EECE 574 - Adaptive Control - Basics of System Identification

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Least-Squares Identification Properties of the Least-Squares Estimates Note that the second term in the RHS is generally non-zero except when 1 {w(t)} is zero mean and uncorrelated, or 2 {w(t)} is independent of {u(t)} and y does not appear in the regressor FIR and step models Laguerre model Output-error models Only in the above cases is the least-squares estimate unbiased. Furthermore, if it is the case as the number of observations increases to infinity, then we say that the method gives consistent estimates. Guy Dumont (UBC) EECE574 - Basics of System Identification 22 / 40
Least-Squares Identification Properties of the Least-Squares Estimates The estimate covariance is then given by cov ˆ θ = E{[X T X] −1 X T WW T X[X T X] −1 } which, if {w(t)} is white, zero-mean gaussian with covariance σ 2 w , becomes cov ˆ θ = σ 2 w[X T X] −1 The matrix (X T X) −1 X is called the pseudo-inverse of X. Because of this, [X T X] −1 is sometimes called the covariance matrix. The condition that X T X be invertible is called an excitation condition. Guy Dumont (UBC) EECE574 - Basics of System Identification 23 / 40
Page 1 and 2: EECE 574 - Adaptive Control Basics
Page 3 and 4: Introduction System Identification
Page 5 and 6: Models for Identification Models fo
Page 15 and 16: Least-Squares Identification Least-
Page 21: Least-Squares Identification Proper
Page 25 and 26: Alternatives to Least-Squares Gener
Page 27 and 28: Alternatives to Least-Squares Gener
Page 29 and 30: Alternatives to Least-Squares Instr
Page 35 and 36: Alternatives to Least-Squares Maxim
Page 37 and 38: Alternatives to Least-Squares Maxim
Page 39 and 40: Practice Identification In Practice

EECE 574 - Adaptive Control - Basics of System Identification

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