A discrete gyroscopic system is characterized by $2n$ first-order ordinary differential equations defined by one symmetric and one skew-symmetric, which system describes the motion of a spinning body containing elastic parts. In this paper, we consider the inverse problems of such system: Given partial spectral data, find a system such that it is of the desired spectral data. The general solution of the problem is given and the best approximation solution to a pair of matrices is provided by QR-decomposition and matrix derivation. In addition, we also consider a special case in which the system operates below the lowest critical speed. The numerical examples show that the proposed method is effective.

离散陀螺系统的特点是 $2n$由一个对称和一个斜对称定义的一阶常微分方程，该系统描述了包含弹性部件的旋转体的运动。在本文中，我们考虑这种系统的逆问题：给定部分光谱数据，找到一个系统，使其具有所需的光谱数据。给出了该问题的一般解，并通过 QR 分解和矩阵推导提供了一对矩阵的最佳近似解。此外，我们还考虑了系统在最低临界速度以下运行的特殊情况。数值例子表明所提出的方法是有效的。