A study on the construction of a framework of control theory based on organic combination of algebraic and analytic methods

这项研究旨在构建一个新的控制系统设计和分析框架，使代数方法和分析方法有机地结合在一起。这里，代数方法包括涉及矩阵的特征值/特征向量分析的方法。算子的谱分析以及矩阵和算子的行列式理论。另一方面，分析方法包括泛函分析、算子理论和复函数理论等方法。本文重点研究了采样数据系统的设计和分析以及线性连续时间周期系统的分析。在采样数据系统的研究中，推导了高效且相当准确的频响增益上下界，并建立了由这些上下界计算任意精度的精确频响增益的二分法，并编制了相应的计算机程序。此外，Positive-re…在线性连续时间周期系统的研究中，首先讨论了在线性连续时间周期系统研究中起关键作用的关联频率响应算子，阐明了系统应满足的条件，从而严格定义了关联频率响应算子。此外，对频率响应算子的性质进行了深入的研究，为后续的研究奠定了坚实的基础。这些结果被推广到使我们能够通过无限维代数方程来分析线性连续时间周期系统的稳定性，我们称之为调和Lyapunov方程。证明了利用该方程进行分析的有效性，并证明了H_2和H_无穷性能分析可以通过我们所说的无限维矩阵的斜截断来进行。还建立了关于该截断的收敛性质。较少

This research aims at constructing a new framework for design and analysis of control systems in such a way that algebraic and analytic methods are combined in an organic fashion. Here, an algebraic method includes such approaches involving eigenvalue/eigenvector analysis of matrices. spectral analysis of operators and determinant theory of matrices and operators. On the other hand, an analytic method includes such approaches involving functional analysis, operator theory and complex function theory. In this research design and analysis of sampled-data systems as well as analysis of linear continuous-time periodic systems are particularly focused on.In the study of sampled-data systems, efficient and fairly accurate upper and lower bounds for the frequency response gain are derived, and a bisection method to compute the exact frequency response gain to any degree of accuracy from those upper and lower bounds is also established, including associated computer programs. Also, positive-re … More alness and Nyquist stability criterion are studied, and spectral properties of operators associated with sampled-data systems are clarified.In the study of linear continuous-time periodic systems, we first dealt with the associated frequency response operator, which plays a key role in the study of linear continuous-time periodic systems, and clarified the conditions that should be satisfied by the system so that the associated frequency response operator can be defined in a rigorous fashion. Furthermore, properties of the frequency response operators are studied thoroughly, by which a solid basis has been established for studies to follow. These results are extended to enable us to analyze stability of linear continuous-time periodic systems through an infinite-dimensional algebraic equation which we call a harmonic Lyapunov equation. Effectiveness of the analysis using this equation is shown, and it is also shown that the H2 and H-infinity performance analysis can be carried out through what we call skew truncation of infinite-dimensional matrices. Convergence properties regarding this truncation are also established. Less