非线性随机切换脉冲系统的稳定性分析与控制研究

Nonlinear stochastic switching and impulsive systems have been widely applied in network control systems, mechanical systems, aviation systems, and so on. Meanwhile, it is greatly complex to study the stability theory of nonlinear stochastic switching and impulsive systems, and the research on the design of observer and controller for it remains open. Based on the stochastic theory, and the results in switching and impulsive systems, the problem of stability and control for nonlinear stochastic switching and impulsive systems will be addressed in this project. The main contents are given as follows: (1) Stochastic stability criteria are derived for nonlinear stochastic switching and impulsive systems with the aid of multiple Lyapunov functions and mode-dependent dwell time; (2) By using Barbalat lemma and persistency of excitation, the adaptive impulsive observer and the switching and impulsive controller are designed; (3) On the basis of finite-time theory, the issue of finite-time analysis and control for nonlinear stochastic switching and impulsive systems is discussed; (4) The model of permanent magnet synchronous machine systems established in the form of nonlinear stochastic switching and impulsive systems is used to illustrate the validity of the obtained results with the combination of stochastic stability theory and the switching and impulsive design method. This project will build a systematic control theory for nonlinear stochastic switching and impulsive systems, and push forward progress in stochastic theory.

非线性随机切换脉冲系统在网络控制系统、机械系统和航空系统等领域有着广泛的应用，同时，非线性随机切换脉冲系统的稳定理论研究极为复杂，关于观测器、控制器设计问题亟需解决。本项目基于随机系统理论和切换脉冲系统的研究成果，研究非线性随机切换脉冲系统的稳定性分析与控制问题。主要研究内容包括：（1）借助于多Lyapunov函数和模态依赖驻留时间法，建立系统稳定的判定条件；（2）应用Barbalat引理和持续激励条件，提出自适应脉冲观测器设计方法，进而考虑系统的切换脉冲控制问题；（3）基于有限时间稳定理论，研究此类系统的有限时间稳定与控制问题；（4）将永磁同步电动机系统建模为随机切换脉冲系统，然后基于随机稳定性理论与切换脉冲控制算法，进行仿真验证。本项目将建立一套较成体系的随机切换脉冲系统的控制理论，推动随机理论的进一步发展。

本项目研究了非线性随机切换脉冲系统的稳定性与控制理论，主要工作包括：（1）给出了具有有色噪声干扰的非线性随机切换脉冲系统的稳定性理论。我们研究了具有有色噪声的非线性时变切换脉冲系统、非线性时滞系统、神经网络、基因调控网络的解的构造，利用Lyapunov函数、矩阵不等式、时间分割技术建立了噪声-状态稳定性与依概率稳定性、几乎必然稳定性判据；（2）给出了具有白噪声干扰的非线性随机切换脉冲系统的稳定性理论。我们考虑了切换与脉冲的同步与异步、干扰与镇定作用，推广了具有符号不定导数的随机Lyapunov理论，建立了具有泊松跳变和具有驻留时间切换脉冲的随机奇异系统的均方稳定性、不含稳定子系统的随机切换基因调控网络系统的均方稳定性与异步切换脉冲非线性随机系统的几乎必然稳定性判据；（3）采用有限时间理论并结合滑膜技术、自适应方法、观测器设计等，研究了几类非线性系统的反馈控制问题，包括切换正时滞系统的有限时间控制、高阶非线性系统的有限时间一致性控制、有限时间下的二阶滑膜变增益控制。