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

点击翻译按钮获取中文摘要

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