OU随机系统的建模与控制: 从噪声模拟到能量整形

The project is to avoid the puzzle---“how to find applications to my controller designed?”. The controller design for nonlinear systems should begin with the modeling to system, or the simulation to the source of disturbances. Using Orstein-Ulenbeck(OU) process to simulate the color noise, produced by filtering the white noise through stable linear sets, is consistent to the requirement of physical right. To achieve the mathematical reality, the stochastic calculus to OU process and the theory on stochastic differential equation are constructed. The research on the existence and uniqueness of solution, the stability of equilibrium, and the dissipativity of the systems are the foundations to controller design. The Lagrange systems have realizable construction and wide applications. The main work of our research consists of the modeling of Lagrange systems, and the controller design based on energy modified. To verify control laws, the practical models such as machinery systems, circuits systems, electronic systems will be used instead of pure mathematical examples. The combination of physical right, controller practices and strict stochastic analysis, leads to the difficulty of the research, which is a great challenge.

本项目努力克服“如何为我的控制器找到应用?”的困惑. 随机非线性控制器的设计, 应该从系统的建模, 甚至外部干扰源的模拟开始. 利用Orstein-Ulenbeck(OU)过程模拟, 由白噪声通过稳定系统后产生的有色噪声, 符合物理正确性的要求. 为了保证数学理想化, 拟构造关于OU过程的随机积分和随机微分方程理论. 讨论解的存在性和唯一性, 研究解的稳定性和耗散性是进行控制器设计的基础. 拉格朗日系统既有理想结构, 又有实用上的普适性. 我们重点研究随机拉格朗日系统的建模, 基于能量整形设计反推控制器. 所有控制策略, 都避免纯粹的数值例子, 而是选取机械, 电路和电力系统模型进行仿真. 需要结合物理正确性, 控制实用性和随机分析的严密性, 使得本项目的研究, 难度大, 有很大的挑战性.

本项目的控制器设计从对外部干扰的模拟开始, 到系统的建模, 然后是微调实际能量变为Lyapunov函数, 设计了满足系统个性的简单的控制器. 将随机干扰设为有色噪声, 保证数学模型的可实现性. 同时认为所有的有色噪声都是, 白噪声通过稳定的线性系统滤波后产生的二阶矩过程. 本项目还研究了干扰信号是OU过程的理想情形, 构造了关于OU过程的随机积分. 建立了随机微分方程理论, 讨论了解的存在性和唯一性, 稳定性和无源性. 重点研究了随机拉格朗日系统. 借助反推的思想, 设计向量型非线性控制. 以能量整形函数为Lyapunov函数, 设计适合拉格朗日系统特点的控制器, 保证了我们的研究成果的实用性.