模态密集柔性空间结构二阶内平衡降阶

 给出一种针对二阶线性系统方程直接进行降阶的二阶系统内平衡降阶方法。大型柔性空间结构动力学方程采用二阶线性微分方程描述，采用传统的一阶内平衡降阶方法降阶后的状态方程是一阶形式，破坏了原系统的二阶结构和物理意义。采用新方法降阶后的系统可以保持原系统二阶结构，同时可以进一步保持原系统的对称和正定特性。柔性空间结构系统级降阶的柔性模态方程通常为对角形式，针对这种特殊形式，系统可控和可观Gramian矩阵存在闭合解析解，给出了闭合解的具体表达形式。数值仿真结果表明，二阶内平衡降阶方法可以达到一阶内平衡方法一样的降阶精度，Gramian矩阵的闭合解析解可以大幅度提高Lyapunov方程求解速度。

Second-order Balanced Reduction for Flexible Space Structures with Close Modes

A new second-order balanced truncation(SOBT) method is presented to obtain the low-order model of large flexible space structures(FSS) which is generally described by second-order linear differential equation. The truncated equation obtained by conventional first-order balanced truncation(FOBT) method,which utilizes the first-order form,destroys the second-order structure and physical meanings of original system.However,the new SOBT method is capable of preserving the second-order structure of the original system,the symmetric and positive properties.The closed-form solution of the controllability and observability Gramians is obtained as the special diagonal form of the system modal equation of FSS.The numerical results show that the SOBT could achieve the accuracy of conventional FOBT and the closed-form solution of Gramians is capable of increasing the solving efficiency of Lyapunov equation.