Topic

# Minimal realization

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

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TL;DR: In this paper, it is shown that principal component analysis (PCA) is a powerful tool for coping with structural instability in dynamic systems, and it is proposed that the first step in model reduction is to apply the mechanics of minimal realization using these working subspaces.

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.

4,861 citations

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TL;DR: 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 as discussed by the authors.

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.

2,123 citations

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01 Jun 1992

TL;DR: In this article, a mathematical notation and review of 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.

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.

1,575 citations

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TL;DR: In this article, 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.

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.

378 citations

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TL;DR: The fuzzy sliding mode control method, developed by the application of fuzzy set theory, provides a simple way to achieve asymptotic stability of the systems and is capable of handling the chattering problem inherent to sliding Mode control simply and effectively.

Abstract: By the application of fuzzy set theory, a method for feedback control of nonlinear systems based on the principle of sliding mode control is developed, called fuzzy sliding mode control. The control method provides a simple way to achieve asymptotic stability of the systems. Other attractive features of the method include a minimal realization of the fuzzy controller and robustness against model uncertainties and external disturbances. In addition, the method is capable of handling the chattering problem inherent to sliding mode control simply and effectively. A simulation study of an inverted pendulum system is presented to demonstrate these features of the method. Also, it is indicated that the method is easy to use for practicing control engineers.

342 citations