1. 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

2. 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

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

4. 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

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

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

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

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

9. 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

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

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

12. 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

13. 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

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

15. Prediction-Error Approach
lecture 8
Prediction-Error Approach 15

16. Prediction-Error Approach
lecture 8
Prediction-Error Approach 16

17. 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

18. 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

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

20. Prediction-Error Approach
lecture 8
Prediction-Error Approach 20

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

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

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

24. 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‘)

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

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

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

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

29. Prediction-Error Approach
lecture 8
Prediction-Error Approach 29

30. 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

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

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

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

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

35. 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

36. 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

37. 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

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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

44. 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

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

46. 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

47. 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

48. 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

49. 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

50. 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

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

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

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

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

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

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

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

58. 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