Applications of the Quillen-Suslin theorem to multidimensional systems theory
Applications of the Quillen-Suslin theorem to multidimensional systems theory
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DOI：
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发表时间：
2007
期刊：
影响因子：
3.8
通讯作者：
A. Fabiańska;A. Quadrat
中科院分区：
文献类型：
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作者：
A. Fabiańska;A. Quadrat
本文的目的是给出Quillen-Suslin定理在数学系统理论中的四个新应用。利用Quillen-Suslin定理的一个构造性版本，也称为Serre猜想，我们展示了如何有效地计算平坦多维线性系统的平坦输出和内射参数化。我们证明了一个平坦的多维线性系统在代数上等价于通过将定义该系统的多项式矩阵中除一个以外的所有泛函算子置零而得到的可控一维线性系统。特别地，我们证明了平坦常微分时滞线性系统与相应的无时滞常微分系统是代数等价的，即通过将所有时滞幅度设为零而得到的可控常微分线性系统。我们还给出了Serre猜想的一个推广的构造性证明，称为Lin-Bose猜想。此外，我们还证明了如何构造性地计算交换多项式环上有理转移矩阵的(弱)左/右/双互质分解。Quillen-Suslin定理也在符号计算文献中研究的所谓线性泛函系统的分解问题中发挥着核心作用。特别地，我们展示了某些自由模的基计算是如何由与系统相关的模的自同态环的射影得到的，使得我们可以得到把系统矩阵变换成等价的块三角形或块对角线形式的么模矩阵。最后，我们演示了QuillenSuslin包，据我们所知，它包含了Quillen-Suslin定理在计算机代数系统中的第一个实现以及本文所开发的不同算法。
The purpose of this paper is to give four new applications of the Quillen-Suslin theorem to mathematical systems theory. Using a constructive version of the Quillen-Suslin theorem, also known as Serre's conjecture, we show how to effectively compute flat outputs and injective parametrizations of flat multidimensional linear systems. We prove that a flat multidimensional linear system is algebraically equivalent to the controllable 1-D dimensional linear systems obtained by setting all but one functional operator to zero in the polynomial matrix defining the system. In particular, we show that a flat ordinary differential time-delay linear system is algebraically equivalent to the corresponding ordinary differential system without delay, i.e., the controllable ordinary differential linear system obtained by setting all the delay amplitudes to zero. We also give a constructive proof of a generalization of Serre's conjecture known as Lin-Bose's conjecture. Moreover, we show how to constructively compute (weakly) left-/right-/doubly coprime factorizations of rational transfer matrices over a commutative polynomial ring. The Quillen-Suslin theorem also plays a central part in the so-called decomposition problem of linear functional systems studied in the literature of symbolic computation. In particular, we show how the basis computation of certain free modules, coming from projectors of the endomorphism ring of the module associated with the system, allows us to obtain unimodular matrices which transform the system matrix into an equivalent block-triangular or a block-diagonal form. Finally, we demonstrate the package QuillenSuslin which, to our knowledge, contains the first implementation of the Quillen-Suslin theorem in a computer algebra system as well as the different algorithms developed in the paper.