Subspace algorithms for the stochastic identification problem

Subspace algorithms for the stochastic identification problem

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Proceedings of the 30th Conference on Declslon and Control Brighton, England - December 1991 TI-IO - 1O:OO SUBSPACE ALGORITHMS FOR THE STOCHASTIC IDENTIFICATION PROBLEM1 Peter Van Overschee2 Bart De Moor ESAT , Department of Electrical Engineering , Katholieke Universiteit Leuven Kardinaal Mercierlaan 94 , 3001 Leuven (Heverlee) , Belgium tel: 32/16/220931 fax: 32/16/221855 email: vanoversQesat.ku1euven.ac.be , demoorQesat.kuleuven.ac.be Abstract In this paper, we derive a new algorithm to consistently iden- tify stochastic state space models from given output data without forming the covariance matrix and using only semi-infinite block Hankel matrices. The algorithm is based on the concept of princi- pal angles and directions. We describe how they can be calculated with only QR and QSVD decompositions. We also provide an interpretation of the principal directions as states of a non-steady state Kalman filter. 1. Introduction Let Yk E @, k = 0, 1,. . . , K be a data sequence that is generated by the following system : Zk+1 = AZk + Wk , Yk = czk t vk (1) where Zk E s" is a state vector. Both wk E 8" and E 8' are assumed to be zero mean, white, gaussian noise sequences with covariance matrix 4: It is assumed that the stochastic process is stationary, i.e. : E[zk] = 0 E[X~Z:] = E, where the state covariance matrix C is independent of the time le. This implies that A is a stable matrix. The central problem discussed in this paper is the identification of a state space model from the data Yk and the determination of the noise covariance matrices. The main contributions of this paper are the following : - Since the pioneering papers by Akaike [l], canonical correla- tions (which were introduced by Hotelling [5] in the statisti- cal community) have been used as a mathematical tool in the stochastic realization problem. In this paper we show how the approach by Akaike and others (e.g. [2],[7]) boils down to applying canonical correlation analysis to two matrices that are double infinite. A careful analysis reveals the nature of this double infinity and we manage to reduce the canonical correlation approach to a semi-infinite matrix problem, i.e. only the number of columns needs to be very large while the number of (b1ock)rows remains sufficiently small. 'The research reported in this paper was partially supported by the Bel- gian Program on Interuniversity Attraction Poles initiated by the Belgian State Science Policy Programming (Prime Minister's Office) and the Eurw pean Community Research Program ESPRIT, Basic Research Action nr. 3280 and by a research project supported by Philips. The scientific responsibility is assumed by its authors. 'Research Assistant of the Belgian National Fund for Scientific Research 8Research Associate of the Belgian National Fund for Scientific Research 'The expected value operator is denoted by E[.].
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This note was uploaded on 01/29/2011 for the course ENGR 52 taught by Professor Mcmillan during the Spring '10 term at Baylor Med.

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Subspace algorithms for the stochastic identification problem

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