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lecture_16 - 2.160 System Identification Estimation and...

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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 ( ˆ 1 1 t y q H t u q G q H t t y − − − + = − [ ] ) ( ) ( ) ( ) ( ) ( ), ( ) 1 ( ˆ 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 ω ω i i e W e W = (2) for almost all π ω π ω ≤ ≤ − Definition2 A quasi-stationary data set ∞ Z is informative enough with respect to model structure M if, for any two models in M ) ( ) ( ) ( ˆ 1 1 1 t z q W t y = θ and ) ( ) ( ) ( ˆ 2 2 2 t z q W t y = θ Condition ] ) ) ( ˆ ) ( ˆ [( 2 2 2 1 1 = − θ θ t y t y E (3) implies ) (4) ( ) ( 2 1 ω ω i i e W e W = for almost all π ω π ω ≤ ≤ − Let us characterize a quasi-stationary data set ∞ Z by power spectrum ) ( ω v Φ (Spectrum Matrix): 2 2 ) ( ) ( ) ( ) ( ) ( × ∈ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ Φ Φ Φ Φ = Φ R y yu uy u z ω ω ω ω ω (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 )) ( ), ( ( ) ( = ω . Proof [ ] ) ( ) ( ) ( ) ( ˆ ) ( ˆ 2 1 2 2 1 1 t z q W q W t y t y − = − θ θ Using eq.11 of Lecture Note 17, (3) can give by [ ] [ ] [ ] ) ( ) ( ) ( ) ( ) ( 2 1 ) ( ) ( 2 1 2 1 2 1 = − Φ − = − ∫ − ω ω π ω ω π π ω ω d e W e W e W e W t z W W E i i z T i i (6) 1 Since ) ( ω z Φ is strictly positive definite for almost all...
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