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

在稳定性分析方面，利用了在频域中定义的采样数据系统的频率响应算子，完成了与之相关的正实理论。例如，澄清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判定正实性的代数方法。此外，对于周期时变系统的Nyquist稳定性判据，利用与之相关的频率响应算子的双正则化行列式，得到了基于张量原理的新结果。这些关于稳定性分析的结果可以很容易地应用于采样数据系统的鲁棒稳定性分析和鲁棒性能分析，可以说，它们通过适当地弥合了代数方法和分析方法之间的差距，为鲁棒稳定性/性能分析的高级方法铺平了道路。同时，对于连续时间控制器的离散化和降阶方法，也进行了理论研究和数值研究。通过这些研究，提出了我们今后的研究方向。此外，对于周期时变系统的分析，介绍了一些不需要解无限维方程组的近似方法，并通过误差分析建立了一些收敛性质，从而验证了它们的有效性。最后，作为我们研究方向有效性的一个例子，给出了一种新的方法，使人们能够以系统和透明的方式处理线性时不变系统的正实分析和有界实分析。

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.