RIA: Coupled State-Space Filtering with Applications

本研究的目的是为自适应滤波、鲁棒估计、控制和结构化矩阵计算等问题开发模块化和并行化的算法。这项工作的主要动机是需要正确地识别、建模和利用可能存在于特定问题中的方便结构，以便开发计算上有效的解决方案。提出的工作的一个内在和相关的部分是首先确定共同的成分，以及明确澄清，在控制、信号处理和数学等学科的几个问题之间存在的相互关系和相互作用。在这方面，提出的研究发展了一个统一的观点，同时解决了估计、控制、自适应滤波和矩阵计算中各种看似无关的问题。它表明，这些领域的许多应用的解决方案都有一个关键和惊人的简单成分，即计算具有结构的矩阵的三角因子。这一特殊事实的意义，更一般地说，作为一个整体提出的研究的意义在于，它使几个很好理解的概念从矩阵理论和线性代数中得到应用，以简化各种算法的推导，提出新的计算效率的变体，并开发新的结构形式。本研究分为理论研究和算法实现两部分。在这项工作中开发的技术在几个领域都有应用，其中包括用于自适应RLS滤波的新窗口方案，基于梯度的算法分析，以及用于工具变量方法的模块化和平方根算法，鲁棒估计和控制。同样重要的是发展一种系统和统一的方法。

The purpose of this research is to develop modular and parallelizable algorithms for problems in adaptive filtering, robust estimation, control, and structured matrix computations. A major motivation for this work is the need to properly identify, model, and exploit convenient structures that might exist in a particular problem in order to develop a computationally effective solution. An intrinsic and relevant part of the proposed work is to first identify common ingredients, as well as to explicitly clarify, the interrelations and interplays that exist among several problems in the disciplines of control, signal processing, and mathematics. In this regard, the proposed research develops a unifying point of view that simultaneously addresses a variety of seemingly unrelated problems in estimation, control, adaptive filtering, and matrix computations. It shows that the solutions to many applications in these areas turn out to share a key and surprisingly simple ingredient, namely, that of computing the triangular factors of matrices that exhibit structure. The significance of this particular fact and, more generally, of the proposed research as a whole, is that it enables several well-understood concepts from matrix theory and linear algebra to be applied in order to simplify the derivation of a variety of algorithms, to propose new computationally efficient variants, and to exploit new forms of structure. The proposed research consists of theoretical investigations and algorithm implementations. The techniques developed in this work have applications in several areas including, among others, new windowing schemes for adaptive RLS filtering, analysis of gradient- based algorithms, and modular and square-root algorithms for instrumental variable methods, robust estimation, and control. Equally important is the development of a systematic and unifying methodology.