切换正奇异系统的稳定分析与控制综合

Switched positive systems have been one of the research hotspots in control area due to their wide applications in the real word. In recent years, the problem of analysis and control for this kind of systems has attracted a great deal of attentions of scholars, and the previous results are mainly on normal systems. However, many practical models can only be described by singular systems and can not be described by normal systems, and a lot of difficulties will be found when consider the problem of stability analysis and control design for this kind of systems. Thus, this project aims to study the problem of stability analysis and control synthesis for switched positive singular systems. Firstly, for switched positive singular systems, by constructing a new linear copositive Lyapunov function, criteria of positivity and stability are established, and an effective L1 control scheme is provided. Secondly, for switched positive singular systems with different cases of time delay (constant delay, time-varying delay, mixed time-varying delay), criteria of positivity and stability as well as an L1 control scheme are proposed. Thirdly, from the linear case deep into the nonlinear case, criteria of positivity and stability as well as an L1 control scheme are proposed for nonlinear switched positive singular systems. Finally, a four-mesh circuit system will be presented as a practical example to verify the effectiveness of the proposed theoretical results. Therefore, the research on this project will further improve the theory of switched positive systems, enlarge the scope of its engineering applications, and undoubtedly have an important theoretical and practical significance.

切换正系统具有广泛的工程背景，是控制领域的研究热点之一。近年来，其分析与控制问题备受学者们的关注，当前结果主要集中于标准系统情形。然而，很多实际对象只能用奇异系统描述而不能用标准系统描述，在对其进行分析与控制时会面临诸多技术瓶颈。为此，本项目拟针对切换正奇异系统的稳定分析与控制综合问题展开研究。首先，针对切换正奇异系统，构造新型的线性余正型Lyapunov函数，建立正性、稳定性判据，提出有效的L1控制方案。其次，针对系统具有不同的时滞情形（常时滞、时变时滞、混合时变时滞），提出正性、稳定性条件及L1控制方案。接着，由线性情形深入至非线性情形，针对非线性切换正奇异系统提出正性、稳定性判据和L1控制方案。最后，以四网孔电路系统作为实例验证所得理论成果。本项目研究将进一步完善切换正系统理论体系，拓展其工程应用范围，具有重要的理论和实际意义。

切换正奇异系统具有广泛的工程背景，近年来，其分析与控制问题备受关注。本项目针对切换正奇异系统的稳定分析与控制综合展开了一系列研究。首先，针对切换正奇异系统，通过奇异值分解技术提出了一个充要的正性条件；基于新型的多线性余正型Lyapunov函数和平均驻留时间方法提出了一个充分的指数稳定性条件，所得的指数衰减率可以根据实际情况进行调整；进一步分析了系统的扰动抑制性能即L1增益性能和L∞增益性能，并通过凸优化方法得到了最优性能水平。然后，分别针对离散时滞、分布时滞和混合时滞情形下的系统，充分利用时滞信息，巧妙地构造了新型的多线性余正型Lyapunov-Krasovskii泛函，同时结合无时滞情形的研究基础，提出了系统的正性判据、稳定性条件和扰动抑制性能条件。注意到，不同于平均驻留时间限制，模态依赖驻留时间限制更符合实际切换情形。因此，进一步，针对不同模态依赖驻留时间限制下的切换正奇异系统，基于新型的离散化线性余正型Lyapunov-Krasovskii泛函方法，提出了系统的稳定性条件和扰动抑制性能条件。接着，针对非线性切换正系统，从其特殊的结构特性入手，巧妙地借鉴T-S模糊建模方法，将其线性化处理转化为T-S模糊系统，同时结合无非线性情形的研究基础，导出了系统的正性判据、稳定性条件和有效的控制方案。最后，以四网孔电路系统作为实例验证了所得理论成果。本研究完善了正系统理论体系，拓展了其工程应用范围，具有重要的理论和实际意义。