线性系统文献综述线系统文献综述.doc

Principal component analysis in linear systems: Controllability, observability, and model reduction 线性系统中主成分分析：可控性，可观性，并减少模型 Kalmans 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, Kalmans 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 ofX_{c}, X_{bar{o}}, 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. 卡尔曼的最小实现理论涉及几何对象（可控，不可观测子空间）所承受的结构性不稳定。具体地，在一个模型中任意小的扰动可能会导致在相关的子空间的尺寸的变化。这种情况表现在它出现在尝试应用课本的算法来计算最小实现计算困难。与几何理论相关的结构不稳定性是不是唯一的控制，它产生于线性方程组的理论也是如此。在这种设置下，计算问题进行了