lecture_15 - 2.160 System Identification Estimation and...

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2.160 System Identification, Estimation, and Learning Lecture Notes No. 1 5 April 12, 2006 Part 3 System Identification Perspective of System Identification Theory True Process S u(t) Experiment Design Data Set {} ) ( ), ( t y t u Z N = Consistency 0 ~ ˆ θ N ? ) ( min arg N V Model Set M y(t) e(t) Key Questions: Q1: Is a given data set informative enough to uniquely determine a model from a given model set? Does Z contain sufficient information to distinguish any two models in M ? Q2: Is merely minimizing ) ( N V good enough to obtain the true (unbiased) model? What if the true model is not involved in the model set? How is the model-data fitting influenced by noise characteristics and input properties? Q3: How accurate is the estimated model? How much variance, expected error, etc.? How much data needed? How to design experiments Key Results Informative experiment and persistent excitation Consistent (unbiased) estimate Signal to noise ration Asymptotic variance Input desig n : Pseudo Random Binary signal Accuracy-variance trade-off System order estimate: Model selection 1
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Mathematical tools for Part 3 system identification Discrete Fourier transform and spectral analysis Central limit theorems Random processes: wide-sense stationary process, ergodic process, etc. 10 Frequency Domain Analysis 10.1 Discrete Fourier Transform and Power Spectrum Discrete Fourier transform of a sampled-data system: () x k , () x k Time ik k X xke ω +∞ =−∞ = (1) Note that () X is a 2 π -periodic function: (2) 2 (2 ) ( ) ( ) ( 1 in k i k i n k kk X Xnx k e x k e eX ωπ ) πω ∞∞ −+ =−∞ =−∞ += = = ∑∑ ±²³ ±´´² ´ ´³ (2) ) ( Xn X ) ( 3 ) A periodic function can be expanded to a Fourier series expansion. Therefore, we can
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lecture_15 - 2.160 System Identification Estimation and...

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