Matrix Factorization Theory for Multidimensional Systems Applications

二维系统稳定化控制器的参数化最早是在1985年报道的，从那时起，即使对于某些可稳定化的传递矩阵不具有右/左互质分解的模型，也取得了相当大的进展。 多元矩阵分解结果的影响最近在多维滤波器的设计中得到了证实，不可分小波的构造已经实现。 感兴趣的潜在应用涉及的分析，处理，编码和压缩的带宽受限的通信信道上的多维多媒体信号的重建，这里提出的研究进一步发展的多元因式分解理论及其变种的适应和使用在具有挑战性的应用在多维系统问题。 研究将进行对多元多项式矩阵的原始因式分解的存在的必要和充分条件有关的问题，以推广的单模完成的建设算法，现在可用。 注意力将被引导到实际构造的因式分解后，它的存在性是通过应用最近的工具，在算法代数，特别是Grobner基理论在多项式维数的情况下，右，左因式分解的存在和建设的条件将被调查，特别是当约化子式的零互素约束不成立时。多元多项式环中的元素矩阵可以表示为可从Suslin稳定性定理导出的初等矩阵的乘积，这为使用这种结构的阶梯实现用于完全重构的双正交多带滤波器组提供了机制，目的是推广它们在滤波器组以及小波构造中的使用。（或其他类似的准单一制）是否强制执行，一般来说仍然是一个悬而未决的问题。 最近的解决方案给出了两个波段的n-D的情况下，使用Grobner基地，将进行研究，以充分了解其能力的局限性，推广到多波段的n-D的情况下，特别注意各种专业化，可能提供完整的建设性的解决方案。

Parameterization of stabilizing controllers for two- dimensional systems was first reported in 1985 and since that time considerable progress has been made even for models in which some stabilizable transfer matrices do not have right/left coprime factorizations. The impact of the multivariate matrix factorization results has recently been demonstrated in the design of multidimensional filter the construction of non- separable wavelets have been realized. Potential applications of interest involved the analysis; processing, coding and compression for reconstruction of multidimensional multimedia signals over bandwidth constrained communication channels.The research proposed here develops further the theory of multivariate factorization and its variants for adaptation and use in challenging applications in multidimensional systems problems. Research will be undertaken towards the of necessary and sufficient conditions for the existence of primitive factorization of multivariate polynomial matrices by relating the problem to generalization of unimodular completion for which construction algorithms are now available. Attention will be directed towards actually constructing the factorization after its existence is guaranteed by applying recent tools in algorithmic algebra, particulary Grobner basis theory over polynomial dimensional cases, the conditions for existence and construction of right and left factorizations will be investigated, especially when the zero coprimenss constraint on the reduced minors fail to hold.The very nonrestrictive conditions under which any square unimodular matrix with entries in a multivariate polynomial ring can be expressed as a product of elementary matrices derivable from Suslin's Stability Theorem provides the machinery for biothogonal multiband filter bank realization for perfect reconstruction using the ladder of such structures with the objective of popularizing their use in filter bank as well as wavelet construction.The tackling of the case when both perfect reconstruction and linear phase constraints (or other like paraunitary) are enforced, is still, in general, an open problem. The recent solution given for the two-band n-D case, using Grobner bases, will be studied to understand fully the limitations of its capability for generalization to the multiband n-D case with particular attention to various specializations that may offer complete constructive solutions.