建議 - 結論與建議 - 數種濾波技術於線性系統識別之可行性探討

第五章結論與建議

5.2 建議

訊號的類型千百種，每一種類型的訊號所適用的濾波方法都不盡相同，除了由理論發展之外，更有不少新的嘗試。在地震反應訊號的領域中，目前對於此類訊號的濾波這部分並沒有很確切的定論及準則，仍屬嘗試階段；本研究採用之三種濾波器為目前普遍使用的三種濾波方法，其濾波的過程中演算法與參數設定，大部分皆遵循前人發展的方法來操作。

在非時變系統中，其濾波效果對於識別分析的影響都有不錯的效果，可以識別出含噪量達到 40%的訊號，然而非時變系統的訊號屬於理想狀態，若是將來想要應用於實際情況，必須要考慮時變系統的訊號，才比較符合工程上的狀況。

時變系統中，本研究所採用的三種濾波方法對於高頻部分(第四模態和第五模態)

的識別分析其成效都不彰。三種濾波器對於時變系統中的高頻部分之識別效果皆不甚理想，推測可能是由於濾波器在處理此類型資料時的過程中，會過濾掉過多的高頻信號，導致訊號在高頻部分失真，因此許多高頻部分的點識別不出來。可以歸納出這三種使用較為廣泛的濾波方法並不適用於此類地震反應的訊號，建議可以改良濾波器的操作演算過程或是深入探討參數的設定，進行比較或分析，盡可能保留較多的高頻的訊號，以期對系統識別分析有所助益。

參考文獻

1. Kalman, R. E.,“A New Approach to Linear Filtering and Prediction Problems”, Transactions of the ASME - Journal of Basic Engineering Vol. 82, pp. 35-45, 1960.

2. Kalman, R. E., Bucy, R.S., “New Results in Linear Filtering and Prediction Theory”, Transactions of the ASME - Journal of Basic Engineering, Vol. 83: pp. 95-107, 1961.

3. The Technical Staff, The Analytic Sciences Corporation, Applied Optimal Estimation, Gelb, A., MIT Press, Cambridge, MA., 1974.

4. Lewis, F. L., Optimal Estimation : with An Introduction to Stochastic Control Theory, John Wiley & Sons, Inc., New York, 1986.

5. Brown, R. G., Hwang, P. Y. C., Introduction to Random Signals and Applied Kalman Filtering, 2^nd Edition, John Wiley & Sons, Inc., 1992.

6. Jacobs, O. L. R., Introduction to Control Theory, 2^nd Edition, Oxford University Press, 1993.

7. Harvey, A. C., Forecasting, Structural Time Series Models and the Kalman Filter, Cambridge University Press, Cambridge, 1989.

8. Welch, G., Bishop, G., “An Introduction to the Kalman Filter”, Computer and Information Science, 7(1), pp.1-16, 2001.

9. Ljung, L., “Asymptotic Behavior of the Extended Kalman Filter as a Parameter Estimator for Linear Systems”, IEEE Transactions on Automatic Control, 24(1), pp.

36~50, 1979.

10. Hoshiya, M., Saito, E., “Structural Identification by Extended Kalman Filter”, Journal of Engineering Mechanics, ASCE, 110(12), pp.1757-1771, 1984.

11. Roman, S., “Adaptive Extended Kalman Filter with Recursive Noise Covariance Identification Applied to the Fermentation Processes”, Modeling, Simulation & Control C：Environmental, Biomedical, Human & Social System, 10(2), pp.52-64, 1987.

12. Julier, S. J., Uhlmann, J. K., “A New Extension of the Kalman Filter to Nonlinear Systems”, The University of Oxford, 1997.

13. Van der Merwe, R., Wan, E. A., “The Square-Root Unscented Kalman Filter for State and Parameter Estimation”, pp. 3461-3464, Salt Lake City, UT , USA, May 2001.

14. 張滿生、張學莊，「卡爾曼濾波器及其工程應用」，計算技術與自動化，第二十七卷，第一期，136~139頁，2008年3月。

15. Bednar, J. B., “Applications of Median Filtering to Deconvolution, Pulse Estimation, and Statistical Editing of Seismic Data”, Geophysics, 48(12), pp. 1598-1610, 1983.

16. Brownrigg, D. R. K.,” The Weighted Median Filter”, Communications of the ACM, 27(8), pp.807-818, Aug 1984.

17. Duncan, G., Beresford, G., “Some Analyses of 2-D Median F-K Filters”, Geophysics, 60(4), pp. 1157-1168, Aug 1995.

18. Duncan, G., Beresford, G., “Median Filter Behavior with Seismic Data”, Geophysical Prospecting, 43(3), pp.329-345, Apr 1995.

19. Liu, C., Liu, Y., Yang, B., Wang, D., Sun, J., “A 2-D Multistage Median Filter to Reduce Random Seismic Noise”, Geophysics, 71(5), pp. 105-110, Sep 2006.

20. Liu, Y., Liu, C., Wang, D., “A 1-D Time-Varying Median Filter for Seismic Random, Spike-Like Noise Elimination”, Geophysics, 74(1), pp.17-24, 2009.

21. Bednar, J. B., Watt, T., “Alpha-Trimmed Means and Their Relationship to Median Filters”, IEEE Transactions on Acoustics, Speech and Signal Processing, 32(1), pp.145-153,1984.

22. Lee,Y. H., Kassam, S. A., “Generalized Median Filtering and Related Noninear Filtering Techniques”, IEEE Transactions on Acoustics, Speech and Signal Processing, 33(3), pp.672~683, Jun 1984.

23. Fitch, J. P., Coyle, E. J., Gallagher, N. C., “The Analog Median Filter”, IEEE Transactions on Circuits and System, 33(1), pp.94~102, Jan 1986.

24. Mallat S., Hwang W. L., “Singularity Detection and Processing with Wavelets”, IEEE Transactions Information Theory, 38(2), pp.617-643, 1992.

25. Mallat S., Zhong S., “Characterization of Signals from Multiscale Edges”, IEEE Transactions Pattern Analysis and Machine Intelligence , 45(7), pp.710-732, 1992.

26. Lu J., et al. “Noise Reduction with Multiscale Edge Representation and Perceptual Criteria”, In:Proceedings of IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis, pp.555~558, Victoria , BC, Canada , Oct 1992.

27. Lu J., et al. “MR Image Filtering with a Multiscale Edge Representation” In:

Proceedings of the 11^th Annual Meeting , Society of Magnetic Resonance in Medicine, pp. 471, Berlin,1992.

28. Simoncelli E. P., et al. “Shiftable Multiscale Transforms”, IEEE Transactions Information Theory, 38(2), pp.587~607 ,1992.

29. Moulin, P., “Wavelet Thresholding Techniques for Power Spectrum Estimation”, IEEE

Transaction on Signal Processing, 42(11), pp.3126-3136, 1994.

30. Dohono, D. L., “Wavelet Shrinkage and W.V.D-a Ten Minute Tour”, Stanford University, Progress in Wavelet Analysis and Applications, pp.109-128, 1993.

31. Dohono, D. L., Johnstone I. M.,” Ideal Spatial Adaptation by Wavelet Shrinkage”, Biometrika, 81(3), pp.425-455, 1994.

32. Dohono, D. L., “De-Noising by Soft-Thresholding”, IEEE Transactions Information Theory, 41(30), pp.613-627, 1995.

33. Dohono D. L., Johnstone I. M., “Adapting to Unknown Smoothness via Wavelet Shrinkage”, Journal of the American Statistical Association, 90(432), pp.1200-1224 , 1995.

34. Dohono D. L., Johnstone I. M., “Minimax Estimation via Wavelet Shrinkage”, Annals of Statistics, 26(3), pp.879-921 , 1998.

35. Dohono D. L., Johnstone I. M., “Ideal De-noising in an Orthonormal Basis Chosen from a Library of Bases”, Comptes Rendus Academies of Sciences, Series I, Paris, Vol.319, pp.1317-1322, 1994.

36. Kirm, H., Pesqueet, J. C., “On the Statistics of Best Bases Criteria”, Wavelets in Statistic of Lecture Notes in Statistic, Spring-Verlag, New York, pp.193~207, 1995.

37. Moulin, P., “Wavelet Thresholding Techniques for Power Spectrum Estimation”, IEEE Transactions Signal Processing, 42(11), pp.3126-3136, 1994.

38. Loupas, T., McDicken, W. N., Allan, P. L., “An Adaptive Weighted Median Filter for Speckle Suppression in Medical Ultrasonic Images”, IEEE Transactions on Circuits and Systems, 36(1), pp.129-135, Jan 1989.

39. 趙彥青、吳繼偉、蕭蘊詩，「最優中值濾波跨度計算及其應用」，計算機輔助工程，

第十五卷，第一期，59~62頁，2006年3月。

40. 熊翥，「地震數據數字處理應用技術」，北京:石油工業出版社，192~194頁，1993 年。

41. 李小紅、汪曉兵、肖都建，「中值在地震資料處理中的應用」，石油物探，第四十三卷，第三期，248~250頁，2004年。

42. 馮暄、劉財、楊寶俊，「中值濾波器對信號相位和形狀影響的研究」，石油物探，

第四十一卷，第一期，37~41頁，2002年。

43. 柳建新、韓世禮、馬捷，「小波分析在地震資料去噪中的應用」，地球物理學進展，

第二十一卷，第二期，541~545頁，2006年6月。

44. 張磊、潘泉、張洪才、戴冠中，「小波域濾波閾值參數c的選取」，電子學報，第

41 二十九卷，第三期，400~402頁，2001年。

45. Zhang L., Bao P., “Denoising by Spatial Correlation Thresholding”, IEEE Transactions on Circuits and Systems for Vedio Technology, 13(6), pp.535-538, 2003.

46. Zhang, X. P., “Adaptive Denoising Based on SURE Risk”, IEEE Signal Processing Letters, 5(10), pp.265-267, Oct 1998.

47. Nason G. P., “Wavelet Shrinkage Using Cross-Validation”, Journal of the Royal Statistical Society, Series B, Vol.58, pp.463-479, 1996.

48. Jansen, M., et al. “Generalized Cross Validation for Wavelet Thresholding”, Signal Processing, 56(1), pp.33-34, 1997.

49. Jansen, M., Bultheel, A., “Multiple Wavelet Threshold Estimation by Generalized Cross Validation for Images with Correlated Noise”, IEEE Transactions on Image Processing, 8(7), pp. 947-953, Jul 1999.

50. Gao, H. Y., Bruce A. G., “WaveShrink and Semisoft Shrinkage”, StatSci Reserch Report No.39, 1995.

51. Gao, H. Y., Bruce A. G., “WaveShrink with Firm Shrinkage”, Statistica Sinica, 7(4), pp.885-874, 1997.

52. Daubechies, I., Ten Lectures on Wavelets, Society for Industrial & Applied, USA, May 1992.

53. Chui, C. K., An Introduction to Wavelets, Academic Press, Inc. 1992.

54. 蘇威智，「以時間序列模型識別結構之模態參數」，博士論文, 土木工程研究所，

國立交通大學，2008年。

55. Huang, C. S., Yang, Y. B., Lu, L. Y., and Chen, C. H., “Dynamic Testing and System Identification of a Multi-Span Highway Bridge”, Earthquake Engineering & Structural Dynamics, 28(8), pp. 857-878, Aug 1999.

56. Huang, C. S., “Structural Identification from Ambient Vibration Measurement Using the Multivariate AR Model”, Journal of Sound and Vibration, 241(3), pp.337-359, Mar 2001.

57. 林志嘉，「橋梁現地實驗與動力特性系統識別」，碩士論文，土木工程研究所，國立台灣大學，2000年。

58. Marmarelis, V. Z., “Nonlinear and Nonstationary Modeling of Physiological System: an Overview in Advanced Methods of Physiological System Modeling”, Biomedical Simulations Resource, Vol. 1, pp.1-24,1987.

59. Tsatsanis, M. K., Giannakis, G. B., “Time-Varying System Identification and Model

Validation Using Wavelets”, IEEE Transactions on Acoustics,Speech and Signal Process, 41(12), pp.3512-3523, 1993.

60. Niedwiecki, M., Identification of Time-Varying Processes, John Wiley & Sons, Inc., New York, USA, 2000.

61. Zou, R., Wang, H., Chon, K.H., “A Robust Time-Varying Identification Algorithm Using Basis Function”, Annals of Biomedical Engineering, 31(7), pp.840-853, 2003.

62. Liu, G. R., Mesh Free Methods- Moving Beyond the Finite Element Method, CRC Press, New York, 2003.

63. Grimble, M. J., et al., “System Identification”, Prentice Hall, New York, 1988.

附表

表 4.1 七層樓剪力構架之非時變系統反應其理論值的系統識別結果

7 DOF Mode

1 2 3 4 5 6 7

頻率 f (Hz)

理論值 0.67 1.97 3.18 4.26 5.15 5.82 6.23 識別值 0.67 1.97 3.18 4.26 5.15 5.82 6.23 阻尼比 ξ

(%)

理論值 5.00% 5.00% 5.00% 5.00% 5.00% 5.00% 5.00%

識別值 5.00% 5.00% 5.00% 5.00% 5.00% 5.00% 5.00%

MAC 值 1.00 1.00 1.00 1.00 1.00 1.00 1.00

表 4.2 七層樓剪力構架含噪訊之非時變系統反應經卡氏濾波過後之噪訊比 NSR 10% 20% 30% 40%

GR=10%

7F 5.56% 9.01% 12.87% 16.83%

6F 5.72% 8.95% 12.82% 16.76%

5F 5.88% 9.14% 12.80% 16.71%

4F 6.05% 9.20% 12.91% 16.89%

3F 6.21% 9.39% 13.03% 17.19%

2F 6.98% 9.78% 13.35% 17.28%

1F 8.99% 11.46% 14.51% 18.32%

ag 8.29% 10.97% 14.27% 17.67%

NSR 10% 20% 30% 40%

GR=20%

7F 8.97% 10.44% 11.76% 14.68%

6F 9.11% 10.33% 12.08% 14.61%

5F 9.84% 11.05% 12.46% 15.02%

4F 10.46% 11.40% 12.59% 15.55%

3F 11.24% 12.38% 12.76% 16.26%

2F 13.37% 14.12% 14.25% 17.78%

1F 18.82% 19.56% 20.56% 22.28%

a_g 16.58% 17.41% 18.70% 20.13%

NSR 10% 20% 30% 40%

GR=30%

7F 12.77% 13.62% 14.69% 16.05%

6F 12.86% 13.43% 14.72% 15.98%

5F 14.06% 14.63% 15.60% 16.87%

4F 15.02% 15.38% 16.47% 17.91%

3F 16.40% 16.97% 17.55% 19.15%

2F 19.60% 19.89% 20.55% 22.01%

1F 27.80% 28.15% 28.61% 29.62%

a_g 24.15% 24.57% 25.24% 26.10%

NSR 10% 20% 30% 40%

GR=40%

7F 16.26% 16.85% 17.49% 18.40%

6F 16.26% 16.58% 17.47% 18.27%

5F 17.91% 18.21% 18.84% 19.59%

4F 19.19% 19.36% 20.07% 21.11%

3F 21.13% 21.47% 21.75% 22.88%

2F 25.29% 25.45% 25.78% 26.86%

1F 35.87% 36.07% 36.38% 37.02%

ag 30.86% 31.12% 31.52% 32.20%

表 4.3 七層樓剪力構架含 10%噪訊之訊號與經卡氏濾波後之系統識別結果

GR NSR=10% Mode

1 2 3 4 5 6 7

10%

(I,J) (5,5) (3,3) (3,3) (2,2) (3,3) (4,4) (5,5) f 0.68 2.00 3.18 4.26 5.17 5.89 6.17 ξ 5.88% 5.01% 4.94% 5.85% 5.64% 5.95% 5.56%

20%

(I,J) (4,4) (2,2) (2,2) (2,2) (2,2) (2,2) (4,4) f 0.68 2.00 3.18 4.26 5.17 5.87 6.21 ξ 5.58% 5.24% 4.96% 5.19% 5.68% 5.12% 5.38%

30%

(I,J) (3,3) (2,2) (2,2) (2,2) (2,2) (2,2) (3,3) f 0.67 1.98 3.18 4.26 5.17 5.83 6.17 ξ 5.58% 4.97% 4.94% 5.13% 5.44% 4.20% 5.64%

40%

(I,J) (3,3) (2,2) (2,2) (2,2) (2,2) (3,3) (3,3) f 0.68 1.97 3.18 4.26 5.17 5.77 6.15 ξ 5.23% 4.88% 4.93% 5.11% 5.34% 4.21% 5.40%

未濾波

(I,J) (8,8) (4,4) (4,4) (4,4) (8,8) (11,11) (10,10) f 0.68 2.00 3.20 4.27 5.14 5.76 6.21 ξ 5.02% 5.56% 5.00% 5.23% 5.01% 5.30% 4.71%

不含噪訊

(I,J) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) f 0.67 1.97 3.18 4.26 5.15 5.82 6.23 ξ 5.00% 5.00% 5.00% 5.00% 5.00% 5.00% 5.00%

表 4.4 七層樓剪力構架含 20%噪訊之訊號與經卡氏濾波後之系統識別結果

GR NSR=20% Mode

1 2 3 4 5 6 7

10%

(I,J) (6,6) (4,4) (4,4) (4,4) (6,6) (4,4) (8,8) f 0.68 1.97 3.19 4.25 5.18 5.84 6.40 ξ 5.96% 4.76% 5.04% 5.09% 4.90% 4.47% 5.91%

20%

(I,J) (6,6) (4,4) (2,2) (2,2) (4,4) (4,4) (8,8) f 0.68 2.00 3.19 4.30 5.19 5.88 6.31 ξ 5.16% 5.01% 4.90% 5.77% 5.47% 4.75% 5.59%

30%

(I,J) (5,5) (4,4) (2,2) (2,2) (4,4) (2,2) (4,4) f 0.68 1.98 3.19 4.28 5.17 5.84 6.29 ξ 5.74% 4.85% 4.86% 5.20% 5.08% 5.76% 4.22%

40%

(I,J) (5,5) (2,2) (2,2) (2,2) (2,2) (2,2) (3,3) f 0.68 2.01 3.19 4.27 5.20 5.79 6.26 ξ 5.33% 4.80% 4.82% 5.06% 5.88% 5.15% 5.31%

未濾波

(I,J) (10,10) (6,6) (4,4) (4,4) (7,7) (7,7) (12,12) f 0.68 1.99 3.20 4.28 5.19 5.89 6.14 ξ 5.67% 5.03% 5.80% 5.09% 5.62% 5.48% 4.68%

不含噪訊

(I,J) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) f 0.67 1.97 3.18 4.26 5.15 5.82 6.23 ξ 5.00% 5.00% 5.00% 5.00% 5.00% 5.00% 5.00%

表 4.5 七層樓剪力構架含 30%噪訊之訊號與經卡氏濾波後之系統識別結果

GR NSR=30% Mode

1 2 3 4 5 6 7

10%

(I,J) (7,7) (4,4) (3,3) (3,3) (6,6) (3,3) (6,6) f 0.68 1.98 3.21 4.27 5.08 5.91 6.36 ξ 5.63% 5.05% 5.34% 5.02% 5.09% 5.23% 4.22%

20%

(I,J) (6,6) (3,3) (2,2) (3,3) (4,4) (3,3) (4,4) f 0.68 2.03 3.20 4.25 5.21 5.86 6.23 ξ 5.35% 5.76% 5.45% 5.00% 5.20% 5.62% 4.02%

30%

(I,J) (6,6) (4,4) (3,3) (3,3) (3,3) (3,3) (7,7) f 0.68 1.97 3.20 4.25 5.20 5.91 6.26 ξ 5.03% 4.64% 5.06% 5.03% 5.94% 5.19% 5.28%

40%

(I,J) (5,5) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) f 0.68 1.99 3.20 4.25 5.17 5.88 6.12 ξ 5.71% 4.91% 4.95% 5.07% 5.74% 5.06% 5.81%

未濾波

(I,J) (11,11) (7,7) (6,6) (4,4) (8,8) (7,7) (10,10) f 0.68 1.98 3.20 4.28 5.13 5.76 6.12 ξ 5.79% 4.81% 5.58% 5.14% 5.83% 4.76% 4.10%

不含噪訊

(I,J) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) f 0.67 1.97 3.18 4.26 5.15 5.82 6.23 ξ 5.00% 5.00% 5.00% 5.00% 5.00% 5.00% 5.00%

表 4.6 七層樓剪力構架含 40%噪訊之訊號與經卡氏濾波後之系統識別結果

GR NSR=40% Mode

1 2 3 4 5 6 7

10%

(I,J) (7,7) (4,4) (4,4) (3,3) (6,6) (3,3) (5,5) f 0.68 1.99 3.19 4.25 5.11 5.71 6.20 ξ 5.93% 5.20% 5.57% 4.84% 5.13% 5.96% 5.03%

20%

(I,J) (6,6) (4,4) (2,2) (3,3) (5,5) (3,3) (4,4) f 0.68 1.97 3.20 4.21 5.06 5.88 6.18 ξ 5.48% 4.83% 5.75% 4.61% 4.72% 3.40% 4.83%

30%

(I,J) (6,6) (3,3) (2,2) (3,3) (6,6) (2,2) (3,3) f 0.68 2.01 3.20 4.21 5.26 5.80 6.28 ξ 5.00% 5.86% 5.54% 4.08% 5.39% 5.53% 5.75%

40%

(I,J) (6,6) (3,3) (3,3) (3,3) (4,4) (2,2) (3,3) f 0.68 2.00 3.19 4.25 5.10 5.73 6.22 ξ 4.82% 5.52% 5.38% 4.41% 5.24% 4.87% 5.26%

未濾波

(I,J) (12,12) (7,7) (6,6) (4,4) (8,8) (7,7) (8,8) f 0.68 1.99 3.20 4.28 5.13 5.86 6.25 ξ 5.63% 5.04% 5.94% 4.92% 5.62% 4.68% 5.84%

不含噪訊

(I,J) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) f 0.67 1.97 3.18 4.26 5.15 5.82 6.23 ξ 5.00% 5.00% 5.00% 5.00% 5.00% 5.00% 5.00%

表 4.7 七層樓剪力構架含噪訊之非時變系統反應經中值濾波過後之噪訊比

NSR

Mode 10% 20% 30% 40%

7^th floor 9.04% 13.26% 17.76% 22.63%

6^thfloor 9.04% 13.03% 17.92% 22.64%

5^th floor 9.62% 13.58% 17.98% 22.58%

4^th floor 10.31% 14.16% 18.62% 23.08%

3^rd floor 11.43% 14.72% 19.17% 23.68%

2^nd floor 12.85% 16.15% 19.93% 24.15%

1^st floor 17.40% 19.61% 23.32% 27.10%

ag earth 15.29% 18.03% 21.28% 26.10%

表 4.8 七層樓剪力構架含 10%噪訊之訊號與經中值濾波後之系統識別結果

NSR=10% Mode

1 2 3 4 5 6 7

中值濾波

(I,J) (6,6) (4,4) (2,2) (2,2) (4,4) (4,4) (6,6) f 0.68 1.97 3.19 4.26 5.17 5.74 6.23 ξ 5.35% 4.81% 5.20% 5.49% 5.94% 5.80% 5.79%

未濾波

(I,J) (8,8) (4,4) (4,4) (4,4) (8,8) (11,11) (10,10) f 0.68 2.00 3.20 4.27 5.14 5.76 6.21 ξ 5.02% 5.56% 5.00% 5.23% 5.01% 5.30% 4.71%

不含噪訊

(I,J) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) f 0.67 1.97 3.18 4.26 5.15 5.82 6.23 ξ 5.00% 5.00% 5.00% 5.00% 5.00% 5.00% 5.00%

表 4.9 七層樓剪力構架含 20%噪訊之訊號與經中值濾波後之系統識別結果

NSR=20% Mode

1 2 3 4 5 6 7

中值濾波

(I,J) (7,7) (4,4) (2,2) (3,3) (6,6) (4,4) (8,8) f 0.68 1.98 3.20 4.28 5.20 5.88 6.12 ξ 5.91% 4.96% 5.98% 5.63% 5.38% 5.98% 3.52%

未濾波

(I,J) (10,10) (6,6) (4,4) (4,4) (7,7) (7,7) (10,10) f 0.68 1.99 3.20 4.28 5.19 5.89 6.14 ξ 5.67% 5.03% 5.80% 5.09% 5.62% 5.48% 4.68%

不含噪訊

(I,J) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) f 0.67 1.97 3.18 4.26 5.15 5.82 6.23 ξ 5.00% 5.00% 5.00% 5.00% 5.00% 5.00% 5.00%

表 4.10 七層樓剪力構架含 30%噪訊之訊號與經中值濾波後之系統識別結果

NSR=30% Mode

1 2 3 4 5 6 7

中值濾波

(I,J) (10,10) (4,4) (3,3) (3,3) (7,7) (4,4) (7,7) f 0.68 1.99 3.21 4.28 5.17 5.86 6.14 ξ 5.38% 5.09% 5.82% 5.25% 5.12% 5.06% 4.62%

未濾波

(I,J) (11,11) (7,7) (6,6) (4,4) (8,8) (7,7) (10,10) f 0.68 1.98 3.20 4.28 5.13 5.76 6.12 ξ 5.79% 4.81% 5.58% 5.14% 5.83% 4.76% 4.10%

不含噪訊

(I,J) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) f 0.67 1.97 3.18 4.26 5.15 5.82 6.23 ξ 5.00% 5.00% 5.00% 5.00% 5.00% 5.00% 5.00%

表 4.11 七層樓剪力構架含 40%噪訊之訊號與中值濾波後之系統識別結果

NSR=40% Mode

1 2 3 4 5 6 7

中值濾波

(I,J) (10,10) (4,4) (4,4) (3,3) (7,7) (4,4) (6,6) f 0.68 2.00 3.20 4.26 5.17 5.90 6.29 ξ 5.35% 5.40% 5.90% 4.63% 4.25% 5.73% 5.72%

未濾波

(I,J) (12,12) (7,7) (6,6) (4,4) (8,8) (7,7) (8,8) f 0.68 1.99 3.20 4.28 5.13 5.86 6.25 ξ 5.63% 5.04% 5.94% 4.92% 5.62% 4.68% 5.84%

不含噪訊

(I,J) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) f 0.67 1.97 3.18 4.26 5.15 5.82 6.23 ξ 5.00% 5.00% 5.00% 5.00% 5.00% 5.00% 5.00%

表 4.12 七層樓剪力構架含噪訊之非時變系統反應經離散小波濾波過後之噪訊比

NSR

Mode 10% 20% 30% 40%

7^th floor 4.59% 7.76% 11.09% 14.48%

6^thfloor 4.88% 7.75% 11.22% 14.43%

5^th floor 5.02% 7.91% 11.06% 14.31%

4^th floor 5.16% 8.07% 11.17% 14.51%

3^rd floor 5.29% 8.08% 11.35% 15.03%

2^nd floor 5.84% 8.43% 11.48% 14.64%

1^st floor 8.45% 10.41% 13.10% 16.28%

ag earth 4.59% 7.76% 11.09% 14.48%

表 4.13 七層樓剪力構架含 10%噪訊之訊號與離散小波濾波後之系統識別結果

NSR=10% Mode

1 2 3 4 5 6 7

小波濾波

(I,J) (2,2) (2,2) (2,2) (2,2) (2,2) (3,3) (4,4) f 0.68 1.97 3.19 4.26 5.17 5.74 6.23 ξ 5.35% 4.81% 5.20% 5.49% 5.94% 5.80% 5.79%

未濾波

(I,J) (8,8) (4,4) (4,4) (4,4) (8,8) (11,11) (10,10) f 0.68 2.00 3.20 4.27 5.14 5.76 6.21 ξ 5.02% 5.56% 5.00% 5.23% 5.01% 5.30% 4.71%

不含噪訊

(I,J) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) f 0.67 1.97 3.18 4.26 5.15 5.82 6.23 ξ 5.00% 5.00% 5.00% 5.00% 5.00% 5.00% 5.00%

表 4.14 七層樓剪力構架含 20%噪訊之訊號與離散小波濾波後之系統識別結果

NSR=20% Mode

1 2 3 4 5 6 7

小波濾波

(I,J) (2,2) (2,2) (2,2) (2,2) (2,2) (3,3) (4,4) f 0.68 1.98 3.20 4.28 5.20 5.88 6.12 ξ 5.91% 4.96% 5.98% 5.63% 5.38% 5.98% 3.52%

未濾波

(I,J) (10,10) (6,6) (4,4) (4,4) (7,7) (7,7) (10,10) f 0.68 1.99 3.20 4.28 5.19 5.89 6.14 ξ 5.67% 5.03% 5.80% 5.09% 5.62% 5.48% 4.68%

不含噪訊

(I,J) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) f 0.67 1.97 3.18 4.26 5.15 5.82 6.23 ξ 5.00% 5.00% 5.00% 5.00% 5.00% 5.00% 5.00%

表 4.15 七層樓剪力構架含 30%噪訊之訊號與離散小波濾波後之系統識別結果

NSR=30% Mode

1 2 3 4 5 6 7

小波濾波

(I,J) (2,2) (2,2) (2,2) (2,2) (2,2) (3,3) (4,4) f 0.68 1.99 3.21 4.28 5.17 5.86 6.14 ξ 5.38% 5.09% 5.82% 5.25% 5.12% 5.06% 4.62%

未濾波

(I,J) (11,11) (7,7) (6,6) (4,4) (8,8) (7,7) (10,10) f 0.68 1.98 3.20 4.28 5.13 5.76 6.12 ξ 5.79% 4.81% 5.58% 5.14% 5.83% 4.76% 4.10%

不含噪訊

(I,J) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) f 0.67 1.97 3.18 4.26 5.15 5.82 6.23 ξ 5.00% 5.00% 5.00% 5.00% 5.00% 5.00% 5.00%

表 4.16 七層樓剪力構架含 40%噪訊之訊號與離散小波濾波後之系統識別結果

NSR=40% Mode

1 2 3 4 5 6 7

小波濾波

(I,J) (4,4) (2,2) (3,3) (2,2) (2,2) (2,2) (2,2) f 0.68 2.00 3.20 4.26 5.17 5.90 6.29 ξ 5.35% 5.40% 5.90% 4.63% 4.25% 5.73% 5.72%

未濾波

(I,J) (12,12) (7,7) (6,6) (4,4) (8,8) (7,7) (8,8) f 0.68 1.99 3.20 4.28 5.13 5.86 6.25 ξ 5.63% 5.04% 5.94% 4.92% 5.62% 4.68% 5.84%

不含噪訊

(I,J) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) f 0.67 1.97 3.18 4.26 5.15 5.82 6.23 ξ 5.00% 5.00% 5.00% 5.00% 5.00% 5.00% 5.00%

表 4.17 七層樓剪力構架含噪訊之非時變系統反應訊號經三種濾波方法處理後的系識別結果綜合比較

NSR Mode 1 2 3 4 5 6 7

10%

Kalman filter (3,3) (2,2) (2,2) (2,2) (2,2) (2,2) (3,3) Median filter (6,6) (4,4) (2,2) (2,2) (4,4) (4,4) (6,6) Wavelet (2,2) (2,2) (2,2) (2,2) (2,2) (3,3) (4,4)

未濾波 (8,8) (4,4) (4,4) (4,4) (8,8) (11,11) (10,10)

20%

Kalman filter (5,5) (2,2) (2,2) (2,2) (2,2) (2,2) (3,3) Median filter (7,7) (4,4) (2,2) (3,3) (6,6) (4,4) (8,8) Wavelet (2,2) (2,2) (2,2) (2,2) (2,2) (3,3) (4,4)

未濾波 (10,10) (6,6) (4,4) (4,4) (7,7) (7,7) (10,10)

30%

Kalman filter (5,5) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) Median filter (10,10) (4,4) (3,3) (3,3) (7,7) (4,4) (7,7) Wavelet (2,2) (2,2) (2,2) (2,2) (2,2) (3,3) (4,4)

未濾波 (11,11) (7,7) (6,6) (4,4) (8,8) (7,7) (10,10)

40%

Kalman filter (6,6) (3,3) (3,3) (3,3) (4,4) (2,2) (3,3) Median filter (10,10) (4,4) (4,4) (3,3) (7,7) (4,4) (6,6) Wavelet (4,4) (2,2) (3,3) (2,2) (2,2) (2,2) (2,2)

未濾波 (12,12) (7,7) (6,6) (4,4) (8,8) (7,7) (8,8)

0% 理論值 (3,3) (3,3) (3,3) (3,3) (3,3) (3,3) (3,3)

附圖

圖 2.1 含噪訊號的濾波流程

圖 2.2 卡氏濾波器之模型示意圖

(資料來源：http://zh.wikipedia.org/wiki/File:Kalman_filter_model.png 作者：Ciphergoth , translated by Jacktance)

真實訊號

噪訊

含噪訊號

濾波器

真實訊號

噪訊

可見隱藏

圖 2.3 卡爾曼濾波器之分析流程 𝒙̂_{𝑡 | 𝑡−1} = 𝑭_𝑡𝒙̂_{𝑡−1 | 𝑡−1}+ 𝑩_𝑡𝒖_𝒕−𝟏

預測狀態

預測估計協方差矩陣 𝑷_{𝑡|𝑡−1} = 𝑭_𝑡𝑷_{𝑡−1|𝑡−1}𝑭_𝑡^𝑇+ 𝑸_𝑡

預測

𝒙̂_{𝑡 |𝑡} = 𝒙̂_{𝑡 | 𝑡−1}+ 𝑲_𝑡𝒚̃_𝑡

𝑷_𝑡|𝑡 = (𝑰 − 𝑲_𝑡𝑯_𝑡) 𝑃_{𝑡|𝑡−1}

𝑲_𝑡= 𝑷_{𝑡|𝑡−1}𝑯_𝑡^𝑇𝑺_𝑡⁻¹ 更新估計模型

更新的協方差矩陣

卡爾曼增益更新

圖 2.4 小波分析樹狀圖

原始訊號 X

cA1 cD1

cD2 cA2

cD3 cA3

cDn cAn

n=1

n=2

n=3

第 n 階

圖 2.5 小波重構示意圖

原始訊號 X

cA 1

cD 1

cD 2 cA

cD 3 cA

cDn cAn

n=1

n=2

n=3

第 n 階

圖 4.1 濾波與系統識別分析流程

非時變系統

濾波

時變系統

系統識別

圖 4.2 a 七層樓之剪力構架示意圖 m₂=0.1 ton

m1=0.1 ton m₅=0.1 ton

m4=0.1 ton m₃=0.1 ton

3 floor

7 6

m₆=0.1 ton m7=0.1 ton

k₇=80 kN/m

k6=80 kN/m 5

a_g

k5=80 kN/m

k4=80 kN/m k3=80 kN/m

k2=80 kN/m

k1=80 kN/m

圖 4.2 b 七層樓之剪力構架示意圖 m2=0.1 ton

m1=0.1 ton m5=0.1 ton

m4=0.1 ton

m3=0.1 ton m6=0.1 ton m7=0.1 ton k7=80 kN/m

k6=80 kN/m

k5=80 kN/m

k₄=80 kN/m k3=80 kN/m

k2=80 kN/m

k1=80 kN/m

圖 4.3 a 七層樓剪力構架第一層樓之地震歷時反應圖(非時變系統)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

無雜訊之地震歷時反應圖 7 DOF 1 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 10 % 7 DOF 1 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 20 % 7 DOF 1 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 30 % 7 DOF 1 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 40 % 7 DOF 1 F

0 8 16 24 32 40 48 56 64 72 80

0 8 16 24 32 40 48 56 64 72 80 0 8 16 24 32 40 48 56 64 72 80

0 8 16 24 32 40 48 56 64 72 80

Time(sec)

圖 4.3 b 七層樓剪力構架第二層樓之地震歷時反應圖(非時變系統)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

無雜訊之地震歷時反應圖 7 DOF 2 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 10 % 7 DOF 2 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 20 % 7 DOF 2 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 30 % 7 DOF 2 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 40 % 7 DOF 2 F

0 8 16 24 32 40 48 56 64 72 80

0 8 16 24 32 40 48 56 64 72 80 0 8 16 24 32 40 48 56 64 72 80

0 8 16 24 32 40 48 56 64 72 80

Time(sec)

圖 4.3 c 七層樓剪力構架第三層樓之地震歷時反應圖(非時變系統)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

無雜訊之地震歷時反應圖 7 DOF 3 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 10 % 7 DOF 3 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 20 % 7 DOF 3 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 30 % 7 DOF 3 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 40 % 7 DOF 3 F

0 8 16 24 32 40 48 56 64 72 80

0 8 16 24 32 40 48 56 64 72 80 0 8 16 24 32 40 48 56 64 72 80

0 8 16 24 32 40 48 56 64 72 80

Time(sec)

圖 4.3 d 七層樓剪力構架第四層樓之地震歷時反應圖(非時變系統)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

無雜訊之地震歷時反應圖 7 DOF 4 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 10 % 7 DOF 4 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 20 % 7 DOF 4 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 30 % 7 DOF 4 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 40 % 7 DOF 4 F

0 8 16 24 32 40 48 56 64 72 80

0 8 16 24 32 40 48 56 64 72 80 0 8 16 24 32 40 48 56 64 72 80

0 8 16 24 32 40 48 56 64 72 80

Time(sec)

圖 4.3 e 七層樓剪力構架第五層樓之地震歷時反應圖(非時變系統)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

無雜訊之地震歷時反應圖 7 DOF 5 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 10 % 7 DOF 5 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 20 % 7 DOF 5 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 30 % 7 DOF 5 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 40 % 7 DOF 5 F

0 8 16 24 32 40 48 56 64 72 80

0 8 16 24 32 40 48 56 64 72 80 0 8 16 24 32 40 48 56 64 72 80

0 8 16 24 32 40 48 56 64 72 80

Time(sec)

圖 4.3 f 七層樓剪力構架第六層樓之地震歷時反應圖(非時變系統)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

無雜訊之地震歷時反應圖 7 DOF 6 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 10 % 7 DOF 6 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 20 % 7 DOF 6 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 30 % 7 DOF 6 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 40 % 7 DOF 6 F

0 8 16 24 32 40 48 56 64 72 80

0 8 16 24 32 40 48 56 64 72 80 0 8 16 24 32 40 48 56 64 72 80

0 8 16 24 32 40 48 56 64 72 80

Time(sec)

圖 4.3 g 七層樓剪力構架第七層樓之地震歷時反應(非時變系統)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

無雜訊之地震歷時反應圖 7 DOF 7 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 10 % 7 DOF 7 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 20 % 7 DOF 7 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 30 % 7 DOF 7 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -40

-20 0 20 40

NSR = 40 % 7 DOF 7 F

0 8 16 24 32 40 48 56 64 72 80

0 8 16 24 32 40 48 56 64 72 80 0 8 16 24 32 40 48 56 64 72 80

0 8 16 24 32 40 48 56 64 72 80

Time(sec)

圖 4.4 a 五層樓之剪力構架示意圖 ag

3 floor

m=0.1 ton

m=0.1 ton m=0.1 ton

m=0.1 ton m=0.1 ton 5

k₁=80 kN/m k(t)

k(t) k(t) k(t)

圖 4.4 b 五層樓之剪力構架示意圖 m=0.1 ton

m=0.1 ton m=0.1 ton

m=0.1 ton

m=0.1 ton k(t)

k(t)

k(t) k(t)

k1=80 kN/m

圖 4.5 a 五層樓剪力構架第一層樓之地震歷時反應圖(時變系統)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -1

0 1

無雜訊之地震歷時反應圖 5 DOF 1 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -1

0 1

NSR = 1 % 5 DOF 1 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -1

0 1

NSR = 3 % 5 DOF 1 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -1

0 1

NSR = 5 % 5 DOF 1 F

0 8 16 24 32 40 48 56 64 72 80

Time(sec)

圖 4.5 b 五層樓剪力構架第二層樓之地震歷時反應圖(時變系統)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -1

0 1

無雜訊之地震歷時反應圖 5 DOF 2 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -1

0 1

NSR = 1 % 5 DOF 2 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -1

0 1

NSR = 3 % 5 DOF 2 F

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

x 10⁴ -1

0 1

NSR = 5 % 5 DOF 2 F

0 8 16 24 32 40 48 56 64 72 80

Time(sec)

圖 4.5 c 五層樓剪力構架第三層樓之地震歷時反應圖(時變系統)

在文檔中數種濾波技術於線性系統識別之可行性探討 (頁 47-99)

建議

第五章 結論與建議

5.2 建議

參考文獻

附表

附圖

非時變系統

濾波

時變系統

系統識別

無雜訊之地震歷時反應圖 7 DOF 2 F

NSR = 10 % 7 DOF 2 F

NSR = 20 % 7 DOF 2 F

NSR = 30 % 7 DOF 2 F

NSR = 40 % 7 DOF 2 F

第五章結論與建議