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Unformatted text preview: 2.160 System Identification, Estimation, and Learning Lecture Notes No. 1 6 April 19, 2006 11 Informative Data Sets and Consistency 11.1 Informative Data Sets Predictor: [ ] ) ( ) ( 1 ) ( ) ( ) ( ) 1 ( 1 1 t y q H t u q G q H t t y + = [ ] ) ( ) ( ) ( ) ( ) ( ), ( ) 1 ( t z q W t y t u q W q W t t y y u = = (1) Definition1 Two models W 1 (q) and W 2 (q) are equal if frequency functions ) ( ) ( 2 1 i i e W e W = (2) for almost all Definition2 A quasi-stationary data set Z is informative enough with respect to model structure M if, for any two models in M ) ( ) ( ) ( 1 1 1 t z q W t y = and ) ( ) ( ) ( 2 2 2 t z q W t y = Condition ] ) ) ( ) ( [( 2 2 2 1 1 = t y t y E (3) implies ) (4) ( ) ( 2 1 i i e W e W = for almost all Let us characterize a quasi-stationary data set Z by power spectrum ) ( v (Spectrum Matrix): 2 2 ) ( ) ( ) ( ) ( ) ( = R y yu uy u z (5) Theorem 1 A quasi-stationary data set Z is informative if the spectrum matrix for is strictly positive definite for almost all T t y t u t z )) ( ), ( ( ) ( = . Proof [ ] ) ( ) ( ) ( ) ( ) ( 2 1 2 2 1 1 t z q W q W t y t y = Using eq.11 of Lecture Note 17, (3) can give by [ ] [ ] [ ] ) ( ) ( ) ( ) ( ) ( 2 1 ) ( ) ( 2 1 2 1 2 1 = = d e W e W e W e W t z W W E i i z T i i (6) 1 Since ) ( z is strictly positive definite for almost all...
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.160 taught by Professor Harryasada during the Spring '06 term at MIT.
- Spring '06