代数几何在控制理论中的方法（Part II：多变量线性系统和投影代数几何）

English | PDF | 2018 | 384 Pages | ISBN : 3319965735 | 24.56 MB

"An introduction to the ideas of algebraic geometry in the motivated context of system theory." This describes this two volume work which has been specifically written to serve the needs of researchers and students of systems, control, and applied mathematics. Without sacrificing mathematical rigor, the author makes the basic ideas of algebraic geometry accessible to engineers and applied scientists. The emphasis is on constructive methods and clarity rather than on abstraction. While familiarity with Part I is helpful, it is not essential, since a considerable amount of relevant material is included here. Part I, Scalar Linear Systems and Affine Algebraic Geometry, contains a clear presentation, with an applied flavor , of the core ideas in the algebra-geometric treatment of scalar linear system theory. Part II extends the theory to multivariable systems. After delineating limitations of the scalar theory through carefully chosen examples, the author introduces seven representations of a multivariable linear system and establishes the major results of the underlying theory. Of key importance is a clear, detailed analysis of the structure of the space of linear systems including the full set of equations defining the space. Key topics also covered are the Geometric Quotient Theorem and a highly geometric analysis of both state and output feedback. Prerequisites are the basics of linear algebra, some simple topological notions, the elementary properties of groups, rings, and fields, and a basic course in linear systems. Exercises, which are an integral part of the exposition throughout, combined with an index and extensive bibliography of related literature make this a valuable classroom tool or good self-study resource. The present, softcover reprint is designed to make this classic textbook available to a wider audience. "The exposition is extremely clear. In order to motivate the general theory, the author presents a number of examples of two or three input-, two-output systems in detail. I highly recommend this excellent book to all those interested in the interplay between control theory and algebraic geometry." —Publicationes Mathematicae, Debrecen "This book is the multivariable counterpart of Methods of Algebraic Geometry in Control Theory, Part I…. In the first volume the simpler single-input–single-output time-invariant linear systems were considered and the corresponding simpler affine algebraic geometry was used as the required prerequisite. Obviously, multivariable systems are more difficult and consequently the algebraic results are deeper and less transparent, but essential in the understanding of linear control theory…. Each chapter contains illustrative examples throughout and terminates with some exercises for further study." —Mathematical Reviews

“系统理论背景下对代数几何思想的介绍。”这描述了该两卷本著作，它专门编写以满足系统、控制和应用数学研究人员与学生的需要。在不牺牲数学严谨性的情况下，作者将代数几何的基本想法对工程师和技术人员进行了普及。重点放在构造方法上而非抽象化。虽然熟悉第一部分是很有帮助的，但并不必需，因为这里包含了大量的相关材料。第一卷“标量线性系统和仿射代数几何”包含了清晰、带应用色彩的核心思想，这是在处理标量线性系统理论时使用代数-几何方法时所涉及的基本想法。第二卷将该理论推广到了多变量系统。通过精心挑选的例子阐明了标量理论的局限性之后，作者介绍了七种表示法来说明一个多变量线性系统的概念，并建立了该基础理论中的主要结果。关键的重要性在于对系统的空间结构进行了清晰、详细的分析，包括定义这个空间集的所有方程。该书还涵盖了几何商定理和用于状态反馈与输出反馈的极为几何化的分析。先修条件是线性代数的基本知识，一些简单的拓扑概念，群、环和域的简单性质以及一个基本的线性系统课程。附有贯穿始终的练习，结合了索引和相关文献详尽的参考书目，使这本书成为了一种优秀的教科书或自学资源。“说明过程非常清晰。为了激励一般理论，作者详细地呈现了两个或者三个输入、两个输出系统的例子。我强烈推荐这本书给所有对控制理论与代数几何之间的相互作用感兴趣的人。” ——《德布勒森数学期刊》“该书是代数几何方法在控制系统中的应用（第一卷）的多变量版本……第一卷考虑的是更简单的单输入-单输出、时不变线性系统，并将其对应的简化的仿射代数几何作为所需的预备知识。明显地，多变量系统更加复杂，因此基于代数的结果更深且不够透明，但对理解线性控制理论是至关重要的…每章都包含详尽的示例并以一些练习结束，供进一步研究。” ——《数学评论》

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