Development and Application of control theory based on organic combination of algebraic and analytic methods

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Regarding stability analysis, the frequency response operators of sampled-data systems defined in the frequency domain have been utilized, and an associated theory for positive-realness has been completed. For example, the relationship between positive-realness and some eigenvalue conditions has been clarified, and an algebraic method for checking positive-realness has been established based on a inertia law about the eigenvalues of operators. Furthermore, regarding the Nyquist stability criterion for periodically time-varying systems, the two-regularized determinant about the associated frequency response operator was employed to lead to a new result based on the principle of argument. These results about stability analysis can readily be applied to robust stability analysis and robust performance analysis of sampled-data systems ; it can be said that they have laid a path to advanced methods for robust stability/performance analysis by properly bridging the gap between algebraic methods and analytic methods.As a side remark, some novel ideas have been derived from such fundamental analysis, and their basic properties have been partially derived. Also, regarding methods for discretization and reduction of continuous-time controllers, theoretical studies as well as numerical studies have been carried out. Through such studies, our future research direction has been suggested. Furthermore, regarding the analysis of periodically time-varying systems, some approximate methods have been introduced which does not require the solution of infinite-dimensional equations, and their effectiveness was clarified by establishing some convergence property via error analysis. Finally, as an example for demonstrating the effectiveness of our research direction, a new method has been shown that enables one to deal with the positive-realness analysis and the bounded-realness analysis of linear time-invariant systems in a systematic and transparent way.