SYSTEMS Identification Ali Karimpour Assistant Professor Ferdowsi University of Mashhad <<<1.1>>> ###Control System Design### {{{Control, Design}}} Reference: “System Identification Theory For The User” Lennart Ljung

Convergence & Consistency lecture 8 Lecture 8 Convergence & Consistency Topics to be covered include: Conditions on the Data Set Prediction-Error Approach Consistency and Identifiability LTI Models: A Frequency-Domain Description of the Limit Model The Correlation Approach 2

lecture 8 Introduction A number of different methods to determine models from data are described in chapter 7. 3 3

Convergence & Consistency lecture 8 Lecture 8 Convergence & Consistency Topics to be covered include: Conditions on the Data Set Prediction-Error Approach Consistency and Identifiability LTI Models: A Frequency-Domain Description of the Limit Model The Correlation Approach 4

Conditions on the Data Set lecture 8 Conditions on the Data Set 5

Conditions on the Data Set lecture 8 Conditions on the Data Set A Technical Condition D1 6

Conditions on the Data Set lecture 8 Conditions on the Data Set Remind: 7

Conditions on the Data Set lecture 8 Conditions on the Data Set A True System S 8

Conditions on the Data Set lecture 8 Conditions on the Data Set When S1 holds, a more explicit version of conditions D1 can be given Exercise: Prove it. 9

Conditions on the Data Set lecture 8 Conditions on the Data Set Information Content in the Data Set Recall: 10

Conditions on the Data Set lecture 8 Conditions on the Data Set Information Content in the Data Set 11

Conditions on the Data Set lecture 8 Conditions on the Data Set The concept of informative data Set is very close to Persistently exciting inputs General enough inputs Exercise: Prove it. 12

Convergence & Consistency lecture 8 Lecture 8 Convergence & Consistency Topics to be covered include: Conditions on the Data Set Prediction-Error Approach Consistency and Identifiability LTI Models: A Frequency-Domain Description of the Limit Model The Correlation Approach 13

Prediction-Error Approach lecture 8 Prediction-Error Approach Using D1 Condition: Uniformly Stable 14

Prediction-Error Approach lecture 8 Prediction-Error Approach 15

Prediction-Error Approach lecture 8 Prediction-Error Approach 16

Prediction-Error Approach lecture 8 Prediction-Error Approach Ensemble- and Time-averages By Lemma 8-2 we have: But for quasi stationary process by theorem 2.3 we have: 17

Prediction-Error Approach lecture 8 Prediction-Error Approach Ensemble- and Time-averages By Lemma 8-2 we have: The General Case In summary, we have 18

Prediction-Error Approach lecture 8 Prediction-Error Approach Example 8.1 Bias in ARX Structures 19

Prediction-Error Approach lecture 8 Prediction-Error Approach 20

Prediction-Error Approach lecture 8 Prediction-Error Approach Calculation of (I) 21

Prediction-Error Approach lecture 8 Prediction-Error Approach Calculation of From relation (I) 22

Prediction-Error Approach lecture 8 Prediction-Error Approach Calculation of steady state: y(t) starts at t = - ∞ 23

Prediction-Error Approach clear; close all; clc a0 = 0.5; b0 = 1; c0 = 1; r0 = (b0^2 + c0*(c0-a0) - a0*c0 + 1) / (1-a0^2); a1 = a0 - (c0 / r0); b1 = b0; a2 = a0; b2 = b0; y(1) = 0; u = randn(1,100); e = randn(1,100); for k = 2:100 y(k) = -a0 * y(k-1) + b0 * u(k-1) + e(k) + c0 * e(k-1); yh1(k) = -a1 * y(k-1) + b1 * u(k-1); yh2(k) = -a2 * y(k-1) + b2 * u(k-1); end MSE_ratio = mean((y-yh1).^2) / mean((y-yh2).^2) k = 1:100; hold on; plot(k,y); plot(k,yh1,'r');plot(k,yh2,'g‘)

Prediction-Error Approach u(t) and e(t) are random signals MSE_ratio = 0.7638 u(t) and e(t) are random signals

Prediction-Error Approach u(t) and e(t) are random signals MSE_ratio = 0.8897 u(t) and e(t) are random signals

Prediction-Error Approach u(t) and e(t) are random signals MSE_ratio = 0.7511 u(t) and e(t) are random signals

Prediction-Error Approach lecture 8 Prediction-Error Approach Example 8.2 Wrong Time Delay 28

Prediction-Error Approach lecture 8 Prediction-Error Approach 29

Convergence & Consistency lecture 8 Lecture 8 Convergence & Consistency Topics to be covered include: Conditions on the Data Set Prediction-Error Approach Consistency and Identifiability LTI Models: A Frequency-Domain Description of the Limit Model The Correlation Approach Ali Karimpour Nov 2009

Consistency and Identifiability lecture 8 Consistency and Identifiability The first condition: Exercise: Prove the above-mentioned theorem. Ali Karimpour Nov 2009

Consistency and Identifiability lecture 8 Consistency and Identifiability Ali Karimpour Nov 2009

Consistency and Identifiability lecture 8 Consistency and Identifiability Exercise: Prove it. Ali Karimpour Nov 2009

Consistency and Identifiability lecture 8 Consistency and Identifiability Example 8.3 First Order Output Error Model Ali Karimpour Nov 2009

Convergence & Consistency lecture 8 Lecture 8 Convergence & Consistency Topics to be covered include: Conditions on the Data Set Prediction-Error Approach Consistency and Identifiability LTI Models: A Frequency-Domain Description of the Limit Model The Correlation Approach Ali Karimpour Nov 2009

LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 LTI Models: A Frequency-Domain Description of the Limit Model Ali Karimpour Nov 2009

LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 LTI Models: A Frequency-Domain Description of the Limit Model An Expression for Ali Karimpour Nov 2009

LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 LTI Models: A Frequency-Domain Description of the Limit Model Ali Karimpour Nov 2009

LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 LTI Models: A Frequency-Domain Description of the Limit Model Ali Karimpour Nov 2009

LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 LTI Models: A Frequency-Domain Description of the Limit Model Ali Karimpour Nov 2009

LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 LTI Models: A Frequency-Domain Description of the Limit Model Open Loop Case Ali Karimpour Nov 2009

LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 LTI Models: A Frequency-Domain Description of the Limit Model Ali Karimpour Nov 2009

LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 LTI Models: A Frequency-Domain Description of the Limit Model Ali Karimpour Nov 2009

LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 LTI Models: A Frequency-Domain Description of the Limit Model Closed Loop Case Ali Karimpour Nov 2009

LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 LTI Models: A Frequency-Domain Description of the Limit Model Example 8.5 Approximation in the Frequency Domain Ali Karimpour Nov 2009

LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 LTI Models: A Frequency-Domain Description of the Limit Model Ali Karimpour Nov 2009

LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 LTI Models: A Frequency-Domain Description of the Limit Model Ali Karimpour Nov 2009

LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 LTI Models: A Frequency-Domain Description of the Limit Model Ali Karimpour Nov 2009

LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 LTI Models: A Frequency-Domain Description of the Limit Model Ali Karimpour Nov 2009

Convergence & Consistency lecture 8 Lecture 8 Convergence & Consistency Topics to be covered include: Conditions on the Data Set Prediction-Error Approach Consistency and Identifiability LTI Models: A Frequency-Domain Description of the Limit Model The Correlation Approach Ali Karimpour Nov 2009

The Correlation Approach lecture 8 The Correlation Approach Basic Convergence Result Ali Karimpour Nov 2009

The Correlation Approach lecture 8 The Correlation Approach Ali Karimpour Nov 2009

The Correlation Approach lecture 8 The Correlation Approach Ali Karimpour Nov 2009

The Correlation Approach lecture 8 The Correlation Approach Instrumental-variable Methods Ali Karimpour Nov 2009

The Correlation Approach lecture 8 The Correlation Approach Ali Karimpour Nov 2009

The Correlation Approach lecture 8 The Correlation Approach Ali Karimpour Nov 2009

The Correlation Approach lecture 8 The Correlation Approach Ali Karimpour Nov 2009

LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 LTI Models: A Frequency-Domain Description of the Limit Model Ali Karimpour Nov 2009