基于二阶矩阵微分方程的机械振动系统线性二次型调节器设计

本文讨论机械振动系统线性二次型状态调节器（LQR）问题，直接针对系统二阶运动微分方程，性能指标为一个依赖于二阶导数的泛函。由欧拉-拉格朗日方程得出一个系统矩阵增广的二阶线性微分方程，指出该方程稳定的特征对就是最优控制振动系统闭环特征对，并给出求解最优控制状态反馈矩阵的方法，另外，由本文方法还可得出基于速度和加速度反馈的最优控制反馈矩阵。这里不涉及求解代数矩阵Riccati方程。

A LQR Design for Mechanical Vibration Systems Based on Second_order Matrix Differential Equations

A linear quadratic regulator for mechanical vibration systems is studied based on second_order matrix equations.The performance index is a functional depending on second derivatives by introducing co_state vector.The Euler_Lagrange equations lead to a system matrix_augmenting linear second_order differential equations whose stable eigenpairs are just the eigenpairs of the closed_loop optimal systems.The optimal feedback constant matrices are determined by aforecited stable eigenpairs?control input matrix and control weight matrix explicitly.In addition,with the method proposed here,the optimal feedback matrices based on the velocity and acceleration feedback can also be obtained.The traditional algebraic matrix Riccati equation is not concerned here.