具摩擦非线性系统随机振动分析与最优控制

项目名称： 具摩擦非线性系统随机振动分析与最优控制

项目编号： No.11472240

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 王永

作者单位： 浙江大学

项目金额： 75万元

中文摘要： 非线性随机振动与最优控制的研究集中于光滑系统，并已取得丰硕成果。在非光滑系统方面，仅涉及碰撞-振动和滞迟系统，对摩擦系统的研究极少，且基本采用数值分析手段。本项目研究具摩擦非线性系统的随机振动分析与最优控制策略。在振动分析方面，针对两大类摩擦模型：静态摩擦和动态摩擦，分别建立摩擦非光滑因素的光滑化技术，而后采用传统随机分析方法从近似光滑系统导出响应统计量。并研究该技术对具体摩擦模型的适用性和精度。在最优控制方面，结合光滑化技术、随机平均技术、随机动态规划原理或随机极大值原理，通过求解随机动态规划方程或前向-后向随机微分方程组导出最优控制策略。研究其效果、效率及对各类不确定参数的鲁棒性。对随机振动的研究，可用于评估摩擦对工程结构响应的影响以及摩擦阻尼器的性能；随机最优控制的研究工作，对需要极高定位精度和低速性能的伺服系统的控制设计具有重要意义。

中文关键词： 随机振动；随机最优控制；非线性系统；摩擦模型；光滑化技术

英文摘要： The researches on nonlinear stochastic vibration and optimal control have scored fruitful achievements on the smooth system, while only the vibro-impact system and hysteretic system have been involved in the aspect of the non-smooth system. The studies on the frictional system keep rare and almost adopt numerical methods. This project focuses on the random vibration analysis and optimal control strategy for nonlinear system with friction. The smoothing techniques are first developed for the static friction models and dynamic friction models, respectively. Based on the approximate smooth system of the original non-smooth system, the statistics of the random responses are derived through the traditional random analysis techniques. The applicability and precision of this procedure for all kinds of specific friction models are investigated. Combining the smoothing techniques, stochastic averaging, stochastic dynamical programming principle or stochastic maximum principle, the optimal control strategy is established by solving the Hamilton-Jacobi-Bellman equation or the forward-backward stochastic differential equations. Furthermore, the control effectiveness, control efficiency and the robustness of the optimal control to various uncertain parameters are discussed. The results on random vibration can be used to evaluate the influence of the friction on structural responses and the performance of frictional dampers, while those on the stochastic optimal control have an important meaning on the control design of servo systems which have high requirements on positioning accuracy and low speed performance.

英文关键词： Random vibration;Stochastic optimal control;Nonlinear system;Friction model;Smoothing technique