多自由度强非线性随机振动系统的直接控制方法

The optimal control of multi-degree-of-freedom strongly nonlinear stochastic vibrating system has always been hot and difficult in the scientific and engineering communities. Due to the difficulty to solve the dynamic programing equation or forward-backward stochastic differential equations, the stochastic optimal control theory based on the optimality principles is confined to the research on low-dimensional and weakly nonlinear systems. Nonlinear stochastic optimal control strategy in the Hamiltonian framework derives the control force by combining the averaged system with lower dimension and the optimality principles. This strategy overcomes the difficulty of solving equations to some extent, while cannot include the displacement feedback mechanism. By compulsively separating the control force to conservative mechanism and dissipative mechanism and combining the stochastic averaging technique, this proposal transforms the functional extreme value problem to a multivariable function extreme value problem. The direct control is then determined by numerical optimization technique. This method not only avoids the difficulty to solve complex equations, but also includes the whole feedback information of system displacement and velocity. It establishes the unite theoretical framework for response control, stability control, reliability control and targeted probability density tracking.

多自由度强非线性随机振动系统的最优控制问题一直以来都是科学与工程领域的研究热点和难点。基于最优性原理的随机最优控制理论，由于方程求解困难，仅限于分析低维及弱非线性系统。哈密顿框架下的非线性随机最优控制理论，从低维平均系统出发建立了高维强非线性系统的控制策略，它在一定程度上解决了求解困难问题，却失之于未能在控制中包含位移反馈机制。本项目将数学与物理相结合，从物理直观上强制分离控制力为保守机制和耗散机制，结合随机平均法将确立最优控制策略的泛函极值问题转化为多变量函数极值问题，从而以优化技术解决之。该方法不但回避了复杂的方程求解过程，而且在控制中包含了关于位移和速度的完整的反馈信息，从而建立了响应、稳定性、可靠性和概率密度控制的统一的理论框架。

本项目以多自由度强非线性随机振动系统为研究对象，以控制其动态行为为终极目标。在模型驱动层面，建立了以响应最小、稳定裕度最大、可靠性最高和距离指定概率密度最近为目标的统一的直接控制理论框架，成功规避了经典随机最优控制理论对系统高维和强非线性的限制，以及非线性随机最优控制理论未能包含刚度反馈机制的缺陷。在数据驱动层面，建立了从系统识别、系统降维、响应分析直到控制设计的完整方法论，在工程动力学领域成功实现数字孪生，即仅从激励和响应数据出发，快速识别系统方程、即时设计控制算法，并实时更新方程和算法，保证对目标系统的实时监测和控制。本项目相关研究成果发表SCI论文24篇，授权国际专利1项。