引用本文:刘凯,符方舟,屠园园.基于互质分解的控制系统可重构性评价[J].控制理论与应用,2026,43(5):1093~1100.[点击复制]
LIU Kai,FU Fang-zhou,TU Yuan-yuan.Reconfigurability evaluation method for control systems based on coprime factorization[J].Control Theory & Applications,2026,43(5):1093~1100.[点击复制]
基于互质分解的控制系统可重构性评价
Reconfigurability evaluation method for control systems based on coprime factorization
摘要点击 353  全文点击 14  投稿时间：2024-09-17  修订日期：2025-09-10
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DOI编号  10.7641/CTA.2025.40500
  2026,43(5):1093-1100
中文关键词  开环不稳定系统  可重构性  Gramian矩阵  互质分解  Riccati方程
英文关键词  open-loop unstable systems  reconfigurability  Gramian matrix  coprime factorization  Riccati equations
基金项目  国家自然科学基金项目(U22B6001, 62388101, 62403047, 62473389), 广东省自然科学基金项目(2025B1515020097, 2024A1515011730), 空间智 能控制技术全国重点实验室开放基金课题项目(2023–JCJQ–LB–006–16)资助.
作者单位E-mail
刘凯 北京空间飞行器总体设计部 15031246@buaa.edu.cn 
符方舟* 中山大学航空航天学院  
屠园园 北京空间飞行器总体设计部  
中文摘要
      可控性Gramian矩阵是评价控制系统可重构性的重要工具, 能够有效反映系统对故障的自主恢复能力. 然 而, 对于工程中常见的开环不稳定系统, 传统的基于Lyapunov方程的Gramian矩阵求解方法存在固有局限性: 既无 法保证方程有解, 其所求结果也与真实的Gramian矩阵存在本质差异. 为此, 本文提出了一种基于互质分解的求解方 法: 依据Riccati 方程的解设计状态反馈, 构建含有内分母的互质分解形式, 而后通过求解闭环稳定系统的 Lyapunov方程即可获得与原系统等价的可控性Gramian矩阵. 在此基础上, 利用可控性Gramian矩阵对该类控制系 统的可重构性进行量化评价, 并以卫星姿态控制系统为例验证了所提方法的有效性.
英文摘要
      The controllability Gramian matrix is commonly used in the reconfigurability evaluation of control systems to acquire the system’s self-recovery capability in response to faults. However, for open-loop unstable systems commonly encountered in engineering, traditional Gramian matrix solution methods based on the Lyapunov equation have inherent limitations: they cannot guarantee the existence of a solution, and the results they produce differ fundamentally from the true Gramian matrix. To address this, this paper proposes a solution method based on coprime factorization: by designing state feedback through the solution of the Riccati equation, a coprime factorization form with internal denominators is constructed. Subsequently, the controllability Gramian matrix, which is identical to the original system, can be obtained by solving the Lyapunov equation for the closed-loop stable system. On this basis, the reconfigurability of such control systems is quantitatively evaluated using the controllability Gramian matrix, and the effectiveness of the proposed method is validated through an example of a satellite attitude control system.