lecture_15 - 2.160 System Identification, Estimation, and...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 7

lecture_15 - 2.160 System Identification, Estimation, and...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online