现代控制理论中线性代数与矩阵分析的融合路径研究

Research on the Integration Path of Linear Algebra and Matrix Analysis in Modern Control Theory

摘要: 现代控制理论作为控制科学的重要分支，在工业自动化、航空航天、机器人技术等众多领域发挥着关键作用。它以状态空间法为核心，突破了经典控制理论仅基于输入输出关系的局限性，能够更深入地描述和分析系统的内部状态变化。而线性代数与矩阵分析作为重要的数学工具，为现代控制理论的发展和应用提供了坚实的理论基础和强大的分析手段。文章首先对线性代数与矩阵分析的理论耦合基础进行了阐述，然后分析了矩阵分析方法在控制理论中的融合路径，以及融合方法在工程实现中的应用。

Abstract: As an important branch of control science, modern control theory plays a crucial role in numerous fields such as industrial automation, aerospace, and robotics. Centered around the state-space method, it breaks through the limitations of classical control theory, which is solely based on input-output relationships, enabling a more in-depth description and analysis of the internal state changes of systems. Linear algebra and matrix analysis, as important mathematical tools, provide a solid theoretical foundation and powerful analytical means for the development and application of modern control theory. This paper first expounds on the theoretical coupling basis of linear algebra and matrix analysis, then analyzes the integration paths of matrix analysis methods in control theory, as well as the applications of these integrated methods in engineering implementation.