Mathematical Sciences:Topis in Interpolation and System Theory

这项研究将探索系统理论的一些基本的新方向。 一段时间以来，人们已经知道H ∞控制的标准问题可以转换为Nevanlinna-Pick插值问题。 最近的工作已经照亮了一个意义深远的模拟点评估和一个完整的理论相关的模拟Nevanlinna-Pick插值的时变系统。这项研究将进一步阐述这些想法，特别是，参数化的系统中的零和极点数据的时变情况下，和连接Nevanlinna-Pick插值的时变系统与Nevanlinna-Pick插值的时不变系统具有无限维的输入-输出和状态空间。 此外，PI将继续研究非线性系统和非线性H ∞控制。特别是，一个非线性工厂的内部-外部因式分解的建设是一个基本的要求，在一些重要的应用，但只有在特殊情况下才能很好地理解。 最后，将进行涉及闭黎曼曲面上亚纯矩阵函数的各种插值问题的工作。 零极点插值问题与经典的Riemann-Hilbert问题、Wiener-Hopf分解以及曲面上全纯向量丛的分类密切相关。此外，这种零极点插值的想法提供了一个新的工具，完全理解的工作在20世纪70年代的Nevanlinna-Pick插值函数解析的一个连通的平面域。一个基本工具是代数曲线上的亚纯丛映射的实现理论，其行列式表示为二维线性系统的传递函数。 控制理论中的一个基本范例就是现在所谓的H ∞控制。 这种方法提供了一种方法来制定控制目标数学的方式，保证可靠的结果时，所考虑的物理系统的数学模型是不是一个完全准确的表示真实的系统。这些差异可能是由于某些物理参数的不确定性或数学模型中未考虑的外部干扰的影响造成的。 最简单的H ∞控制问题的最早解决方案之一是通过Nevanlinna-Pick插值的经典数学理论。 在新兴的H无穷理论的需求的刺激下，各种研究人员改进，扩展并使这个经典理论在计算上实用。 PI在将这种方法扩展到时变系统和非线性系统（其中基本物理定律不再满足叠加原理）方面发挥了重要作用。 虽然它现在已经知道如何统一的时变理论的概念与时不变的情况下，需要进一步的工作，以获得一个更好的理解分解和鲁棒稳定性最终时不变的线性系统和非线性系统。 这项研究有望有助于的应用包括飞机飞行控制、汽车悬架系统和光盘播放器的改进以及化学过程控制，仅举几例。 研究插值问题的功能定义在拓扑上更复杂的表面称为黎曼曲面也将进行。 这项工作预计将是有用的应用程序中的控制系统涉及的传递函数具有代数奇点。

This research will explore some basic new directions in systems theory. It has been known for some time that the standard problem of H-infinity control can be converted to a Nevanlinna- Pick interpolation problem. Recent work has illuminated a far-reaching analogue of point evaluation and a whole theory for associated analogues of Nevanlinna-Pick interpolation for time-varying systems. This investigation will pursue further elaboration of these ideas, in particular, the parametrization of a system in terms of zero and pole data for the time-variant case, and the connections of Nevanlinna-Pick interpolation for time-variant systems with Nevanlinna-Pick interpolation for time-invariant systems having infinite -dimensional input-output and state spaces. In addition, the PI will continue work on nonlinear systems and nonlinear H-infinity control. In particular, the construction of an inner-outer factorization for a nonlinear plant is a basic requirement in a number of important applications but is well-understood only in special situations. Finally, work on various interpolation problems involving meromorphic matrix functions on closed Riemann surfaces will be carried out. Zero-pole interpolation problems are closely related to classical Riemann-Hilbert problems and Wiener-Hopf factorization as well as to the classification of holomorphic vector bundles over the surface. In addition, such zero-pole interpolation ideas provide a new tool for completely understanding of work done in the 1970's on Nevanlinna-Pick interpolation for functions analytic on a finitely connected planar domain. A basic tool is the realization theory for meromorphic bundle maps on an algebraic curve with determinantal representation as the transfer function of a 2-dimensional linear system. A basic paradigm in control theory is what is now called H-infinity control. This approach provides a way to formulate control objectives mathematically in a way which guarantees reliable results when the mathematical model of the physical system under consideration is not a completely accurate representation of the true system. The discrepancies may be caused by uncertainty in certain physical parameters, or the effect of outside disturbances not taken into account in the mathematical model. One of the earliest solution procedures for the simplest H- infinity control problems was through the classical mathematical theory of Nevanlinna-Pick interpolation. Spurred by the needs of the emerging H-infinity theory, various researchers improved, extended and made this classical theory computationally practical. The PI has been instrumental in extending this approach to time-variant systems and nonlinear systems (where the basic physical law s no longer satisfy a superposition principle). Although it has now known how to unify time-variant theory conceptually with the time-invariant case, further work is needed to obtain a better understanding of decomposability and robust stability properties for eventually time-invariant linear systems and for nonlinear systems. Applications to which this research is expected to contribute include airplane flight control, improvements in automobile suspension systems and compact disk players, and chemical process control, to mention just a few. Research on interpolation problems for functions defined on topologically more complicated surfaces called Riemann surfaces will also be carried out. This work is expected be useful in applications to control systems involving transfer functions having algebraic singularities.