6.435, System Identification

Transcription

1 System Identification SET 3 Nonparametric Identification Munther A. Dahleh 1

2 Nonparametric Methods for System ID Time domain methods Impulse response Step response Correlation analysis / time Frequency domain methods Sine-wave testing Correlation analysis / Frequency Fourier-analysis Spectral analysis 2

3 Problem Formulation Black Box Actual system is Linear time-invariant stable. Process: Time domain methods estimates of Frequency-domain methods estimates of 3

4 Tests: a) at each freq. b) c) d) 4

5 Time-Domain Methods Impulse response estimate: Analysis: small if Practicality: not very useful. 5

6 Step response estimate: Analysis: Practicality: Not good for determining. Good for determining delays, modes. 6

7 Methods (Continued) Correlation Analysis Assume u is quasi-stationary are uncorrelated. 7

8 Case I: If To estimate: 8

9 Case II: Input is not white. Using the approximation In matrix form: notice 9

10 Estimate Question: Under what conditions the above system has a unique solution? Persistency of excitation! Note that you get the same estimate regardless of the spectrum of the noise. 10

11 Analysis of Correlation Method Estimate Need to determine the covariance of for a fixed large N. 11

12 Covariance, proportional to 12

13 Frequency-Response Analysis Input Extract How do you measure in the presence of noise? A good approach is correlation. 13

14 Define 14

15 Estimate: Comment: 15

16 Empirical Transfer Function Estimate (ETFE) For an arbitrary input Recall: Correlation analysis If, then the previous analysis shows that Similarly for u = White input. 16

17 General Procedure 1. Calculate 2. Obtain the inverse DFT: 3. Define The algorithm is quite efficient; requires only the computation of the Inverse DFT. Note also that the algorithm is Linear. 17

18 Properties of EFTE Theorem: Given: With: s(t) is stationary, zero mean with spectrum 18

19 Then:

20 Proofs Bias Covariance 1 st : Compute 20

21 21

22 22

23 Put together Now: 23

24 Comments on EFTE Suppose U = periodic and zero for others increases as a function of N for some EFTE is defined for a fixed number of frequencies, i.e. independent of N. At these frequencies, ETFE is unbiased and Covariance decays as. (Recall ). 24

25 Suppose V is a stochastic process, uncorrelated with v in dist. (a bounded function) ETFE is asymptotically unbiased, with increasingly more welldefined frequencies (as N ). The variance does not decrease as N. Estimates are asymptotically uncorrelated. 25

26 Spectral Estimation Traditionally N-Long time series estimate In here, different context. Theme: Show the mechanics Importance of windowing, tradeoffs Relate to spectral estimation {smaller variance} 26

27 Spectral Estimation: Non Std (Ljung) Idea: the actual function is smooth. The values of should be related for small intervals ω. According to previous analysis, is uncorrelated with and has variance Suppose satisfies 27

28 Define the estimate (new) at as follows: Where are chosen so that is minimized. Solution: 28

29 As N, the sums integrals 29

30 Equivalently: Let be a window function. Then, If is unknown, but slowly varying in frequency 30

31 Relations to Traditional Spectral Analysis Recall: in distribution estimate of 31

32 Define: Similarly: Conclusion 32

33 Efficient Computation Of course for large enough but not as large as N. Example is: (Bartlett) Similarly for 33

34 Analysis of Spectral Estimation and 34

35 Write Recall: 35

36 Numerator Denominator Numerator: Denominator: for each finite, the estimate is biased. 36

37 For a fixed Improved variance on the expense of the biase. 37

38 Estimating the Disturbance Spectrum If was measurable, then Bias: Variance Problem: is not readily measurable. 38

39 The residual spectrum. is the estimate Define 39