可变时间型切换系统的稳定性与镇定性研究

Variable-time switched systems have the foundational engineering background. The current research results only focus on the case with fixed switching instants, and there exist few results on the case with variable-time switching instants. However, the switching instants often depend on the system states in the switching process, which generates the unavailability of the existing result to investigate the dynamic properties. Two crucial problems: the numbers of hits between trajectory and switching surface and the investigation of precise upper and lower bound of dwell time, restrict the research for the dynamics for variable-time switched systems. This project investigates the stability of a category of special variable-time switched systems. Firstly, by virtue of introducing and analyzing the value of hit function on the switching surface, we will find the conditions which guarantee the trajectory hits each switching surface exactly once. Secondly, we will obtain the precise upper and lower bound of dwell time by establishing optimization model or the theoretical analysis. Thirdly, by structured approach and mode-dependent average dwell time, we will establish the criteria of stability for variable-time switched systems. Lastly, we will investigate the stabilization of variable-time switched systems by designing the switching rule and feedback controller. This project could not only indicate the fundamental property of variable-time switched systems, but also deepen the theory of switched systems. Therefore, it has importantly theoretical significance and application value.

切换系统具有深厚的工程应用背景，现有研究成果只针对固定时刻型切换系统，对切换时刻可变的情形研究较少。而实际切换过程中切换发生时刻常依赖于系统状态，现有研究成果难以分析此类系统的动力学性质。两个关键性难题，即系统轨迹与切换面的碰撞次数和驻留时间上下界的精确求取，制约了可变时间型切换系统的动力学研究。本项目将研究一类可变时间型切换系统的稳定性与镇定性。通过引入碰撞函数并分析其在切换面处的取值，得出系统轨迹碰撞切换面一次仅一次和多次的条件；再采取优化方法建立优化模型，或利用理论分析，求得驻留时间的精确上下界；然后通过构造法和模式相关的平均驻留时间等方法，建立可变时间型切换系统稳定的充分条件；最后，设计切换规则和反馈控制器实现切换系统的镇定。本项目的研究可揭示可变时间型切换系统动力学的基本属性，是对切换系统理论的深化与完善，因此，具有重要的理论意义和应用价值。

可变时间切换系统具有深厚的工程应用背景，其切换发生时刻与系统状态密切相关。本项目针对可变时间切换系统的稳定性与镇定性展开研究工作，为切换系统的动力学行为分析提供新的理论与研究方法。首先，运用延拓法与压缩映射原理，建立了可变时间切换系统解的存在性与唯一性的基本条件；其次，通过引入碰撞函数，采用单调性理论分析其在切换面处的取值，得出系统积分曲线碰撞切换面一次仅且一次和碰撞切换面多次的条件；再次，引入反映系统积分曲线与切换面在时间轴上的距离函数，估计相邻两次切换发生的时刻范围，进而求取驻留时间的上界与下界；然后以驻留时间上下界为基础，运用构造法、平均驻留时间等切换系统的稳定性分析方法，建立可变时间切换全局稳定的充分条件；最后，以可变时间切换系统稳定性条件为基础，通过设计切换规则和引入反控制器，建立系统镇定的基本规范。本项目的研究揭示了可变时间型切换系统动力学的基本属性，是对切换系统理论的深化与完善，因此，具有重要的理论意义和应用价值。