A Framework for Subspace Identification Methods

A Framework for Subspace Identification Methods -...

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Proceedings of the American Control Conference Arlington, VA June 25-27, 2001 A Framework for Subspace Identification Methods Ruijie Shi and John F. MacGregor Dept. of Chemical Engineering, McMaster University, Hamilton, ON L8S 4L7, Canada Email: shir@mcmaster.ca, macgreg@mcmaster.ca ' Abstract Similarities and differences among various subspace identification methods (MOESP, N4SID and CVA) are examined by putting them in a general regression framework. Subspace identification methods consist of three steps: estimating the predictable subspace for multiple future steps, then extracting state variables from this subspace and finally fitting the estimated states to a state space model. The major differences among these subspace identification methods lie in the regression or projection methods used in the first step to remove the effect of the future inputs on the future outputs and thereby estimate the predictable subspace, and in the latent variable methods used in the second step to extract estimates of the states. This paper compares the existing methods and proposes some new variations by examining them in a common framework involving linear regression and latent variable estimation. Limitations of the various methods become apparent when examined in this manner. Simulations are included to illustrate the ideas discussed. 1. Introduction Subspace identification methods (SIMs) have become quite popular in recent years. The key idea in SIMs is to estimate the state variables or the extended observability matrix directly from the input and output data. The most influential methods are CVA (Canonical Variate Analysis, Larimore, 1990), MOESP (Multivariable Output Error State space, Verhaegen and Dewilde, 1992) and N4SID (Numerical Subspace State-Space System IDentification, Van Overschee and De Moor, 1994). These methods are so different in their algorithms that it is hard to bring them together and get more insights on the essential ideas and the connections among them. However, some effort has been made to contrast these methods. Viberg (1995) gave an overview of SIMs and classified them into realization-based direct types, and also pointed out the different ways to get system matrices via estimated states extended observability matrix. Van Overschee and De Moor (1995) gave a unifying theorem based on lower order approximation of an oblique projection. Here different methods are viewed as different choices row and column weighting matrices for the reduced rank oblique projection. The basic structure and idea of their theorem is based on trying to cast these methods into the N4SID algorithm. It focuses on the algorithms instead of concepts and ideas behind of these methods. In this paper, SIMs are compared by casting them into a general statistical regression framework. The fundamental similarities and differences among these SIMs is clearly shown in this statistical framework. All the discussion in this paper is limited to the open loop case of linear time invariant &TI) system.
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A Framework for Subspace Identification Methods -...

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