PRINCIPAL COMPONENT ANALYSIS IN LINEAR-SYSTEMS - CONTROLLABILITY, OBSERVABILITY, AND MODEL-REDUCTION
PRINCIPAL COMPONENT ANALYSIS IN LINEAR-SYSTEMS - CONTROLLABILITY, OBSERVABILITY, AND MODEL-REDUCTION
复制标题
DOI：
10.1109/tac.1981.1102568
复制
发表时间：
1981-01-01
影响因子：
6.8
通讯作者：
MOORE, BC
中科院分区：
文献类型：
--
作者：
MOORE, BC
卡尔曼的最小实现理论涉及几何对象(可控的、不可观测的子空间)，这些对象容易受到结构不稳定性的影响。具体地说，模型中任意微小的扰动都可能导致关联子空间的维度发生变化。这种情况表现在试图应用教科书算法来计算最小实现时出现的计算困难。与几何理论相关的结构不稳定性并不是控制所特有的；它也出现在线性方程理论中。在这种背景下，计算问题已经被研究了几十年，并开发了优秀的工具来应对这种情况。本文的主要目的之一是引起人们对主成分分析(Hotling，1933)和计算矩阵奇异值分解的算法(Golub和Reinsch，1970)的关注。它们共同构成了处理动态系统中结构不稳定性的强大工具。如本文所述，主成分分析是一种信号分析技术。(奇异值分解提供了计算机制。)为此，卡尔曼的最小实现理论被重新定义为对注入信号的响应。将信号分析应用于可控性和可观性，得到了一个“内部平衡”模型具有特殊性质的坐标系。对于渐近稳定的系统，这给出了可控子空间和不可观测子空间的工作近似。有人提出，模型降阶的一个自然的第一步是利用这些工作子空间应用最小实现机制。
Kalman's minimal realization theory involves geometric objects (controllable, unobservable subspaces) which are subject to structural instability. Specifically, arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces. This situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization. Structural instability associated with geometric theories is not unique to control; it arises in the theory of linear equations as well. In this setting, the computational problems have been studied for decades and excellent tools have been developed for coping with the situation. One of the main goals of this paper is to call attention to principal component analysis (Hotelling, 1933), and an algorithm (Golub and Reinsch, 1970) for computing the singular value decompositon of a matrix. Together they form a powerful tool for coping with structural instability in dynamic systems. As developed in this paper, principal component analysis is a technique for analyzing signals. (Singular value decomposition provides the computational machinery.) For this reason, Kalman's minimal realization theory is recast in terms of responses to injected signals. Application of the signal analysis to controllability and observability leads to a coordinate system in which the "internally balanced" model has special properties. For asymptotically stable systems, this yields working approximations of, the controllable and unobservable subspaces. It is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces.