本研究建立了一套時變線性系統之識別方法，以估算時變線性系統之瞬時模態參數。
研究中採用了移動最小平方差法結合多項式基底函數架構TVARX 模型時變係數之形狀函 數，再以最小平方差法反算各組形狀函數所對應之係數。利用形狀函數及其對應之係數建 立出結構之系統矩陣，經由系統識別程序，獲得系統之瞬時模態參數如自然振動頻率、阻 尼比、以及振動模態。整個識別程序如圖5.1 所示。
本研究以單自由度時變線性系統之數值模擬驗證此識別流程之可行性，與一些現有識 別流程比較，並進行各項參數探討以確實掌握此識別方法之特性。本研究所發展方法較遞 迴最小平方差法、直接傳統基底函數展開法與權重基底函數展開法為優。遞迴最小平方差 法追蹤時變特性之能力教差，尤其當特性隨時間變化較快時或含有雜訊之情況。傳統基底 函數展開法架構TVARX 模型則須引入較高之多項式階數，但過高之多項式階數易容易於 估算過程造成病態矩陣。而以權重基底函數展開法架構TVARX 模型僅須較低之多項式階 數即可獲得準確之結果，但其必須逐步取窗計算每一時刻之模態特性，故過程很耗時。
最後此識別流程應用於國家地震工程研究中心所進行的鋼筋混凝土門型架構之振動台 試驗，驗證此識別方法能應用於實測資料上。識別所得之瞬時自然振動頻率和阻尼比能真 實反應實驗觀察所得之物理現象。
參考文獻
Charbonnier, R., Barlaud, M., Alengrin, G., and Menez, J. (1987), ‘Results on AR-modeling of nonstationary signals,’ Signal Processing, 12, pp. 143-151.
Choi, B. Y. and Bien, Z. (1989), ‘Sliding-windowed weighted recursive least-squares method for parameter estimation,’ Electron Letter, 25(20), pp. 1381-1382.
Chui, C. K. (1992) An Introduction to Wavelets. Wavelet Analysis and its Application volume 1.
Boston: Academic Press.
Cole Jr., H. A. (1971) ‘Methods and apparatus for measuring the damping characteristics of a structure,’ United State Patent No. 3620069.
Fortescue, T. R., Kershenbaum, L. S. and Ydstie, B. E. (1981) ‘Implementation of self-tuning regulators with variable forgetting factors,’ Automatica, 17(6), pp.831-835.
Gordon, C. G. (1991), ‘Generic criteria for vibration sensitive equipment,’ Vibration Control in Microelectronics, Optics and Metrology, SPIE Proceedings, 1619, pp. 71-85.
Grenier, Y. (1983), ‘Time-dependent ARMA modeling nonstationary signals,’ IEEE Trans. on Acoustics, Speech, and Signal Processing, ASSP-31, pp. 899-911.
Huang, C. S. (2001), ‘Structural identification from ambient vibration measurement using the multivariate AR model,’ Journal of Sound and Vibration, 241(3), pp. 337-359.
Huang, C. S. and Su, W. C. (2007), ‘Identification of modal parameters of a time invariant linear system by continuous wavelet transformation,’ Mechanical Systems and Signal Processing, 21(4), pp. 1642-1664.
Huang, C. S. and Yeh, C. H. (1999), ‘Some properties of randomdec signatures,’ Mechanical Systems and Signal Processing, 13(3), pp. 491-507.
Jiang, J. and Cook, R. (1992), ‘Fast parameter tracking RLS algorithm with high noise immunity,’ Electron Letter, 28(22), pp. 2042-2045.
Kalouptsidis , N., and Theodoridis, S. (1993), Adaptive System Identification and Signal Processing Algorithm, Prentice-Hall, Englewood Cliffs, NJ.
Lee, Z.-K., Wu, T.-H. and Loh, C.-H. (2003), ‘System identification on the seismic behavior of an isolated bridge,’ Earthquake Engineering and Structural Dynamics, 32, pp. 1797-1812.
Liu, G. R. (2003), Mesh Free Methods-Moving beyond the Finite Element Method, CRC Press, NY.
Ljung L. (1987), System Identification, Theory for the User, Prentice-Hall, Englewood Cliffs, NJ.
Ljung, S. and Ljung, L. (1985), ‘Error propagation properties of recursive least-squares adaptation algorithms,’ Automatica, 21, pp. 157-167.
Ljung, L., and Soderstrom, T. (1983), Theory and Practice of Recursive Identification, MIT Press, Cambridge, MA.
Lynch, J. P. (2007), ‘An overview of wireless structural health monitoring for civil structures,’
Philosophical Transactions of the Royal Society A, 365, pp. 345–372.
Marmarelis, V. Z. (1987), ‘Nonlinear and nonstationary modeling of physiological system: An overview,’ in Advanced Methods of Physiological System Modeling, edited by V. Z.
Marmarelis, Los Angeles: Biomechanical Simulations Resource, Univ. Southern California, 1, pp. 1-24.
Niedźwiecki, M. (1988), ‘Functional series modelling approach to identification of nonstationary stochastic systems,’ IEEE Transactions Automatic Control, 33, pp. 955-961.
Niedźwiecki, M. (2000), Identification of Time-Varying Processes, John Wiley & Sons, LTD, NY.
Park, D. J. and Jun, B. E. (1992), ‘Self-perturbing recursive least squares algorithm with fast tracking capability,’ Electron Letter, 28(6), pp. 558-559.
Parkum, J. E., Poulsen, N. K. and Holst, J. (1992), ‘Recursive forgetting algorithm,’
Rajan, J. J., and Rayner, P. J. W. (1996), ‘Generalized feature extraction for time-varying autoregressive models,’ IEEE Trans. on Signal Processing, 44, pp. 2498-2507.
Toplis, B. and Pasupathy, S. (1988), ‘Tracking improvements in fast RLS algorithm using a variable forgetting factor,’ IEEE Trans. on Acoustics, Speech, and Signal Processing, ASSP-36(2), pp. 206-227.
Tsatsanis, M. K. and Giannakis, G. B. (1993), ‘Time-varying system identification and model validation using wavelets,’ IEEE Transactions signal Processing, 41, pp. 3512-3523.
Watson, G. A. (1980), Approximation Theory and Numerical Methods, John Wiley & Sons, NY.
Wei, H.-L. and Billings, S. A. (2002), ‘Identification of time-varying systems using multiresolution wavelet models,’ International Journal of Systems Science, 33(15), pp.
1217-1228.
Zou, R., Wang, H., and Chon, K. H. (2003), ‘A robust time-varying identification algorithm using basis functions,’ Annals of Biomedical Engineering, 31, pp. 840-853.
黃炯憲，(1999)，「微動量測分析工具探討 (二) 時間序列法」，國家地震工程研究中心 報告NCREE–99–018。
表 2.1 CC2430/31 感應模組規格表
表2.2 SPC51 規格介紹
型號 SPC51
頻道數 16
A/D 轉換 16bit
最大輸出電壓 ±10V
取樣頻率 可調式(10,20,50,100,200,500,1000Hz) 放大倍率(Gain) 1、2、10、100
啟動方式 手動、自動、時間設定
高通濾波器 0.1Hz 或 1Hz
低通濾波器 1
3 取樣頻率
紀錄長度 可調式(最多 99999999 點/頻道)
記憶體 硬碟9.34G
表2.3 速度計規格
型號 VSE15D
頻率範圍 0.1~70Hz 量測範圍 ±10cm/sec(kine)
靈敏度 1V/kine 或 10V/kine
最大輸出電壓 ±10V
圖2.1 工程二館簡易示意圖
圖2.2 無線監測系統-集錄器
圖2.3 無線監測系統-CC2430/31 感應模組
圖2.4 SPC51 集錄系統
圖2.5 VSE15D 速度計
圖2.6 感應計擺設方位
圖2.7 現地量測照片
0 2 4 6 8 10 12 14 16 18 20
0 2 4 6 8 10 12 14 16 18 20
0 2 4 6 8 10 12 14 16 18 20
0 2 4 6 8 10 12 14 16 18 20
圖3.1 指數型權重函數。
圖4.1 輸入地震歷時及其頻譜反應
圖4.2 各種時變系統之輸出歷時及其頻譜反應
(a)
(b)
(c)
(d)
圖4.3 以不同支撐參數與節點數識別之誤差：(a) case 1；(b) case 2；(c) case 3；(d) case 4
(a)
(b)
(c)
(d)
圖4.4 以不同多項式階數與節點數識別之誤差：(a) case 1；(b) case 2；(c) case 3；(d) case 4
圖4.5 識別結果與理論值比較
case 1
case 1
case 2 case 2
case 3 case 3
case 4
case 4
圖4.6 不同 TVARX (I,J)下之 AIC 值 ( case 2)
圖4.7 不同 TVARX (I,J)下之 FPE 值 (case 2)
圖4.8 各組支撐參數於不同模型階數下之識別誤差 (case 2，含雜訊)。
圖4.9 於訊號中加入噪訊比 5%之雜訊所得識別結果 (case 2)
圖 4.10 與遞迴識別法識別 case 1 之結果比較
圖 4.11 與遞迴識別法識別 case 2 之結果比較
圖 4.12 利用傳統基底函數展開法與權重基底函數展開法識別 case 2 之 f n and
圖 4.13 利用權重基底函數展開法識別 case 2 之 f and n
(含雜訊)圖4.14 待測結構物示意圖
45 15
17
40
100
40
44
112.5 20
Load Cells
a
g36 160
20
27
Shaking Table
( Front View ) ( Side View )
(a)
(b)
(c)
圖4.15 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4.16 不同輸入下之輸出頻譜反應
(a)
(b)
(c)
圖4.17 不同 TVARX (I,J)對應之 AIC 值：
(a)破壞前、(b)地震中與(c)破壞後。
(a)
(b)
(c)
圖4.18 不同 TVARX (I,J)對應之 FPE 值：
圖4.19 由實驗資料識別所得之瞬時模態參數
(a)
(b)
圖4.20 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5.1 線性時變系統之識別流程 Start
Input measure responses y(t) and input force f(t)
Estimate coefficient matrices through a least squares approach
Estimate instantaneous modal parameters from the true value and shape function
End
Construct shape function ( p,i
t ~p,j
t~ ,
Θ
Φ
) for time varying coefficients (Φ
i
t ,Θ
j t ) via amoving least squares approach Choose basis function (polynomial or Haar wavelet packet) and control parameters: number
of basis (N or p M ), support length of weight l function (d ) and number of nodal points(m l ) n
Choose order of TVARX model
Are the identified results convergent?
Yes
No
Increase order of TVARX Choose other control parameters