Mathematical Sciences:Topis in Interpolation and System Theory

本研究将探索系统理论的一些基本新方向。人们早就知道，h∞控制的标准问题可以转化为Nevanlinna- Pick插值问题。最近的工作已经阐明了一个深远的模拟点评估和整个理论的相关类似的Nevanlinna-Pick插值时变系统。本研究将进一步阐述这些思想，特别是在时变情况下，根据零点和极点数据对系统进行参数化，以及时变系统的Nevanlinna-Pick插值与具有无限维输入输出和状态空间的时不变系统的Nevanlinna-Pick插值的联系。此外，PI将继续研究非线性系统和非线性h∞控制。特别是，在许多重要的应用中，构造非线性对象的内外分解是一项基本要求，但只有在特殊情况下才能很好地理解。最后，将讨论闭合黎曼曲面上涉及亚纯矩阵函数的各种插值问题。零极插值问题与经典的Riemann-Hilbert问题和Wiener-Hopf分解以及曲面上全纯向量束的分类密切相关。此外，这种零极插值思想为完全理解20世纪70年代在有限连通平面域上解析函数的Nevanlinna-Pick插值所做的工作提供了一个新的工具。一个基本的工具是用行列式表示作为二维线性系统传递函数的代数曲线上的亚纯束映射的实现理论。控制理论中的一个基本范例就是现在所说的h∞控制。当所考虑的物理系统的数学模型不能完全准确地表示真实系统时，这种方法提供了一种以数学方式制定控制目标的方法，以保证可靠的结果。这种差异可能是由某些物理参数的不确定性引起的，或者是数学模型中未考虑到的外部干扰的影响。最早解决最简单的H-∞控制问题的方法之一是通过经典数学理论Nevanlinna-Pick插值。在新兴的h∞理论需求的刺激下，许多研究者改进、扩展并使这一经典理论具有计算实用性。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。虽然现在已经知道如何将时变理论与定常情况在概念上统一起来，但需要进一步的工作来更好地理解最终定常线性系统和非线性系统的可分解性和鲁棒稳定性。这项研究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。在拓扑上更复杂的曲面上定义的函数的插值问题也将进行研究。这项工作有望在应用于控制系统涉及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.