switching signals in the theory of stability and stabilization for generalized positive switched systems. It can unify and deepen the existing results in the literature, complement the theory of positive switched systems, and provide the new idea and method to study consensus of multi-agent systems.

切换系统理论在复杂系统控制领域占有非常重要的地位。近十年来，正切换系统由于其特殊性得到了许多研究者的密切关注。本项目针对正切换系统稳定性理论研究中存在的问题和不足，综合运用微分方程、矩阵分析、代数和几何等多种理论和方法，对几类广义正切换系统的稳定性和镇定性进行系统而深入地研究，探索多种形式的公共（多重）余正(copositive)李雅普诺夫函数的代数和几何结构，研究系统在任意（特定）切换信号下的稳定性和基于切换信号的可镇定性，深刻揭示时滞、脉冲和扰动等复杂因素对系统稳定性和镇定性的影响机理，并将研究成果应用到多智能体系统的一致性问题。本项目所开展的研究对于揭示切换信号在广义正切换系统稳定性和镇定性理论中的主导地位具有重要意义，可以统一和深化已有的研究成果，进一步完善正切换系统的理论体系，并为多智能体系统一致性研究提供新的思路和方法。

切换系统是一类非常重要的系统，其理论已得到了系统而深入的研究。作为一类特殊的切换系统，正切换系统在工程领域中有广泛的应用，其稳定性理论有待于更加深入的研究。本项目运用正系统理论、余正李雅普诺夫函数和积分不等式等技术方法，对几类正切换系统和广义正切换系统的稳定性和镇定性进行深入研究，并设法将相关结果运用到多智能体系统的一致性问题。主要研究结果包括：提出了弱线性余正李雅普诺夫函数和联合线性余正李雅普诺夫函数的概念，建立了正切换系统的新的稳定性和镇定性准则；基于模型变换和正系统的理论方法，得到了切换时变系统指数稳定的Metzler矩阵形式的准则；采用非李雅普诺夫函数方法，明确估计了具有扰动输入的切换齐次正系统的可达集；基于改进的时滞积分不等式，给出了几类切换非线性系统的稳定和有限时间稳定性准则。所得结果丰富和完善了正切换系统稳定性理论体系，并为多智能体系统一致性研究提供新的思路和方法。