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SYSTEMS Identification

SYSTEMS Identification. Ali Karimpour Assistant Professor Ferdowsi University of Mashhad a_karimpoure@yahoo.com. Reference: “System Identification Theory For The User” Lennart Ljung(1999). Lecture 6. Nonparametric Time and Frequency domain methods. Topics to be covered include :

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SYSTEMS Identification

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  1. SYSTEMSIdentification 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 Nonparametric Time and Frequency domain methods Topics to be covered include: • Transient-Response Analysis and Correlation Analysis • Frequency Response Analysis • Fourier Analysis • Spectral Analysis • Estimating the Disturbance Spectrum

  3. Nonparametric Time and Frequency domain methods Topics to be covered include: • Transient-Response Analysis and Correlation Analysis • Frequency Response Analysis • Fourier Analysis • Spectral Analysis • Estimating the Disturbance Spectrum

  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. 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. 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 • If the input is not white noise Then if the input is white noise An estimate of the impulse response is thus obtained from

  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 • If the input is not white noise Then if the input is not white noise Choose the input so that (I) and (II) become easy to solve.

  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 d) Derive with correlation analysis with non white noise input 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

  9. Nonparametric Time and Frequency domain methods Topics to be covered include: • Transient-Response Analysis and Correlation Analysis • Frequency Response Analysis • Fourier Analysis • Spectral Analysis • Estimating the Disturbance Spectrum

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

  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. Nonparametric Time and Frequency domain methods Topics to be covered include: • Transient-Response Analysis and Correlation Analysis • Frequency Response Analysis • Fourier Analysis • Spectral Analysis • Estimating the Disturbance Spectrum

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

  15. Claim: Fourier Analysis Properties of ETFE Now suppose Remember: Let By above claim

  16. Fourier Analysis Properties of ETFE Since v(t) is assumed zero mean So

  17. Fourier Analysis Properties of ETFE So Lemma

  18. Nonparametric Time and Frequency domain methods Topics to be covered include: • Transient-Response Analysis and Correlation Analysis • Frequency Response Analysis • Fourier Analysis • Spectral Analysis • Estimating the Disturbance Spectrum

  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. Spectral Analysis For large N we have If transfer function is not constant

  21. Spectral Analysis If noise spectrum is not known and don’t change very much over frequency intervals Then

  22. Spectral Analysis Connection with the Blackman-TukeyProceture If If

  23. Spectral Analysis If noise spectrum don’t change much over frequency intervals similarly

  24. Spectral Analysis The fouriercofficients for periodogram The fouriercofficients of the function

  25. Spectral Analysis smooth function is chosen so that its fourier coefficients vanish for Weighting Function: The Frequency Window

  26. Spectral Analysis

  27. Spectral Analysis Asymptotic Properties of the Smoothed Estimate 1-Bias 2-Variance Here

  28. Spectral Analysis Value of the width parameter that minimizes the MSE is The optimal choice of width parameter leads to a MSE error that decays like

  29. Spectral Analysis Smoothing the ETFE Exercise 3:

  30. Spectral Analysis Another Way of Smoothing the ETFE 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

  31. Or one that is weighted according to the inverse variances: with

  32. Estimating The Disturbance Spectrum Suppose 1-Bias 2-Variance

  33. The Residual Spectrum

  34. We have Coherency Spectrum: Denote Then

  35. Nonparametric Time and Frequency domain methods Topics to be covered include: • Transient-Response Analysis and Correlation Analysis • Frequency Response Analysis • Fourier Analysis • Spectral Analysis • Estimating the Disturbance Spectrum

  36. Estimating the Disturbance Spectrum Properties of ETFE

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