1. SYSTEMS Identification Ali Karimpour Assistant Professor Ferdowsi University of Mashhad Reference: “System Identification Theory For The User” Lennart Ljung

2. lecture 1 Ali Karimpour Nov 2010 2 Lecture 8 Convergence & Consistency Topics to be covered include: v Conditions on the Data Set v Prediction-Error Approach v Consistency and Identifiability v LTI Models: A Frequency-Domain Description of the Limit Model v The Correlation Approach lecture 8

3. lecture 1 Ali Karimpour Nov 2010 33 Introduction lecture 8 Previously different methods to determine models from data are described. To use these methods in practice, we need insight into their properties: How well will the identified model describe the actual system? Are some identification methods better than others? How should the design variables be chosen? The questions depend to mapping: 1- Simulation. 2- Analysis.

4. lecture 1 Ali Karimpour Nov 2010 4 Lecture 8 Convergence & Consistency Topics to be covered include: v Conditions on the Data Set v Prediction-Error Approach v Consistency and Identifiability v LTI Models: A Frequency-Domain Description of the Limit Model v The Correlation Approach lecture 8

5. lecture 1 Ali Karimpour Nov 2010 5 lecture 8 Conditions on the Data Set Data generation configuration Let Analysis means:

6. lecture 1 Ali Karimpour Nov 2010 6 Conditions on the Data Set lecture 8 D1 A Technical Condition D1

7. lecture 1 Ali Karimpour Nov 2010 7 Conditions on the Data Set lecture 8 Remind:

8. lecture 1 Ali Karimpour Nov 2010 8 Conditions on the Data Set lecture 8 S A True System S

9. lecture 1 Ali Karimpour Nov 2010 9 Conditions on the Data Set lecture 8 Exercise1: Prove the lemma. When S1 holds, a more explicit version of conditions D1 can be given

10. lecture 1 Ali Karimpour Nov 2010 10 Conditions on the Data Set lecture 8 Information Content in the Data Set. Whether the data set Z  allows us to distinguish between different models in the set? The data set is informative if it is capable of distinguishing between different models. Definition: A quasi-stationary data set Z  is informative enough with respect to the model set M * if, for any two models W 1 (q) and W 1 (q) in the set, Definition: A quasi-stationary data set Z  is informative if it is informative enough with respect to the model set L *, consisting of all linear, time invariant models. Recall:

11. lecture 1 Ali Karimpour Nov 2010 11 Conditions on the Data Set lecture 8 Remember Theorem: A quasi-stationary data set Z  is informative if the spectrum matrix for z(t)=[u(t) y(t)] T is strictly positive definite for almost all . Proof: We need to show 2-65

12. lecture 1 Ali Karimpour Nov 2010 12 Lecture 8 Convergence & Consistency Topics to be covered include: v Conditions on the Data Set v Prediction-Error Approach v Consistency and Identifiability v LTI Models: A Frequency-Domain Description of the Limit Model v The Correlation Approach lecture 8

13. lecture 1 Ali Karimpour Nov 2010 13 Prediction-Error Approach lecture 8 In PEM Clearly it depends to For quadratic criterion and a linear, uniformly stable model structure M, we have Using D1 Condition: Uniformly Stable since M is stable

14. lecture 1 Ali Karimpour Nov 2010 14 Prediction-Error Approach lecture 8

15. lecture 1 Ali Karimpour Nov 2010 15 Prediction-Error Approach lecture 8 Since D M is compact It may happen that does not have a unique/global minimum.

16. lecture 1 Ali Karimpour Nov 2010 16 Prediction-Error Approach lecture 8

17. lecture 1 Ali Karimpour Nov 2010 17 Prediction-Error Approach lecture 8 Example 8.1 Bias in ARX Structures Suppose that the system is given by Where {u(t)} and {e 0 (t)} are independent white noises with unit variances. Let the model structure be given by The prediction-error variance is Exercise2 : Proof I

18. lecture 1 Ali Karimpour Nov 2010 18 Prediction-Error Approach lecture 8 Example 8.1 Bias in ARX Structures The prediction-error variance is But for true value of parameters When we apply PEM the estimates will converge, according to pervious theorem. But one of the parameters is biased. It is clear that bias is beneficial for the prediction performance of the model. So it gives a strictly better predictor. Exercise3 : Proof and explain the example by suitable simulation.

19. lecture 1 Ali Karimpour Nov 2010 19 Prediction-Error Approach lecture 8 Example 8.2 Wrong Time Delay Consider the system {e(t)} and {w(t)} are independent white-noise sequence with unit variances. Let the model structure be given by The associated prediction-error variance is: Exercise4 : Proof I

20. lecture 1 Ali Karimpour Nov 2010 20 Prediction-Error Approach lecture 8 Example 8.2 Wrong Time Delay Consider the system Hence Now the predictor Is a fairly reasonable one for

21. lecture 1 Ali Karimpour Nov 2010 21 Lecture 8 Convergence & Consistency Topics to be covered include: v Conditions on the Data Set v Prediction-Error Approach v Consistency and Identifiability v LTI Models: A Frequency-Domain Description of the Limit Model v The Correlation Approach lecture 8 Ali Karimpour Nov 2009

22. lecture 1 Ali Karimpour Nov 2010 22 The first condition: lecture 8 Ali Karimpour Nov 2009 Consistency and Identifiability Exercise5: Prove the above-mentioned theorem. Suppose that assumption S1 holds so that we have a true system. Will PE identification recover the true plant? So

23. lecture 1 Ali Karimpour Nov 2010 23 lecture 8 Ali Karimpour Nov 2009 Consistency and Identifiability

24. lecture 1 Ali Karimpour Nov 2010 24 Consistency and Identifiability lecture 8 Ali Karimpour Nov 2009

25. lecture 1 Ali Karimpour Nov 2010 25 Consistency and Identifiability lecture 8 Ali Karimpour Nov 2009 Example 8.3 First Order Output Error Model Suppose that the true system is given by Let we define a first-order output error model as: By pervious theorem the estimates of and will converges to the true value of and.

26. lecture 1 Ali Karimpour Nov 2010 26 Lecture 8 Convergence & Consistency Topics to be covered include: v Conditions on the Data Set v Prediction-Error Approach v Consistency and Identifiability v LTI Models: A Frequency-Domain Description of the Limit Model v The Correlation Approach lecture 8 Ali Karimpour Nov 2009

27. lecture 1 Ali Karimpour Nov 2010 27 LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009

28. lecture 1 Ali Karimpour Nov 2010 28 LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009 An Expression for Under assumption S1 we have

29. lecture 1 Ali Karimpour Nov 2010 29 LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009 An Expression for Monic Independent So Overbar is complex conjugate.

30. lecture 1 Ali Karimpour Nov 2010 30 LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009 An Expression for Let So

31. lecture 1 Ali Karimpour Nov 2010 31 LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009 We now have a characterization of in the frequency domain. with indicated weightings.

32. lecture 1 Ali Karimpour Nov 2010 32 LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009 Open Loop Case

33. lecture 1 Ali Karimpour Nov 2010 33 LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009 Open Loop Case Consider an independently noise model with Spectral factorization

34. lecture 1 Ali Karimpour Nov 2010 34 LTI Models: A Frequency-Domain Description of the Limit Model Open Loop Case =

35. lecture 1 Ali Karimpour Nov 2010 35 LTI Models: A Frequency-Domain Description of the Limit Model Open Loop Case

36. lecture 1 Ali Karimpour Nov 2010 36 LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009 Closed Loop Case

37. lecture 1 Ali Karimpour Nov 2010 37 LTI Models: A Frequency-Domain Description of the Limit Model Closed Loop Case =

38. lecture 1 Ali Karimpour Nov 2010 38 LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009 Example 8.5 Approximation in the Frequency Domain

39. lecture 1 Ali Karimpour Nov 2010 39 LTI Models: A Frequency-Domain Description of the Limit Model Example 8.5 Approximation in the Frequency Domain

40. lecture 1 Ali Karimpour Nov 2010 40 LTI Models: A Frequency-Domain Description of the Limit Model Example 8.5 Approximation in the Frequency Domain

41. lecture 1 Ali Karimpour Nov 2010 41 LTI Models: A Frequency-Domain Description of the Limit Model Example 8.5 Approximation in the Frequency Domain

42. lecture 1 Ali Karimpour Nov 2010 42 Lecture 8 Convergence & Consistency Topics to be covered include: v Conditions on the Data Set v Prediction-Error Approach v Consistency and Identifiability v LTI Models: A Frequency-Domain Description of the Limit Model v The Correlation Approach lecture 8 Ali Karimpour Nov 2009

43. lecture 1 Ali Karimpour Nov 2010 43 LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009 In section 7.5 we defined the correlation approach to identification,with the special cases of PLR and IV methods.

44. lecture 1 Ali Karimpour Nov 2010 44 LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009 Basic Convergence Result Consider the function And correlation vector ξ(t,θ) is obtained by linear filtering of past data: where

45. lecture 1 Ali Karimpour Nov 2010 45 LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009

46. lecture 1 Ali Karimpour Nov 2010 46 LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009

47. lecture 1 Ali Karimpour Nov 2010 47 LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009 Instrumental-variable Methods

48. lecture 1 Ali Karimpour Nov 2010 48 The Correlation Approach lecture 8 Ali Karimpour Nov 2009

49. lecture 1 Ali Karimpour Nov 2010 49 The Correlation Approach lecture 8 Ali Karimpour Nov 2009

50. lecture 1 Ali Karimpour Nov 2010 50 The Correlation Approach lecture 8 Ali Karimpour Nov 2009

51. lecture 1 Ali Karimpour Nov 2010 51 LTI Models: A Frequency-Domain Description of the Limit Model lecture 8 Ali Karimpour Nov 2009