第五章 結論與建議
5.2 建議
1. 對於遺忘因子以及卡式濾波器之控制因子r t1( )之選擇，應找出一個定量 之方法，避免進行太多之識別分析比較，進而能迅速得到吾人理想之識 別結果。
2. 本文僅針對單自由度系統作數值模擬分析，亦可改變輸入為多自由度訊 號輸入，將其推廣至多自由度系統。以便應用於分析實際之結構物之動 態反應量測。
3. 比較蘇(2008)中以多項式基底配合移動最小平方差之識別結果，發現其 識別效果除了跳躍變化系統均明顯優於遞迴最小平方差法，尤其在分析 具雜訊之反應；故蘇(2008)之方法或許是估算時變系統之瞬時模態參數 之較佳選擇。
參考文獻
Arunachalam, K. G. and Chesmore, E. D., “System identification techniques for adaptive recursive filtering.”, International Journal of Electronics, 74(3), pp. 381-399, 1993.
Atsuhiko, S. and Hiroshi, F., “Transient response analysis with extended kalman filter for recursive self-identification using semi-active devices.”, Collection of Technical Papers - 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, pp. 8337-8344, 2007.
Belge, M. and Miller, E. L., “A sliding window RLS-like adaptive algorithm for filtering alpha-stable noise.”, IEEE Signal Process Letter, 7(4), pp. 86-89, 2000.
Campi, M., “Performance of RLS identification algorithms with forgetting factor: A Φ-Mixing approach.”, Journal of Mathematical Systems, Estimation, and Control, 4(3), pp.
1-25, 1994.
Cao, L. and Schwartz, M. H., “The kalman filter based recursive algorithm: windup and its avoidance.”, Proceedings of the American Control Conference Arlington, pp. 25-27,June 2001.
Chen, H. U., Identification and Stochastic Adaptive Control., Boston, Birkhauser, 1991.
Chen, H., Liu, H., Li, D. and Dai, J., “Frequency estimation based on extended RLS with variable forgetting factor.”, Electric Power Automation Equipment, 28(7), pp. 45-48,2008.
Choi, B. Y. and Bien, Z., “Sliding-windowed weighted recursive least-squares method for parameter estimation.”, Electron Letter, 25(20), pp. 1381-1382, 1989.
Cho, Y. S., Kim, S. B. and Powers, E. J., “Time-frequency analysis using AR models with variable forgetting factors.”, Proceedings - ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 90(5), pp. 2479-2482, 1990.
Choi, H. I. and Williams, W. J., “Improved time-frequency representation of multicomponent signals using exponential kernels.”, IEEE Transactions on Acoustics, Speech and Signal Processing, 17(6), pp. 862-871, 1989.
Cohen, L., “Time-frequency distributions－A review.”, Proceedings of the IEEE, 77(7), pp.
941-981, 1989.
Costa, E. and Astolfi, A., “A necessary and sufficient condition for semi-stability of the Kalman filter.”, Proceedings of the American Control Conference, pp. 1280-1285, 2008.
Enbin, S., Zhu, Y. and You, Z., “The kalman type recursive state estimator with a finite-step
correlated process noises.”, Proceedings of the IEEE International Conference on Automation and Logistics, pp. 196-200, 2008.
Fortescue, T. R., Kershenbaum, L. S. and Ydstie, B. E., “Implementation of self-tuning regulators with variable forgetting factors.”, Automatica, 17(6), pp. 831-835, 1981.
Ma, H. and Girgis, A., “Identification and tracking of harmonic sources in a power system using a kalman filter.”, IEEE Transactions on Power Delivery, 11(3), pp. 1659-1665, 1996.
Huang, N. E., Shen, Z., Long, S. R., Wu, M. C. and Shin, H. H. “The empirical mode decomposition and Hilbert spectrum for nonlinear and nonstationary time series analysis.”, Proceedings of the Royal Society of London, A454, pp. 903-995, 1998.
Rene, V., “Recursive identification of switched ARX systems.”, Automatica, 44(9), pp.
2274-2287, September 2008.
Jiang, J. and Cook, R., “Fast parameter tracking RLS algorithm with high noise immunity.”, Electron Letter, 28(22), pp. 2042-2045,1992.
Johansson, R., System Modeling and Identification, Prentice-Hall, NJ, 1993.
Jones, D. L. and Baraniuk, R. G.,“An adaptive optimal-kernel time-frequency representation.”, IEEE Transactions on Signal Processing, 43 (10), pp. 2361-2371, 1995.
Kadrikamanathan, V., Jaward, M. H., Fabri, S. G., and Kadrikamanathan, M., “Particle filters for recursive model selection in linear and nonlinear system identification.”, Proceedings of the IEEE Conference on Decision and Control, 3, pp. 2391-2396, 2000.
Kalouptsidis, N. and Theodoridis, S., Adaptive System Identification and Signal Processing Algorithm., Prentice-Hall, UK, 1993.
Kijewski, T. and Kareem, A., “Wavelet transforms for system identification in civil engineering.”, Computer-aided Civil and Infrastructure Engineering, 18(5), pp. 339-355, 2003.
Leung, S. H. and So, C. F., “Gradient-based variable forgetting factor RLS algorithm in time-varying environments.”, IEEE Transactions on Signal Processing, 53(8), pp.3141-3150, 2005.
Ljung, L. and Soderstrom, T., “Theory and Practice of Recursive Identification.” MIT Press, Cambridge, MA, 1983.
Loh, C. H., Lin, C. Y. and Huang, C. C., “Time domain identification of frames under earthquake loadings.”, Journal of Engineering Mechanics, ASCE 126(7), pp.693-703,
2000.
Martin, W. and Flandrin, P., “Wigner-Ville spectral analysis of non-stationary processes.”, IEEE Transactions on Acoustics, Speech and Signal Processing, ASSP 33, pp. 1461-1470, 1985.
Marmarelis, V. Z., “Nonlinear and nonstationary modeling of physiological system: an overview in advanced methods of physiological system modeling, edited by V. Z.
Marmarelis.”, Biomedical Simulations Resource, 1, pp. 1-24, 1987.
Niedwiecki, M., Identification of Time-Varying Processes, John Wiley & Sons, Inc. New York. NY, USA 2000.
Niedwiecki, M. and Klaput, T., “Fast recursive basis function algorithms for identification of time-varying processes.”, Proceedings of the 40th IEEE conference on Decision and Control, 2001.
Pang, S. H., Yu, K. P. and Zou, J. X., “Time-varying system identification using recursive subspace method based on free response data.”, Journal of Vibration Engineering, 18(2), pp. 233-237, June 2005.
Parkum, J. E., Poulsen, N. K. and Holst, J., “Recursive foretting algorithm.”, Internation Journal of Control, 55(1), pp. 109-128, 1992.
Park, D. J. and Jun, B. E., “Self-perturbing recursive least squares algorithm with fast tracking capability.”, Electron Letter, 28(6), pp.558-559, 1992.
Qian, S., Introduction to Time-frequency and Wavelet Transforms. Prentice-Hall PTR, NJ, 2002.
Roman, S., “Adaptive extended kalman filter with recursive noise covariance identification applied to the fermentation processes.”, Modelling, Simulation & Control C:
Environmental, Biomedical, Human & Social Systems, 10(2), pp.52-64, 1987.
Schoukens, J., Identification of Linear Systems : A Practical Guideline to Accurate Modeling.
Oxford : Pergamon Press, c1991.
Seibold, S. and Fritzen, C. P., “Recursive identification of nonlinear systems by means of the extended kalman filter.”, Proceedings of the International Modal Analysis Conference, IMAC-1, pp. 802-809, 1991.
Soderstrom, T. and Stoica, P., System Identification, Prentice Hall. Englewood Cliffs NJ,1988.
Sinha, N. K., Modeling and Identification of Dynamic Systems. New York : Van Nostrand Reinhold, c1983.
Song, S. and Lim, J. S., “Gauss Newton variable forgetting factor recursive least squares for time varying parameter tracking.”, Electronics Letters, 36(11), pp. 988-990, May 2000.
Song, S., Lim, J. S., Baek, S. J., and Sung, K. M., “Variable forgetting factor linear least squares algorithm for frequency selective fading channel estimation.”, IEEE Transactions on Vehicular Technology, 51(3), pp. 613-616, May 2002.
Sugisaki, K. and Ohmori, H., “Online estimation of complexity using variable forgetting factor.”, SICE Annual Conference, pp. 1-6, 2007.
Toplis, B. and Pasupathy, S., “Tracking improvements in fast RLS algorithm using a variable forgetting factor.”, IEEE Transactions on Acoustics, Speech and Signal Processing, ASSP 36(2), pp. 206-227, 1988.
Tsatsanis, M. K. and Giannakis, G. B., “Time-varying system identification and model validation using wavelets.”, IEEE Transactions on Signal Process, 41, pp. 3512-3523, 1993.
Wang, F., An, L., An, L., and Xu, Y., “Applying adaptive recursive filtering to system identification.”, International Conference on Signal Processing Proceedings, ICSP-1, pp.250-253, 1996.
Wang, L. and Reza, L., “Variable forgetting factor RLS algorithm with application to fuzzy time-varying systems identification.”, International Journal of Systems Science, 27(2), pp.
205-214, Feb 1996.
Wu, C. L., Yang, Y. S. and Loh, C. H., “Dynamic gravity load collapse of non-ductile RC frames I: experimental approach.”, Proceedings of the 8th U.S. National Conference on Earthquake Engineering, pp. 750, 2006.
Zhang, B. M. and Lo, K. L., “A recursive measurement error estimation identification method for bad data analysis in power system state estimation.”, IEEE Transactions on Power Systems, 6(2), pp.191-198, Feb 1991.
Zhang, Z. G.., “A recursive frequency estimator using linear prediction and a Kalman filter based iterative algorithm.”, IEEE Transactions on Circuits and Systems, 55(6), pp.
576-580, June 2008.
Zhao, W. X. and Chen, H. F., “Recursive identification for Hammerstein system with ARX subsystem.” , PESC 07 - IEEE 38th Annual Power Electronics Specialists Conference, pp.
2478-2484, 2007.
Zou, R., Cupples, W. A., Yip, K. P., Holstein-Rathlou, N. H. and Chon, K. H., “Time-varying properties of renal autoregulatory mechanisms.”, IEEE transactions on bio-medical
engineering, 49, pp.1112-1120, 2002.
Zou, R., Wang, H. and Chon, K. H., “A robust time-varying identification algorithm using basis function.”, Annals of Biomedical Engineering, 31, pp.840-853, 2003.
林沛暘等編著，「標竿鋼構樓房震動台試驗」，國家地震工程研究中心報NCREE-06-020，
2006。
吳子修，「結構振動訊號之識別與分析」，國立臺灣大學，博士論文，1997。
蘇威智，「以時間序列模型識別結構之模態參數」，國立交通大學，博士論文，2008。
表2.1 平緩變化系統之識別誤差結果(常數形遺忘因子)
表2.3 跳躍變化系統之識別誤差結果(常數形遺忘因子)
表2.5 平緩變化系統之識別誤差結果(變數形遺忘因子)
表2.7 跳躍變化系統之識別誤差結果(變數形遺忘因子)
表2.9 平緩變化系統之識別誤差結果(卡式濾波器)
表2.13 含 2%雜訊平緩變化系統反應之識別誤差結果(常數型遺忘因子)
λ 0.96 0.97
(I,J) em a x , f em a x , f em a x , f em a x , f
(20,20) 29.6 148 2.26 25.2 21.2 134 1.62 19.3 (25,25) 16.2 137 1.69 20.5 12.7 128 1.32 15.1 (30,30) 24.2 142 1.38 16.2 11.9 90.0 1.15 12.1 (35,35) 16.4 93.2 1.21 13.4 8.96 95.3 1.07 10.2 (40,40) 29.1 116 1.19 12.9 8.81 94.1 1.06 10.01 (45,45) 16.3 96.7 1.14 10.8 8.37 114 1.09 8.49 (50,50) 11.7 102 1.12 10.6 10.8 84.7 1.07 8.71
λ 0.98 0.99
(I,J) em a x , f em a x , f em a x , f em a x , f
(20,20) 9.95 117 1.27 12.5 5.78 80.2 1.58 7.01 (25,25) 5.03 79.6 1.13 9.59 4.68 52.9 1.60 5.66 (30,30) 5.24 64.0 1.14 8.34 5.31 44.7 1.64 5.57 (35,35) 5.11 58.5 1.10 7.18 5.34 39.9 1.65 5.83 (40,40) 5.14 87.0 1.12 7.29 5.53 37.6 1.65 5.57 (45,45) 5.23 86.4 1.18 6.67 5.54 39.9 1.71 5.62 (50,50) 5.88 67.0 1.17 7.24 5.41 40.8 1.70 5.32
表2.14 含 2%雜訊週期變化系統反應之識別誤差結果(常數型遺忘因子)
λ 0.96 0.97
(I,J) em a x , f em a x , f em a x , f em a x , f
(20,20) 75.1 617 6.52 64.9 30.9 592 6.71 54.7 (25,25) 44.5 589 6.68 54.9 40.6 588 6.92 48.3 (30,30) 35.3 664 6.94 51.8 36.8 467 7.13 48.2 (35,35) 48.6 564 7.05 53.1 35.4 368 7.24 49.5 (40,40) 55.9 395 7.22 52.7 57.2 365 7.52 51.7 (45,45) 60.3 388 7.43 52.1 51.3 388 7.67 52.8 (50,50) 57.3 347 8.01 57.5 42.7 388 8.43 59.0
λ 0.98 0.99
(I,J) em a x , f em a x , f em a x , f em a x , f
(20,20) 25.3 248 7.64 44.6 34.1 234 11.4 39.3 (25,25) 30.3 322 7.92 43.0 35.9 321 11.5 43.9 (30,30) 31.2 320 8.17 45.4 40.5 330 11.8 45.5 (35,35) 37.7 358 7.95 47.8 36.3 294 11.3 52.4 (40,40) 43.5 272 8.39 53.2 54.9 222 12.6 56.8 (45,45) 36.9 407 8.43 56.4 36.0 284 12.9 62.6 (50,50) 43.1 384 9.20 59.8 46.5 346 12.4 65.3
表2.15 含 2%雜訊跳躍變化系統反應之識別誤差結果(常數型遺忘因子)
λ 0.96 0.97
(I,J) em a x , f em a x , f em a x , f em a x , f
(20,20) 230 297 7.16 54.7 116 296 5.21 47.2 (25,25) 128 299 5.92 48.8 121 298 4.14 40.7 (30,30) 116 298 4.98 43.4 110 291 3.57 34.7 (35,35) 109 299 4.44 36.3 104 299 3.13 30.1 (40,40) 100 286 3.69 30.9 100 252 2.61 23.6 (45,45) 140 278 3.97 25.2 129 258 2.79 19.4 (50,50) 150 293 3.24 24.4 128 164 2.65 18.8
λ 0.98 0.99
(I,J) em a x , f em a x , f em a x , f em a x , f
(20,20) 108 296 3.53 36.0 101 272 2.14 23.5 (25,25) 110 299 2.78 33.3 101 243 1.75 18.4 (30,30) 105 296 2.12 26.5 100 181 1.59 13.8 (35,35) 101 290 2.19 22.2 100 166 1.92 12.8 (40,40) 100 166 1.96 17.7 100 166 1.90 10.8 (45,45) 112 164 1.63 15.8 100 164 1.54 9.13 (50,50) 109 166 1.80 13.4 109 164 1.61 7.82
表2.16 含 2%雜訊折線變化系統反應之識別誤差結果(常數型遺忘因子)
λ 0.96 0.97
(I,J) em a x , f em a x , f em a x , f em a x , f
(20,20) 62.7 486 4.04 53.1 31.3 409 3.41 41.0 (25,25) 73.2 491 3.56 45.9 29.6 313 3.05 36.0 (30,30) 42.0 273 3.21 39.8 27.8 271 2.92 32.0 (35,35) 35.4 255 2.98 37.3 25.1 269 2.91 31.9 (40,40) 30.0 291 2.85 34.6 28.9 234 2.88 29.8 (45,45) 49.7 338 3.09 32.8 37.5 310 2.94 29.3 (50,50) 57.5 583 3.04 34.1 52.1 286 2.92 30.4
λ 0.98 0.99
(I,J) em a x , f em a x , f em a x , f em a x , f
(20,20) 32.1 280 3.29 29.2 15.7 121 1.61 19.2 (25,25) 16.7 190 3.04 26.1 15.0 135 1.57 19.0 (30,30) 15.6 162 3.01 24.7 16.9 166 1.66 18.9 (35,35) 15.3 169 3.04 25.9 17.7 152 1.78 20.3 (40,40) 19.8 185 3.02 25.4 22.0 178 1.81 20.0 (45,45) 26.0 248 3.16 26.0 26.8 170 1.94 21.9 (50,50) 20.3 247 3.21 27.2 26.2 162 2.09 23.4
表2.17 含 2%雜訊平緩變化系統反應之識別誤差結果(變數型遺忘因子)
λ 0.96 0.97
(I,J) em a x , f em a x , f em a x , f em a x , f
(20,20) 20.5 178 2.21 19.8 11 142 1.88 13,3 (25,25) 22.3 234 2.09 14.6 13 112 1.73 10.2
(30,30) 10.1 112 1.85 11.8 8.82 82 1.63 8.58
(35,35) 12.7 152 1.73 9.71 9.38 168 1.61 7.52 (40,40) 16.5 199 1.72 9.24 9.95 271 1.59 7.24 (45,45) 11.2 124 1.67 8.05 9.33 91.2 1.56 6.71 (50,50) 8.51 124 1.64 8.27 8.91 77.6 1.57 7.16
λ 0.98 0.99
(I,J) em a x , f em a x , f em a x , f em a x , f
(20,20) 6.6 91.9 1.43 8.97 3.80 49.1 0.945 5.57 (25,25) 6.5 67.2 1.48 7.03 4.22 31.8 0.982 4.62 (30,30) 5.6 54.5 1.36 6.15 3.94 31.4 0.919 4.51 (35,35) 5.7 42.5 1.47 5.71 3.92 38.1 0.956 4.84 (40,40) 6.7 41.7 1.34 5.87 4.24 42.9 0.932 5.38 (45,45) 5.3 51.2 1.22 5.73 4.17 40.5 0.924 5.71 (50,50) 8.5 40.8 1.31 6.46 4.34 47.3 0.964 6.42
表2.18 含 2%雜訊週期變化系統反應之識別誤差結果(變數型遺忘因子)
λ 0.96 0.97
(I,J) em a x , f em a x , f em a x , f em a x , f
(20,20) 60.3 404 6.86 52.5 33.2 285 7.51 45.1 (25,25) 48.1 649 7.03 47.7 45.5 433 7.74 44.8 (30,30) 39.6 517 7.35 47.3 29.4 336 7.97 45.6 (35,35) 45.7 365 7.33 48.5 40.8 367 7.84 47.4 (40,40) 54.3 383 7.68 51.2 52.3 368 8.44 52.9 (45,45) 56.2 388 7.72 53.6 45.5 391 8.36 54.7 (50,50) 72.9 393 8.53 58.8 42.4 584 9.12 59.2
λ 0.98 0.99
(I,J) em a x , f em a x , f em a x , f em a x , f
(20,20) 25.5 231 9.25 40.8 35.4 250 14.1 46.1 (25,25) 31.1 310 9.47 41.4 40.3 324 14.3 51.3 (30,30) 35.6 316 9.33 43.6 43.7 333 14.5 51.2 (35,35) 34.7 338 9.38 48.2 51.3 289 14.3 54.3 (40,40) 43.9 235 9.82 53.7 54.9 232 15.7 58.9 (45,45) 35.4 345 9.87 57.8 52.8 409 15.5 66.7 (50,50) 43.6 352 10.9 61.5 47.5 390 15.4 69.4
表2.19 含 2%雜訊跳躍變化系統反應之識別誤差結果(變數型遺忘因子)
λ 0.96 0.97
(I,J) em a x , f em a x , f em a x , f em a x , em a x , f em a x , (20,20) 114 298 4.14 44.7 109 294 3.14 37.8 (25,25) 118 296 3.12 40.3 110 295 2.32 33.4 (30,30) 109 294 2.61 33.4 105 293 1.97 26.2 (35,35) 103 298 2.35 27.2 101 279 1.94 21.6 (40,40) 100 268 2.17 21.8 100 233 1.8 17.4 (45,45) 124 260 2.45 18.5 113 172 2.14 15.3 (50,50) 122 250 2.44 16.2 109 165 2.22 12.2
λ 0.98 0.99
(I,J) em a x , f em a x , em a x , f em a x , em a x , f em a x , f
(20,20) 103 292 2.37 29.1 100 255 1.93 17.5 (25,25) 103 299 1.84 23.7 100 224 1.65 12.1 (30,30) 101 287 1.42 18.4 100 168 1.71 9.72 (35,35) 100 260 1.69 15.3 100 165 2.33 9.87 (40,40) 105 162 1.95 12.2 100 164 2.54 8.99 (45,45) 102 161 2.05 10.7 100 164 2.61 7.84 (50,50) 102 163 2.02 8.72 100 159 2.72 6.82
表2.20 含 2%雜訊折線變化系統反應之識別誤差結果(變數型遺忘因子)
λ 0.96 0.97
(I,J) em a x , f em a x , f em a x , f em a x , f
(20,20) 38.3 298 3.43 36.3 23.9 278 2.13 30.2 (25,25) 32.6 286 3.21 33.1 26.5 243 1.92 27.7 (30,30) 31.3 263 2.33 61.5 19.2 241 1.87 26.4 (35,35) 72.8 235 2.21 31.8 12.3 262 1.62 27.3 (40,40) 19.6 268 1.97 29.7 17.5 204 1.57 26.8 (45,45) 25.3 284 1.82 28.2 18.7 261 1.62 26.1 (50,50) 57.1 250 1.95 30.4 23.3 247 1.61 28.3
λ 0.98 0.99
(I,J) em a x , f em a x , f em a x , f em a x , f
(20,20) 17.1 180 1.52 22.1 20.1 118 2.56 17.2 (25,25) 9.06 150 1.37 21.8 19.8 133 2.59 18.8 (30,30) 10.2 178 1.39 21.6 21.4 161 2.53 18.4 (35,35) 9.75 188 1.33 23.4 22.3 184 2.52 19.2 (40,40) 12.3 202 1.42 22.3 25.8 167 2.79 20.6 (45,45) 19. 260 1.56 24.2 29.3 159 2.83 22.4 (50,50) 17. 273 1.58 25.4 30.4 165 2.86 23.3
表2.21 含 2%雜訊平緩變化反應系統之識別誤差結果(卡式濾波器)
1( )
r t 0.01 0.1
(I,J) em a x , f em a x , f em a x , f em a x , f
(20,20) 7.82 23.1 2.45 9.43 7.96 25 2.01 9.51 (25,25) 7.81 29.5 2.73 11.7 7.23 24 2.25 9.46 (30,30) 8.95 39.3 3.26 17.5 7.58 32 2.53 11.6 (35,35) 11.5 71.2 4.32 26.3 8.43 52 3.22 17.3 (40,40) 15.3 100 6.13 29.9 11.4 68 4.36 23.2 (45,45) 16.2 91.3 7.84 25.5 12.4 78 4.94 24.6 (50,50) 18.8 67.3 8.12 18.2 15.7 103 6.32 26.7
1( )
r t 1 10
(I,J) em a x , f em a x , f em a x , f em a x , f
(20,20) 8.61 56.1 2.16 12.6 12.1 105 2.45 25.1 (25,25) 7.95 31.6 2.12 13.2 7.72 76.1 2.13 26.1 (30,30) 7.63 28.4 2.44 10.4 7.96 38.7 2.51 16.5 (35,35) 8.42 42.3 2.98 13.6 8.34 38.3 2.95 15.3 (40,40) 9.75 44.7 3.53 14.2 9.53 38.1 3.43 13.1 (45,45) 10.1 42.6 3.95 13.7 10. 35.3 3.96 10.9 (50,50) 11.7 52.3 4.82 17.1 11.6 46.5 4.57 13.6
表2.22 含 2%雜訊週期變化系統反應之識別誤差結果(卡式濾波器)
1( )
r t 0.01 0.1
(I,J) em a x , f em a x , f em a x , f em a x , f
(20,20) 58.1 124 20.5 53.4 56.5 132 16.7 39.4 (25,25) 62.4 225 21.3 87.1 59.3 181 18.2 60.9 (30,30) 81.2 322 28.6 116 62.9 243 21.4 88.3 (35,35) 112 384 36.8 122 89.1 380 29.1 116 (40,40) 134 301 39.3 121 128 393 37.5 133 (45,45) 113 416 40.4 118 149 377 41.9 123 (50,50) 135 451 40.2 143 148 431 42.1 139
1( )
r t 1 10
(I,J) em a x , f em a x , f em a x , f em a x , f
(20,20) 58.2 188 15.7 41.1 67.5 306 16.3 77.1 (25,25) 60.3 199 17.8 52.5 62.2 285 17.5 65.4 (30,30) 62.7 212 20.4 64.4 63.8 279 20.2 68.0 (35,35) 69.4 250 23.6 90.2 71.1 259 23.7 81.2 (40,40) 118 424 34.5 118 117 469 34.0 122 (45,45) 149 349 40.3 120. 152 396 40.8 105 (50,50) 125 416 38.1 141. 121 436 36.4 123
表2.23 含 2%雜訊跳躍變化系統反應之識別誤差結果(卡式濾波器)
1( )
r t 0.01 0.1
(I,J) em a x , f em a x , f em a x , f em a x , f
(20,20) 97.3 121 5.21 21.2 97.2 132 6.65 27.5 (25,25) 95.6 122 4.73 13.4 100 107 5.73 13.3 (30,30) 99.1 169 4.91 14.1 101 108 5.52 18.6 (35,35) 99.4 164 5.16 15.5 102 120 5.06 13.1 (40,40) 99.3 165 5.64 16.4 100 147 4.81 11.7 (45,45) 100 166 5.63 18.2 100 148 5.08 13.7 (50,50) 100 166 5.72 20.1 101 166 4.95 14.3
1( )
r t 1 10
(I,J) em a x , f em a x , f em a x , f em a x , f
(20,20) 100 165 9.17 51.3 106 190 11.7 103 (25,25) 104 125 7.04 23.5 113 166 9.29 62.5 (30,30) 104 142 7.12 31.2 108 166 9.86 67.3 (35,35) 105 100 5.63 23.7 108 166 6.93 46.6 (40,40) 100 121 5.02 15.5 101 145 6.05 28.7 (45,45) 100 112 5.31 13.2 101 125 5.82 17.3 (50,50) 101 125 5.13 14.1 101 135 5.55 18.1
表2.24 含 2%雜訊折線變化系統反應之識別誤差結果(卡式濾波器)
1( )
r t 0.01 0.1
(I,J) em a x , f em a x , f em a x , f em a x , f
(20,20) 23.5 69.4 3.39 24.1 26.1 89.3 3.20 20.6 (25,25) 23.3 84.2 3.45 28.3 22.4 74.4 2.96 21.6 (30,30) 24.1 99.7 3.73 37.4 22.0 77.2 3.03 23.0 (35,35) 26.3 107 4.57 52.1 24.5 99.2 3.42 33.2 (40,40) 29.2 175 6.22 65.5 25.3 123 4.54 47.4 (45,45) 33.37 216 7.51 63.5 27.8 151 5.38 52.5 (50,50) 40. 184 9.24 63.7 30.4 168 6.43 51.5
1( )
r t 1 10
(I,J) em a x , f em a x , f em a x , f em a x , f
(20,20) 37.7 159 4.28 45.8 48.1 302 6.19 106 (25,25) 25.4 105 3.35 21.0 29.8 161 3.96 32.2 (30,30) 24.3 88.5 3.02 18.6 27.5 130 3.17 24.5 (35,35) 24.2 89.3 3.27 26.7 27.2 121 3.43 25.6 (40,40) 26.6 106 4.33 39.2 29.7 125 4.58 38.1 (45,45) 26.5 107 4.62 34.4 27.3 92.3 4.61 28.6 (50,50) 28.2 120 5.39 36.6 29.3 99.1 5.29 32.4
表3.1 平緩變化系統之識別誤差結果(常數形遺忘因子)
表3.3 跳躍變化系統之識別誤差結果(常數形遺忘因子)
表3.5 平緩變化系統之識別誤差結果(變數形遺忘因子)
表3.7 跳躍變化系統之識別誤差結果(變數形遺忘因子)
表3.9 平緩變化系統之識別誤差結果(卡式濾波器)
表3.13 含 2%雜訊平緩變化系統反應之識別誤差結果(常數形遺忘因子)
表3.17 含 2%雜訊平緩變化系統反應之識別誤差結果(變數形遺忘因子)
表3.21 含 2%雜訊平緩變化系統反應之識別誤差結果(卡式濾波器)
r t 0.000001 0.000001
(I,J) em a x , f em a x , f em a x , f em a x , f
(a)
(b)
(c)
(d)
圖2.1 各種時變系統之自然振動頻率與阻尼比歷時圖
(a) case 1；(b)case 2；(c)case 3；(d)case 4
圖2.2 輸入地震歷時及其頻譜反應
log(Amp)
0.2
-0.2
圖2.3 各種時變系統之輸出歷時及其頻譜反應
log(Amp) log(Amp)
log(Amp) log(Amp)
(a)
(b)
(c)
(d)
圖2.4 以遺忘因子法配合常數形遺忘因子之識別結果
(a) case 1；(b)case 2；(c)case 3；(d)case 4
(a)
(b)
(c)
(d)
圖2.5 以遺忘因子法配合變數形遺忘因子之識別結果
(a)case 1；(b)case 2；(c)case 3；(d)case 4
(a)
(b)
(c)
(d)
圖2.6 卡式濾波器之識別結果 (a)case 1；(b)case 2；(c)case 3；(d)case 4
(a)
(b)
(c)
(d)
圖2.7 以遺忘因子法配合常數形遺忘因子之識別結果(2%雜訊) (a) case 1 (b)case 2；(c)case 3；(d)case 4
(a)
(b)
(c)
(d)
圖2.8 以遺忘因子法配合變數形遺忘因子之識別結果(2%雜訊) (b) case 1 (b)case 2；(c)case 3；(d)case 4
(a)
(b)
(c)
(d)
圖2.9 卡式濾波器之識別結果(2%雜訊) (c) case 1 (b)case 2；(c)case 3；(d)case 4
(a)
(b)
(c)
(d)
圖3.1 以權重多項式基底函數法配合常數形遺忘因子之識別結果(k=2) (b) case 1；(b)case 2；(c)case 3；(d)case 4
(a)
(b)
(d)
圖3.2 以權重多項式基底函數法配合常數形遺忘因子之識別結果(k=3) (a)case 1；(b)case 2；(c)case 3；(d)case 4
(a)
(b)
(c)
(d)
圖3.3 以權重多項式基底函數法配合變數形遺忘因子之識別結果(k=2) (a)case 1；(b)case 2；(c)case 3；(d)case 4
(a)
(b)
(c)
(d)
圖3.4 權重多項式基底函數法配合變數形遺忘因子之識別結果(k=3) (a) case 1；(b)case 2；(c)case 3；(d)case 4
(a)
(b)
(c)
(d)
圖3.5 權重多項式基底函數法配合卡式濾波器識別結果 (k=2) (a)case 1；(b)case 2；(c)case 3；(d)case 4
(a)
(b)
(c)
(d)
圖3.6 權重多項式基底函數法配合卡式濾波器識別結果(k=3) (a)case 1；(b)case 2；(c)case 3；(d)case 4
(a)
(b)
(c)
(d)
圖3.7 權重多項式基底函數法配合常數形遺忘因子之識別結果(k=2,2%雜訊) (a)case 1；(b)case 2；(c)case 3；(d)case 4
(a)
(b)
(c)
(d)
圖3.8 權重多項式基底函數法配合常數形遺忘因子之識別結果(k=3,2%雜訊) (a)case 1；(b)case 2；(c)case 3；(d)case 4
(a)
(b)
圖3.9 權重多項式基底函數法配合變數形遺忘因子之識別結果(k=2,2%雜訊) (a)case 1；(b)case 2；(c)case 3；(d)case 4
(a)
(b)
(c)
(d)
圖3.10 權重多項式基底函數法配合變數形遺忘因子之識別結果(k=3,2%雜訊) (a) case 1；(b)case 2；(c)case 3；(d)case 4
(a)
(b)
(c)
(d)
圖3.11 權重多項式基底函數法配合卡式濾波器識別結果(k=2,2%雜訊) (a)case 1；(b)case 2；(c)case 3；(d)case 4
(a)
(c)
(d)
圖3.12 權重多項式基底函數法配合卡式濾波器識別結果(k=3,2%雜訊) (a)case 1；(b)case 2；(c)case 3；(d)case 4
圖4.1 待測結構物
圖 4.2 待測結構物之位移計架設圖(NCREE 提供)
圖 4.3 待測結構物之加速度計架設圖 (NCREE 提供)
圖 4.4 待測結構物之基底剪力載重元件圖 (NCREE 提供)
(a)
(b)
(c)
圖4.5 振動台試驗之輸入與輸出。((a)破壞前、(b)地震中與(c)破壞後)
圖4.6 不同輸入下之輸出頻譜反應
圖4.7 迴歸所得之瞬時自然振動頻率
圖4.8 迴歸所得之瞬時阻尼比
圖4.9 識別所得之瞬時自然振動頻率
圖4.10 識別所得之瞬時阻尼比
(a)
圖4.11 由基底剪力與層間位移所得之遲滯迴圈:
(a) 0<t<30 秒. (b) 0<t<3.10 秒
圖4.12 進行振動台試驗之結構模型 (NCREE 提供)
圖4.13 三層樓鋼構架(Benchmark A)之照片 (NCREE 提供)
Benchmark A
圖4.14：三層樓鋼構架之感應子位置示意圖 (NCREE 提供)
圖4.15 Benchmark A 之 X 向白噪輸入及其頻譜圖(50gal)
圖4.16 白噪輸入下(50gal)Benchmark A 三樓位移輸出反應及其頻譜圖
圖4.17 Benchmark A 之 X 向白噪輸入及其頻譜圖(100gal)
圖4.18 白噪輸入下(100gal)Benchmark A 三樓位移輸出反應及其頻譜圖
圖4.19 Benchmark A 之 X 向 EL Centro 地震輸入及其頻譜圖(100gal)
圖4.20 EL Centro 地震輸入下(100gal)Benchmark A 三樓位移輸出反應及其頻譜圖
圖4.21 Benchmark A 之 X 向 EL Centro 地震輸入及其頻譜圖(200gal)
圖4.22 EL Centro 地震輸入下(200gal)Benchmark A 三樓位移輸出反應及其頻譜圖
圖4.23 Benchmark A 之 X 向 EL Centro 地震輸入及其頻譜圖(300gal)
圖4.24 EL Centro 地震輸入下(300gal)Benchmark A 三樓位移輸出反應及其頻譜圖
圖4.25 Benchmark A 之 X 向 EL Centro 地震輸入及其頻譜圖(500gal)
圖4.26 EL Centro 地震輸入下(500gal)Benchmark A 三樓位移輸出反應及其頻譜圖
(a)
(b)
(c)
圖4.27 Benchmark A 線性試驗各模態識別結果 (a) 第一模態 (b)第二模態 (c)第三模態
1st mode
2nd mode 2nd mode
3rd mode 3rd mode
1st mode
(a)
(b)
(c)
圖4.28 Benchmark A 線性與非線性試驗各模態識別結果 (a)第一模態 (b)第二模態 (c)第三模態
2nd mode 2nd mode
3rd mode 3rd mode
1st mode 1st mode