lecture_11 - 2.160 System Identification, Estimation, and...

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2.160 System Identification, Estimation, and Learning Lecture Notes No. March 15, 2006 6.5 Times-Series Data Compression y ( t ) ) b 3 b 2 b 1 FIR u( t t Finite Impulse Response Model Consider a FIR Model ( ( ( ( t y ) = t u b 1) + t u b 2) + ⋅⋅ ⋅ + b t u m ) 1 2 m The transfer functions of (11) are then: G ( q ) = q B ), H ( q ) = 1 ( One-Step-Ahead Predictor 1 ( ı( 1 ) ( t y θ ) = H ( q ) G ( t u q ) + [ 1 H ( q ) ] t y ) ) ( = G ( t u q ) T t y ) = ϕ ( t ) : linear regression Given input data { t u 1 ), Nt } , the least square estimate of the parameter vector ( was obtained as N ı LS 1 ( 1 N = min arg 1 ( t y ) t y | ) ) 2 = ( R ( N ) ) f ( N ) (40) N t = 1 2 where N N T 1 ) ( R ( N ) = 1 ( t ) ( t ) = ΦΦ T and f ( N ) = ( t y t ) (41) N t = 1 N t = 1 Pro’s and Con’s of FIR Modeling Pros. LS ( LSE gives a consistent estimate N = 0 as long as the input sequence { t u ) } is lim ı N →∞ uncorrelated with the noise e(t) , which may be colored. Cons Two failure scenarios The impulse response has a slow decaying mode 1 11
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The sampling rate is high The number of parameters, m, is too large to estimate. The persistently exciting condition rank = Φ rank full can hardly be satisfied. Check the eigenvalues of ΦΦ T (or the singular values of Φ ) λ 2 1 m It is likely n + 1 0 = = 1 2 n m Often m becomes more than 50 and it is difficult to obtain such an input series having 50 non-zero singular values. Time series data compression is an effective method for coping with this difficulty. Before formulating the above least square estimate problem, data are processed so that the information contained in the series of regressor may be represented in compact, low-dimensional form.
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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.

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lecture_11 - 2.160 System Identification, Estimation, and...

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