In this article, the minimum norm partial eigenstructure assignment problem (MNPESAP) is considered in a symmetric high-order system using feedback control law with the highest-order derivative term. Two new orthogonality relations are constructed using the system matrices and undetermined feedback gains. The parametric expressions of the feedback controller and the necessary and sufficient condition for solving the partial eigenstructure assignment problem are derived. The Sylvester matrix equations and gradient formula are presented for solving MNPESAP. An algorithm is proposed to obtain the minimum norm solutions of feedback gains using gradient-based optimization method. Numerical examples are provided to verify the effectiveness of given results.

中文翻译：

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基于反馈控制的高阶系统最小范数部分特征结构分配问题

在本文中，最小范数部分特征结构分配问题（MNPESAP）是在一个对称高阶系统中考虑的，该系统使用具有最高阶导数项的反馈控制律。使用系统矩阵和未确定的反馈增益构建了两个新的正交关系。推导了反馈控制器的参数表达式和求解部分特征结构分配问题的充要条件。给出了求解 MNPESAP 的 Sylvester 矩阵方程和梯度公式。提出了一种利用基于梯度的优化方法获得反馈增益最小范数解的算法。提供了数值例子来验证给定结果的有效性。