Minimal realization

About: Minimal realization is a research topic. Over the lifetime, 720 publications have been published within this topic receiving 19764 citations.

Principal component analysis in linear systems: Controllability, observability, and model reduction

Abstract: 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 X_{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.

Nonlinear controllability and observability

Abstract: The properties of controllability, observability, and the theory of minimal realization for linear systems are well-understood and have been very useful in analyzing such systems. This paper deals with analogous questions for nonlinear systems.

Linear System Theory

Abstract: Mathematical notation and review state equation representation state equation solution transition matrix properties two important cases internal stability Lyapunov stability criteria additional stability criteria controllability and observability realizability minimal realization input-out-put stability controller and observer forms linear feedback state observation polynomial fraction description polynoial fraction applications geometric theory applications of geometric theory.

Stochastic theory of minimal realization

Abstract: In this paper it is shown that a natural representation of a state space is given by the predictor space, the linear space spanned by the predictors when the system is driven by a Gaussian white noise input with unit covariance matrix. A minimal realization corresponds to a selection of a basis of this predictor space. Based on this interpretation, a unifying view of hitherto proposed algorithmically defined minimal realizations is developed. A natural minimal partial realization is also obtained with the aid of this interpretation.