Computer Aided Systems Theory — EUROCAST 2001
Computer Aided Systems Theory — EUROCAST 2001
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计算机辅助系统理论 — EUROCAST 2001
DOI：
10.1007/3-540-45654-6
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
J. L. Freire
中科院分区：
文献类型：
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作者：
R. Moreno;B. Buchberger;J. L. Freire
在本文中，我们简要概述了Gröbner基地的理论，给新手没有事先在该领域的知识。在解释了通过Gröbner方法解决问题的一般策略之后，我们通过研究多项式除法的唯一性（“归约”）来发展Gröbner基的概念。为了明确地构造Gröbner基，引入了S多项式的关键概念，从而得到了构造问题的完整算法解决方案。从多项式方程求解和代数关系两个方面给出了算法的应用实例。在对复杂性问题进行了简短的讨论之后，我们总结了一些历史评论和参考文献。最初，Gröbner基的方法是在[3，4]中引入的，用于交换代数（多项式理想理论，代数几何）中一些基本问题的算法解决方案。1985年，应N. K.玻色，我写了一份调查的Gröbner基地的方法，他的书n维系统理论，见[7]。从那时起，Gröbner基方法在系统理论中得到了相当多的应用。不久，《多维系统与信号处理杂志》的特刊将完全致力于这个主题，见[11]。通过回顾近年来的文献，可以发现系统理论中越来越多的问题可以用Gröbner基方法来解决：多元多项式矩阵的因式分解，单边和双边多项式矩阵方程的可解性检验和解的构造，Bezout恒等式，FIR / IIR多维滤波器组的设计，R. Moreno-Diaz等人（编辑）：EUROCAST 2001，LNCS 2178，pp. 1-19，2001年。Springer-Verlag柏林海德堡2001反馈稳定补偿器/渐近观测器的可稳定性/可检测性测试和合成，无差拍或渐近跟踪控制器/调节器的合成，nD多项式矩阵完成问题的构造性解，最小左零化器/最小右零化器的计算，消除行为的潜变量表示的变量，可控部分的计算;可控性检验，
In this paper, we give a brief overview on Gröbner bases theory, addressed to novices without prior knowledge in the field. After explaining the general strategy for solving problems via the Gröbner approach, we develop the concept of Gröbner bases by studying uniquenss of polynomial division ("reduction"). For explicitly constructing Gröbner bases, the crucial notion of S polynomials is introduced, leading to the complete algorithmic solution of the construction problem. The algorithm is applied to examples from polynomial equation solving and algebraic relations. After a short discussion of complexity issues, we conclude the paper with some historical remarks and references. 1 Motivation for Systems Theorists Originally, the method of Gröbner bases was introduced in [3, 4] for the algorithmic solution of some of the fundamental problems in commutative algebra (polynomial ideal theory, algebraic geometry). In 1985, on the invitation of N. K. Bose, I wrote a survey on the Gröbner bases method for his book on n dimensional systems theory, see [7]. Since then quite some applications of the Gröbner bases method have been found in systems theory. Soon, a special issue of the Journal of Multidimensional Systems and Signal Processing will appear that is entirely devoted to this topic, see [11]. Reviewing the recent literature on the subject, one detects that more and more problems in systems theory turn out to be solvable by the Gröbner bases method: factorization of multivariate polynomial matrices, solvability test and solution construction of unilateral and bilateral polynomial matrix equations, Bezout identity, design of FIR / IIR multidimensional filter banks, R. Moreno-Diaz et al. (Eds.): EUROCAST 2001, LNCS 2178, pp. 1-19, 2001. Springer-Verlag Berlin Heidelberg 2001 stabilizability / detectability test and synthesis of feedback stabilizing compensator / asymptotic observer, synthesis of deadbeat or asymptotic tracking controller / regulator, constructive solution to the nD polynomial matrix completion problem, computation of minimal left annhilators / minimal right annhilators, elimination of variables for latent variable representation of a behaviour, computation of controllable part; controllability test,