混杂系统基于双测度的实用稳定性分析
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上海第二工业大学 数理与统计学院 上海 201209

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国家自然科学青年基金;国家自然科学数学天元基金;上海市自然科学基金(面上项目)


Practical stability analysis for hybrid systems in terms of two measures
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School of Mathematics,Physics and Statistics,Shanghai Polytechnic University

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    摘要:

    混杂系统是一种时间尺度上的动力系统,它的演化依赖于连续变量和离散事件之间的相互作用。在理论探究和工程实践中,控制系统的稳定性研究是一个重要问题。对于混杂系统,即使每一个子系统都是稳定的,整个系统不一定是稳定的;相反地,如果存在一个或某几个子系统不稳定,整个系统也可能是稳定的,这就是混杂系统的复杂性和特殊性。在实际问题中,当所设计的系统处于数学意义上的非稳定状态并且系统充分摆动接近这一状态时,根据实用稳定性的定义认为这一表现是可以接受的。因此,对混杂系统提出时间尺度上的实用稳定性这一概念是非常有意义的。本工作的主要贡献是在双测度条件下对一般时间尺度上的混杂系统进行了实用稳定性分析。首先将双测度实用稳定性概念引入一般时间尺度上的混杂系统的研究。其次利用Lyapunov函数以及比较原理,给出了双测度意义下一般时间尺度混杂系统的实用稳定性判据。上述工作丰富和推进了混杂系统稳定性方面的研究。

    Abstract:

    The hybrid system is a kind of dynamic systems on time scales, its evolution demonstrates the interplay of continuous variables and discrete events. The research of the stability for control systems is an important issue to the theory exploring and engineering practice. For the hybrid system, even if each subsystem is stable, it does not mean that the whole system is stable; on the contrary, if one or several subsystems are unstable, the whole system may also be stable, which is the complexity and specificity of the hybrid system. In practice, when the designed system is in a mathematically unstable state and it swings sufficiently close to this state, this performance is acceptable according to the definition of the practical stability. Therefore, it is meaningful to propose the concept of the practical stability on time scales for hybrid systems. The main contribution of this paper is to present the practical stability analysis of hybrid systems in terms of two measures on time scales. First, some concepts of the practical stability in terms of two measures are introduced into the research of hybrid systems on general time scales. Then, by means of Lyapunov functions and the comparison principle, some sufficient conditions of the practical stability for hybrid systems in terms of two measures on general time scales are obtained. Above work enriches and promotes the research on the stability of hybrid systems.

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  • 收稿日期:2023-11-02
  • 最后修改日期:2024-06-04
  • 录用日期:2025-02-24
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