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

在稳定性分析方面，利用了定义在频域中的采样数据系统的频率响应算子，并完成了正实性的相关理论。例如，阐明了正实性与某些特征值条件之间的关系，并基于算子特征值的惯性定律建立了一种检验正实性的代数方法。此外，对于周期时变系统的Nyquist稳定性判据，利用频率响应算子的双正则化行列式，基于幅角原理得到了新的结果.这些稳定性分析的结果可方便地应用于采样系统的鲁棒稳定性分析和鲁棒性能分析;可以说，他们通过适当地弥合代数方法和解析方法之间的差距，为鲁棒稳定性/性能分析的先进方法铺平了道路。顺便说一下，从这种基本分析中得到了一些新的想法，并部分推导了它们的基本性质。此外，关于连续时间控制器的离散化和减少的方法，理论研究以及数值研究已经进行。通过这些研究，我们未来的研究方向提出了建议。此外，对于周期时变系统的分析，已经引入了一些近似方法，这些方法不需要求解无穷维方程，并且通过误差分析建立了一些收敛性质来阐明它们的有效性。最后，作为一个例子，证明我们的研究方向的有效性，一个新的方法已被证明，使一个处理的正实性分析和有界实性分析的线性时不变系统的系统和透明的方式。

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.