1. SYSTEMS Identification Ali Karimpour Assistant Professor Ferdowsi University of Mashhad a_karimpoure@yahoo.com Reference: “System Identification Theory For The User” Lennart Ljung(1999)

2. lecture 6 Ali Karimpour Nov 2010 2 Lecture 6 Nonparametric Time and Frequency domain methods Topics to be covered include: v Transient-Response Analysis and Correlation Analysis v Frequency Response Analysis v Fourier Analysis v Spectral Analysis v Estimating the Disturbance Spectrum

3. lecture 6 Ali Karimpour Nov 2010 3 Nonparametric Time and Frequency domain methods Topics to be covered include: v Transient-Response Analysis and Correlation Analysis v Frequency Response Analysis v Fourier Analysis v Spectral Analysis v Estimating the Disturbance Spectrum

4. lecture 6 Ali Karimpour Nov 2010 4 Transient-Response Analysis and Correlation Analysis Impulse Response Analysis Suppose Let the input as Then the output will be So we have The error is: This simple idea is impulse Response Analysis. Its basic weakness is that many physical processes do not allow pulse input so the the error be small, moreover that input make the system exhibit nonlinear effects.

5. lecture 6 Ali Karimpour Nov 2010 5 Transient-Response Analysis and Correlation Analysis Step Response Analysis Suppose Let the input as Then the output will be So we have The error is: It would suffer from large error in most practical applications. It is acceptable for delay time, static gain, dominating time constant.

6. lecture 6 Ali Karimpour Nov 2010 6 Transient-Response Analysis and Correlation Analysis Correlation Analysis Suppose If the input is a quasi stationary sequence with Then we have If the input is white noise Then if the input is white noise An estimate of the impulse response is thus obtained from If the input is not white noise

7. lecture 6 Ali Karimpour Nov 2010 7 Transient-Response Analysis and Correlation Analysis Correlation Analysis Suppose If the input is a quasi stationary sequence with Then we have If the input is white noise Then if the input is not white noise Choose the input so that (I) and (II) become easy to solve. If the input is not white noise

8. lecture 6 Ali Karimpour Nov 2010 8 Transient-Response Analysis and Correlation Analysis Exercise 1: Suppose and v(t) is normal noise such that a) Derive with Impulse Response Analysis let u(t)<3 b) Derive with Step Response Analysis let u(t)<3 c) Derive with correlation analysis with white noise input let u(t)<3 d) Derive with correlation analysis with non white noise input u(t)<3

9. lecture 6 Ali Karimpour Nov 2010 9 Nonparametric Time and Frequency domain methods Topics to be covered include: v Transient-Response Analysis and Correlation Analysis v Frequency Response Analysis v Fourier Analysis v Spectral Analysis v Estimating the Disturbance Spectrum

10. lecture 6 Ali Karimpour Nov 2010 10 Frequency-Response Analysis Sine Wave testing Suppose Let the input as Then the output will be This is known as frequency analysis and is a simple method for obtaining detailed information about a linear system. Bode plot of the system can be obtained easily. One may concentrate the effort to the interesting frequency ranges. Many industrial processes do not admit sinusoidal inputs in normal operation. Long experimentation periods. ??

11. lecture 6 Ali Karimpour Nov 2010 11 Frequency-Response Analysis Sine Wave testing ?? Frequency analysis by the correlation method. If v(t) does not contain a pure periodic component of frequency ω. 0 Define 0 0 0

12. lecture 6 Ali Karimpour Nov 2010 12 Transient-Response Analysis and Correlation Analysis Exercise 2: Suppose and v(t) is normal noise such that a) Derive and discuss about the value of N.

13. lecture 6 Ali Karimpour Nov 2010 13 Nonparametric Time and Frequency domain methods Topics to be covered include: v Transient-Response Analysis and Correlation Analysis v Frequency Response Analysis v Fourier Analysis v Spectral Analysis v Estimating the Disturbance Spectrum

14. lecture 6 Ali Karimpour Nov 2010 14 Fourier Analysis In a linear system different frequencies pass through the system independently. So extend the frequency analysis estimate to the case of multifrequency inputs. Empirical Transfer-Function Estimate Properties of ETFE Claim: Remember:

15. lecture 6 Ali Karimpour Nov 2010 15 Fourier Analysis Properties of ETFE Claim: Remember: Now suppose Let By above claim

16. lecture 6 Ali Karimpour Nov 2010 16 Fourier Analysis Properties of ETFE Since v(t) is assumed zero mean So

17. lecture 6 Ali Karimpour Nov 2010 17 Fourier Analysis Properties of ETFE So

18. lecture 6 Ali Karimpour Nov 2010 18 Nonparametric Time and Frequency domain methods Topics to be covered include: v Transient-Response Analysis and Correlation Analysis v Frequency Response Analysis v Fourier Analysis v Spectral Analysis v Estimating the Disturbance Spectrum

19. lecture 6 Ali Karimpour Nov 2010 19 Spectral Analysis Smoothing the ETFE The true transfer function is a smooth function. If the frequrncy distance 2π/N is small compared to how quickly Changes then Are uncorrelated and if we assume To be constant over the interval

20. lecture 6 Ali Karimpour Nov 2010 20 For large N we have If transfer function is not constant Spectral Analysis

21. lecture 6 Ali Karimpour Nov 2010 21 If noise spectrum is not known and don’t change very much over frequency intervals Then Spectral Analysis

22. lecture 6 Ali Karimpour Nov 2010 22 Connection with the Blackman-Tukey Proceture If Spectral Analysis

23. lecture 6 Ali Karimpour Nov 2010 23 If noise spectrum don’t change much over frequency intervals similarly Spectral Analysis

24. lecture 6 Ali Karimpour Nov 2010 24 The fourier cofficients for periodogram The fourier cofficients of the function Spectral Analysis

25. lecture 6 Ali Karimpour Nov 2010 25 smooth function is chosen so that its fourier coefficients vanish for Weighting Function: The Frequency Window Spectral Analysis

26. lecture 6 Ali Karimpour Nov 2010 26 Spectral Analysis

27. lecture 6 Ali Karimpour Nov 2010 27 Asymptotic Properties of the Smoothed Estimate 1-Bias 2-Variance Here Spectral Analysis

28. lecture 6 Ali Karimpour Nov 2010 28 Value of the width parameter that minimizes the MSE is The optimal choice of width parameter leads to a MSE error that decays like Spectral Analysis

29. lecture 6 Ali Karimpour Nov 2010 29

30. lecture 6 Ali Karimpour Nov 2010 30 Spectral Analysis Smoothing the ETFE Exercise 3:

31. lecture 6 Ali Karimpour Nov 2010 31 Another Way of Smoothing the ETFE Spectral Analysis The ETFEs obtained over different data sets will also provide uncorrelated estimates, and another approach would be to form averages over these. Split the data set into M batches, each containing R data (N=R. M). Then form the ETFE corresponding to the kth batch: The estimate can then be formed as a direct average

32. lecture 6 Ali Karimpour Nov 2010 32 Or one that is weighted according to the inverse variances: with

33. lecture 6 Ali Karimpour Nov 2010 Estimating The Disturbance Spectrum Suppose 1-Bias 2-Variance

34. lecture 6 Ali Karimpour Nov 2010 34 The Residual Spectrum

35. lecture 6 Ali Karimpour Nov 2010 35 We have Coherency Spectrum: Denote Then

36. lecture 6 Ali Karimpour Nov 2010 36 Nonparametric Time and Frequency domain methods Topics to be covered include: v Transient-Response Analysis and Correlation Analysis v Frequency Response Analysis v Fourier Analysis v Spectral Analysis v Estimating the Disturbance Spectrum

37. lecture 6 Ali Karimpour Nov 2010 37 Estimating the Disturbance Spectrum Properties of ETFE