## 國立臺灣大學工學院土木工程研究所 碩士論文

### Department of Civil Engineering College of Engineering National Taiwan University

### Master Thesis

## 協方差型隨機子空間識別法之應用

### Application of Covariance Driven Stochastic Subspace Identification Method

## 劉奕成 Yi-Cheng Liu

## 指導教授：羅俊雄 博士 Advisor: Chin-Hsiung Loh, Ph.D.

### 中華民國 100 年 6 月

### June, 2011

ii

**Acknowledgement **

本研究得以順利完成，首先感謝恩師 羅俊雄教授非常熱忱與用心的指 導教誨，讓學生能從中學習到做研究的精神、目的與態度，並且提供了最完善的 資源與諮詢，在此表達由衷的感謝。另外，承蒙 莊哲男教授在口試期間提綱挈 領的指點與寶貴經驗的傳授，讓學生能在系統識別領域上獲益良多，除此之外，

也感謝 田堯彰教授與鍾立來教授在論文口試期間提供寶貴建議，使本研究更加 充實與完整，在此特別致上感謝。

兩年研究期間，感謝書賢學長、丁友學長、健煌學長、恭君學長、謝恭學長、

毓文學姐與佳慧學姐在兩年中的照顧，在研究上給予的解惑與協助，並非常有耐 心與包容常常被打擾。在求學的兩年之中，也感謝嘉明、宜錚與世銘學長給予的 鼓勵，同窗黃今陽、葉少華、李榮桓、吳豐名、李育謙的相互扶持與陪伴。赴美 國開會的過程中，也要特別感謝老師、師母、Ken、謝恭學長與艾倫學姐的關心與 照顧。

來台灣的兩年，感謝哥斯大黎加大學所提供的獎學金，但因為你們看不懂中 文所以只好寫西文： reconocimiento y gran agradecimiento a Universidad de Costa Rica por la beca otorgada y todo el apoyo que me ha brindado permitiéndome pasar estos dos años en Taiwán y tener la oportunidad de concentrarse en el estudio y en la investigación.

最後，要特別感謝在哥斯大黎加的奶奶、爸媽、姊姊與朋友，感謝您們的支 持、包容與承擔，也感謝在台灣一直陪伴著我的親戚朋友，謝謝您們的支持與體 貼，讓我能無後顧之憂地專心學習與研究，謹以此文獻與您們以表我無限的感恩。

### 摘要 摘要 摘要 摘要

本研究目的是探討隨機子空間識別法（Stochastic Subspace Identification, SSI）

在只有結構微震反應的量測下，於土木結構系統識別及損壞診斷上的應用範疇。

在離線分析的應用上，可將於不同矩陣維度識別出之系統極點（system poles）繪 製成穩定圖，以達正確識別結構震態的目的。在此研究的前半段，首先將對隨機 子空間識別法搭配穩定圖的識別效果做研究，在不同情況諸如：訊號之雜訊、非 線性、時變性與間隔緊密頻率等因素之甘擾下，比較各種隨機子空間識別法對此 等甘擾因素之敏感度。接下來，協方差型隨機子空間識別法（Covariance driven Stochastic Subspace Identification, SSI-COV）將應用在廣州電視塔（Canton Tower）

的系統識別工作，其為一座大型挑高細長結構，並為結構健康監測之標杆問題。

除此之外，奇異譜分析法（Singular Spectrum Analysis, SSA）將以「前置子空間濾 波器」的概念與協方差型隨機子空間識別法結合，名為「SSA-SSI-COV」識別法，

除了能有效提昇資料解析能力，更提供一個能決定系統識別之最佳系統維度的做 法。

研究的第二部份是針對系統震態參數之線上識別與損壞診斷技巧的開發，以 遞 迴 式 協 方 差 型 隨 機 子 空 間 識 別 法 （ Recursive Covariance-driven Stochastic Subspace identification, RSSI-COV）為主體，並搭配延伸工具變項─投影近似子空 間追蹤演算法（Extended Instrumental Variable – Projection Approximation Subspace Tracking algorithm, EIV-PAST）達成線上更新子空間的目地。另外，一個可供線上 作業之子空間前置濾波器─「遞迴式奇異譜分析法（recursive Singular Spectrum, rSSA）」的開發與搭配，可有效減低雜訊對實地結構識別品質之影響，提昇線上 分析的穩定性。此兩種子空間技術將透過時變性系統之數值模擬與實地試驗數據 得到驗証，並從中取得可靠的識別模型控制參數。最後，它們將被應用在三個結 構震態追縱的實驗上：（1）三層樓鋼構試體瞬時勁度折減之震動台實驗，（2）

單層雙跨鋼筋混凝土結構之震動台試驗，此兩者皆以結構受到地震作用下之輸出 反應做線上震態識別。最後，（3）橋樑沖刷實驗之損壞診斷與預警之應用。

關鍵詞 關鍵詞 關鍵詞

關鍵詞：：：： 隨機子空間識別、協方差型、系統識別、結構健康監測、遞歸式隨機子 空間識別、遞歸式奇異譜分析、廣州電視塔

iv

**Abstract **

In this research the application of output-only system identification technique known as Stochastic Subspace Identification (SSI) algorithms in civil structures is carried out. With the aim of finding accurate modal parameters of the structure in off-line analysis, a stabilization diagram is constructed by plotting the identified poles of the system with increasing the size of data matrix. A sensitivity study of the implementation of SSI through stabilization diagram is firstly carried out, different scenarios such as noise effect, nonlinearity, time-varying systems and closely-spaced frequencies are considered. Comparison between different SSI approaches was also discussed. In the following, the identification task of a real large scale structure: Canton Tower, a benchmark problem for structural health monitoring of high-rise slender structures is carried out, for which the capacity of Covariance-driven SSI algorithm (SSI-COV) will be demonstrated. The introduction of a subspace preprocessing algorithm known as Singular Spectrum Analysis (SSA) can greatly enhance the identification capacity, which in conjunction with SSI-COV is called the SSA-SSI-COV method, it also allows the determination of the best system order.

The objective of the second part of this research is to develop on-line system parameter estimation and damage detection technique through Recursive Covariance-driven Stochastic Subspace identification (RSSI-COV) approach. The Extended Instrumental Variable version of Projection Approximation Subspace Tracking algorithm (EIV-PAST) is taking charge of the system-related subspace updating task. To further reduce the noise corruption in field experiments, the data pre-processing technique called recursive Singular Spectrum Analysis technique (rSSA)

is developed to remove the noise contaminant measurements, so as to enhance the stability of data analysis. Through simulation study as well as the experimental research, both RSSI-COV and rSSA-SSI-COV method are applied to identify the dynamic behavior of systems with time-varying characteristics, the reliable control parameters for the model are examined. Finally, these algorithms are applied to track the evolution of modal parameters for: (1) shaking table test of a 3-story steel frame with instantaneous stiffness reduction. (2) Shaking table test of a 1-story 2-bay reinforced concrete frame, both under earthquake excitation, and at last, (3) damage detection and early warning of an experimental steel bridge under continuous scour.

**Keywords: Stochastic Subspace Identification, Covariance Driven, System **
Identification, Structural Health Monitoring, Recursive Stochastic Subspace
Identification, Recursive Singular Spectrum Analysis, Canton Tower

vi

**Contents **

口試委員審定書 口試委員審定書 口試委員審定書

**口試委員審定書……….……….. ** **i **

**Acknowledgement ……….……….. ** **ii **

**Abstract (in Chinese) ……….……….. iii **

**Abstract (in English) ……… iv **

**Contents ……… vi **

**Table List ……….. x **

**Figure List ……… xi **

**Chapter 1 Introduction ……… 1 **

1.1 Background ………. 1

1.2 Research Objectives………. 6

**Chapter 2 Stochastic Subspace Identification (SSI) Methods….……….. 9 **

2.1 Introduction ………..…………... 9

2.2 Models of vibrating structures ………. 9

2.2.1 Continuous-time state-space model ……… 9

2.2.2 Discrete-time state-space model ………. 14

2.2.3 Stochastic state-space model ……….. 15

2.3 Covariance-driven Stochastic Subspace Identification (SSI-COV) ……… 17

2.4 Data-driven Stochastic Subspace Identification (SSI-DATA) ……… 19

2.5 Singular Spectrum Analysis (SSA) ………. 23

2.6 Pole discrimination: the stabilization diagram ……… 24

2.6.1 Alternatives to build the stabilization diagram ………... 24 2.6.2 Comparison of stabilization diagram alternatives and influence of the 29

model order determination ………..

2.6.2.1 Simulation example: 6-DOF simulation study ……… 29

2.6.2.2 Experimental example: identification of a 6-story steel frame from shaking table test ……….. 32

**Chapter 3 Simulation study of SSI-based algorithms ………... 34 **

3.1 Noise effect in the identification of modal parameters ……… 34

3.1.1 Addition of a spatially white noise (from 50% to 200%) ………... 34

3.1.2 Addition of a white noise correlated with output (violation to SSI assumption) ………. 35

3.2 Nonlinearity in the signal ……… 36

3.3 Closely-spaced frequencies blended with signals of a time-varying system 38 3.4 Preprocessing with SSA and noise effect in closely-spaced frequencies … 41 3.4.1 Sinusoidal waves ………. 41

3.4.2 Response of a 2-DOF system subjected to white noise excitation ……. 42

**Chapter 4 Application of SSI to the identification of Canton Tower………... 46 **

4.1 Frequency Domain Decomposition (FDD)……….. 47

4.2 SSI-COV and SSI-DATA ……… 47

4.3 SSA-SSI-COV ………. 49

4.3.1 Implementation ………... 49

4.3.2 Canto Tower identification through SSA-SSI-COV ……….. 51

4.3.3 Canto Tower identification through SSA-SSI-DATA ……….... 53

4.4 Low pass filter with SSI-COV ………. 54

4.5 Improve the identification convergence speed with decimation………….. 55

viii

**Chapter 5 Recursive Stochastic Subspace Identification algorithms ……….. 58 **

5.1 Recursive Covariance-driven Stochastic Subspace Identification algorithm (RSSI-COV) ……… 58

5.1.1 Projection Approximation Subspace Tracking (PAST) ……….. 61

5.1.2 Instrumental Variable Projection Approximation Subspace Tracking (IV-PAST) ……….. 64

5.1.3 Extended Instrumental Variable Projection Approximation Subspace Tracking (EIV-PAST) ………. 65

5.1.4 Adaptation of EIV-PAST to RSSI-COV ……… 66

5.2 Recursive Singular Spectrum Analysis (RSSA) ……….. 71

**Chapter 6 Simulation study of RSSI-COV and rSSA-SSI-COV ……….. 77 **

6.1 Implementation of the RSSI-COV and rSSA-SSI-COV algorithm ………. 77

6.2 Simulation study 1: time invariant 6-DOF system ……….. 79

6.3 Simulation study 2: time varying 6-DOF system with sudden stiffness reduction ……….. 80

6.4 Simulation study 3: time-varying 6-DOF system with gradual stiffness reduction ……….. 82

**Chapter 7 Application of recursive SSI algorithms in damage detection and **
**early warning ……….. 87 **

7.1 Application 1: shaking table test of a 3-story steel structure with instantaneous stiffness reduction ………. 87

7.1.1 White noise base excitation ……….. 88

7.1.2 El Centro 100 gal ……….. 89

7.1.3 TCU082 100 gal ………... 93

7.2 Application 2: shaking table test of a 1-story 2-bay RC frame …………... 95

7.3 Application 3: bridge pier scouring experiment ……….. 99

7.3.1 Bridge pier imminent settlement indicator: modal frequency drop …… 100

7.3.1.1 Test conducted in 2011/01/19 with full measurements …………... 100

7.3.1.2 Test conducted in 2011/01/24 with full measurements …………... 104

7.3.1.3 Test conducted in 2011/01/26 with full measurements …………... 105

7.3.1.4 Test conducted in 2011/03/29 with full measurements …………... 106

7.3.2 Damage location indicator: mode shape slope ratio ………... 108

7.3.2.1 Mode shape slope ratio for test conducted in 2011/01/19 ………... 109

7.3.2.2 Mode shape slope ratio for test conducted in 2011/01/26 ………... 111

7.3.2.3 Mode shape slope ratio for test conducted in 2011/03/29 ………... 112

7.3.3 Novelty Index ………. 112

**Chapter 8 Conclusions ………. 115 **

8.1 Research conclusions ………... 115

8.2 Recommendations for future work ……….. 116

**References ………. 121 **

**Appendix A: Frequency Domain Decomposition (FDD) ………. 128 **

**Appendix B: Prediction Error Method through Stochastic Subspace Identification **
(PEM/SSI) ……….………. 130

**Appendix C: Novelty index through Kalman-Filter-based prediction error ……… 135 **

x

**Table List **

**Table 2-1 Comparison of identification results of SSI-COV……….. 139 **
**Table 3-1 Comparison of the identified frequencies of the 6 DOF simulation **

**example ………... 140 **
**Table 3-2 Comparison of the identified damping ratios of the 6 DOF simulation **

**example ……...………..……….. 140 **
**Table 3-3 Different set of frequency and k***3*** values used in the modeling ………. 140 **
**Table 3-4 Comparison of identification results for nonlinear signals ……… 141 **
**Table 3-5 Comparing identification results of two close frequencies with signal **

**generated by ambient vibrations ………..………... 141 **
**Table 4-1 Comparison of the identified modal parameters of Canton Tower…… 142 **
**Table 4-2 Comparison of the identified modal parameters of Canton Tower for **

**different sampling rates………... 142 **
**Table 6-1 Sudden reduction of modal frequencies due to loss of stiffness ……… 143 **
**Table 6-2 Parameters for rSSA and RSSI-COV ……… 143 **
**Table 7-1 Shaking table test analyzed by RSSI-COV and rSSA-SSI-COV …….. 143 **
**Table 7-2 Identified modal frequencies and damping ratios for case AB and NB **

**from white noise excitation ………. 144 **
**Table 7-3 Outlier analysis from the damage detection of a 1-story 2-bay RC frame ** 144
**Table 7-4 Bridge scour test schedule and arrangement ………. 144 **
**Table 7-5 Specification of VSE-15D velocity sensor AS-2000 accelerometer….. 144 **
**Table 7-6 rSSA-SSI-COV and RSSI-COV model parameter for 2011/01/19 test. 145 **
**Table 7-7 rSSA-SSI-COV model parameter for 2011/01/24 test ……….. 145 **
**Table 7-8 rSSA-SSI-COV and RSSI-COV model parameter for 2011/01/26 test. 145 **
**Table 7-9 rSSA-SSI-COV and RSSI-COV model parameter for 2011/03/29 test. 145 **
**Table 7-10 Selected correlation coefficient R for mode discrimination ………… 145 **

**Figure List **

**Figure 2-1 ** Simulated velocity response at 6^{th} DOF, system subjected to white
**noise excitation .………. 146 **
**Figure 2-2 ** Comparison between the 2^{nd} and 3^{rd}** version of stabilization diagram 146 **
**Figure 2-3 ** Simulated acceleration measurement at 6^{th} DOF. The direct

transmitted external acceleration serves as the measurement noise.

* A trend can be observed within the randomness caused by the Du*k

**term .……… 146 **
**Figure 2-4 ** Effects of noise and insufficient Toeplitz matrix columns in the use

**of rectangular Toeplitz matrix ……… 147 **
**Figure 2-5 ** Effects of noise in the stabilization diagram by square Toeplitz

**matrix ………. 147 **
**Figure 2-6 ** Photo of the 6-story structure and its instrumentation. AX are the

**accelerometers………. 148 **
**Figure 2-7 ** Plot of part of the acceleration response measured at 6^{th}** floor.…….. 148 **
**Figure 2-8 ** Singular values determined from SVD of square Toeplitz matrix.

As the matrix size increases, the gap between the first 12 and the
**rest become more clear.………... 149 **
**Figure 2-9 ** Stabilization diagram made using a) square Toeplitz matrix and, b)

**rectangular Toeplitz matrix. ………... 149 **
**Figure 2-10 Square Toeplitz matrix: a) Underestimation, b) Overestimation of **

the system order. Rectangular Toeplitz matrix: c) Underestimation,
**d) Overestimation of the system order. ……….. 149 **
**Figure 3-1 ** Measurement of 6^{th}** DOF with added 100% white noise.…………... 150 **
**Figure 3-2 ** **Stabilization diagram for added 100% white noise.……… 151 **
**Figure 3-3 ** **Stabilization diagram for added 200% white noise.……… 151 **
**Figure 3-4 ** **Stabilization diagram for added noise correlated with output.……… 151 **

xii

**Figure 3-5 ** Iterative procedure to find the secant stiffness and next- step
**displacement.………... 151 **
**Figure 3-6 ** Comparison between linear and nonlinear acceleration response.

*Frequency: 0.1 Hz. k**3*** : -0.26k………. 152 **
**Figure 3-7 ** **Comparison between nonlinear and linear restoring force………….. 152 **
**Figure 3-8 ** Stabilization diagram for 0.1 Hz, k3 : -0.26k. a) SSI-COV, b)

SSI-DATA, c) PEM/SSI. For 1 Hz, k3 : -80k. d) SSI-COV. e)
SSI-DATA. For 10 Hz, k_{3}** : -7300k. f) SSI-COV. g) SSI-DATA…... ** 152
**Figure 3-9 ** Generate signal with two closely-spaced frequency and a

**time-varying frequency………... 153 **
**Figure 3-10 Stabilization diagram built with SSI-COV for different system **

**orders.……….. 153 **
**Figure 3-11 Stabilization constructed with SSI-DATA with different system **

**orders.……….. 154 **
**Figure 3-12 Stabilization constructed with PEM/SSI with different system **

**orders.……….. 154 **
**Figure 3-13 Stabilization diagram obtained by applying directly SVD to Hankel **

**data matrixfor different system orders……… 155 **
**Figure 3-14 Stabilization diagram for signal with 5% noise added……… 155 **
**Figure 3-15 7.99 Hz and 8.00 Hz sine wave with added 10% noise……….. 155 **
**Figure 3-16 Stabilization diagram for signal with added 10% noise, 7.99 Hz and **

**8.00 Hz……… 156 **
**Figure 3-17 Variation of singular values with different dimensions of Hankel **

matrix……….. 156

**Figure 3-18 Stabilization diagram constructed using SSA-SSI-COV with a) 10 **
Singular values from 200x5000 Hankel matrix, and b) 4 Singular
**values from 1000x3000 Hankel matrix, SSA……… ** 156

**Figure 3-19 Simulated system response. a) Noise free acceleration **
**measurements, b) with direct transmission of input acceleration…... ** 157
**Figure 3-20 Stabilization diagram built using a) SSI-COV and b) SSI-DATA, **

10000 columns is used in SSI-DATA and 10000 points were used
**in covariance for SSI-COV.……… 157 **
**Figure 3-21 Stabilization diagram for noisy acceleration measurements, a) **

SSI-COV and b) SSI-DATA. 10000 columns is used in SSI-DATA
**and 10000 points were used in covariance for SSI-COV……… 157 **
**Figure 3-22 Variation of singular values in a 800 block rows by 3000 columns **

**Hankel matrix, applying SSA……….. 158 **
**Figure 3-23 Comparison of the reconstructed signal with the noise free **

**acceleration measurements, 8 Singular Values chosen from SSA….. 158 **
**Figure 3-24 Stabilization diagram built with SSI-COV for different singular **

**values extracted from SSA, system order fixed to 4………... 158 **
**Figure 4-1 ** Locations of accelerometers in the Canton Tower. A floor section

**shows the position of accelerometers………. ……… 159 **
**Figure 4-2 ** Acceleration measurements at the first minutes of the record for a)

1^{st} sensor and b) 20^{th}** sensor……… 159 **
**Figure 4-3 ** Plot of Frequency Domain Decomposition of Canton Tower

**acceleration data, a) whole picture, b) from 0 to 3 Hz……… 160 **
**Figure 4-4 ** Singular Value Decomposition to the square Toeplitz matrix of 300

**block rows………... 160 **
**Figure 4-5 ** **Stabilization diagram constructed by SSI-COV and SSI-DATA…… 161 **
**Figure 4-6 ** Comparing identified frequencies, damping ratios and complex

mode shapes from mode 1 to 8, identified by SSI-COV and
**SSI-DATA………... 162 **
**Figure 4-7 ** Comparing identified frequencies, damping ratios and complex

mode shapes from mode 9 to 18, identified by SSI-COV and
**SSI-DATA………... 163 **

xiv

**Figure 4-8 ** Singular value decomposition of Hankel matrix with 340 block
**rows and 15000 columns………. 164 **
**Figure 4-9 ** Plot the distribution of SV from SSI-COV analysis, data from: a)

312 SV in SSA, b) 134SV in SSA, c) 120 SV in SSA, d) 95 SV in
**SSA, e) 66 SV in SSA, f) 48 SV in SSA………. 164 **
**Figure 4-10 Comparison of stability diagram made with SSA-SSI-COV, 0~1 **

**Hz……… 165 **
**Figure 4-11 Fourier Spectrum of the reconstructed signal with SSA 95 SV…….. 166 **
**Figure 4-12 Fourier Spectrum of the reconstructed signal with SSA 136 SV…… 167 **
**Figure 4-13 Comparison of stability diagram made with SSA-SSI-COV, a) 136 **

**SV and b) 95 SV, 1~5 Hz……… 168 **
**Figure 4-14 Comparing identified frequencies, damping ratios and complex **

mode shapes from mode 1 to 8, identified by SSA-SSI-COV using
**a) 136 SV and b) 95 SV……….. 169 **
**Figure 4-15 Comparing identified frequencies, damping ratios and complex **

mode shapes from mode 9 to 18, identified by SSA-SSI-COV using
**a) 136 SV and b) 95 SV……….. 170 **
**Figure 4-16 Stabilization diagram constructed by SSA-SSI-Data using 154 SV, **

**Order 90 a) 0~1 Hz, b) 1~5 Hz………... 171 **
**Figure 4-17 Fourier Spectrum of acceleration measurement at sensor No. 19…... 171 **
**Figure 4-18 Frequency response function of Butterworth filter. Order: 10, cutoff **

**frequency: 5 Hz………... 172 **
**Figure 4-19 Comparison of the low-pass filtered signal with original signal for **

**various sensors……… 172 **
**Figure 4-20 Comparison between stabilization diagram constructed with **

SSI-COV for a) 0~1Hz, c) 1~5 Hz, and with SSI-DATA for b) 0~1
**Hz, d) 1~5 Hz……….. 173 **
**Figure 4-21 Identified mode shapes with SSI-COV, mode 1~6. ……….. ** 174

**Figure 4-22 Identified mode shapes with SSI-COV, mode 7~12 ……….. 175 **
**Figure 4-23 Identified mode shapes with SSI-COV, mode 13~18………. 176 **
**Figure 4-24 Identified mode shapes with SSI-COV, mode 19~24………. 177 **
**Figure 4-25 Comparison of stabilization diagram for 0~1 Hz, signal **

downsampled to different rates………... 178
**Figure 4-26 Comparison of stabilization diagram for 1~6 Hz, signal **

downsampled to different rates………... 179
**Figure 4-27 Comparison of complex mode shapes for the first 10 modes, signal **

downsampled to different rates………... 180
**Figure 6-1 ** **Flow chart of the implementation of RSSI-COV……… 181 **
**Figure 6-2 ** Frequency tracking by RSSI-COV for time-invariant system. a)

moving window length = 1500 points. b) moving window length =
**3000 points……….. 181 **
**Figure 6-3 ** Damping ratio tracking by RSSI-COV for time-invariant system. a)

moving window length = 1500 points. b) moving window length =
**3000 points……….. 182 **
**Figure 6-4 ** Frequency tracking by RSSI-COV for time-invariant system,

*adding noise correlated with output. a) number of block rows i = 50. *

**b) i = 100 block rows………... 182 ****Figure 6-5 ** Damping ratio tracking by RSSI-COV for time-invariant system,

*adding noise correlated with output. a) number of block rows i = 50. *

**b) i = 100 block rows………... 182 ****Figure 6-6 ** Frequency tracking by RSSI-COV for a 6-DOF system with sudden

*stiffness reduction. a) moving window length L = 2500 points. b) L *

**= 4000 points………... 183 **
**Figure 6-7 ** Damping ratio tracking by RSSI-COV for a 6-DOF system with

*sudden stiffness reduction. a) moving window length L = 2500 *
* points. b) L = 4000 points………... * 183

xvi

**Figure 6-8 ** a) Modal frequency, and b) damping ratio tracking by RSSI-COV
for a 6-DOF system with sudden stiffness reduction. System order:

**16………. 183 **
**Figure 6-9 ** Examples of mode shapes computed by RSSI-COV for a 6-DOF

system with sudden stiffness reduction. a) at point 4000 (100%

stiffness), b) at point 12000 (75% stiffness), c) at point 15000 (50%

**stiffness), d) at point 19000 (25% stiffness) ……….. 184 **
**Figure 6-10 Frequency tracking by RSSI-COV for addtion of a noise correlated **

*with output. a) Order 12, 150 number of block rows i. b) i=120, *
**Order 18………... 184 **
**Figure 6-11 Frequency tracking by RSSI-COV for a 6-DOF system with slow **

*stiffness reduction. Noise free. a) moving window length L = 2500 *
*points. b) L = 4000 points. c) L = 5000 points, Order = 12. d) L = *
**5000 points, Order = 18………... 185 **
**Figure 6-12 Frequency tracking by RSSI-COV considering the time-varying **

*effect. a) number of block rows i = 100, Order 12, b) i = 130, Order *
* 12, c) i = 130, Order 18, d) i = 130, Order = 24……… * 185

**Figure 6-13 Frequency tracking by RSSI-COV for a 6-DOF system with slow**

*stiffness reduction. Noise correlated with output. a) i = 70, b) i = *
**120………... 186 **
**Figure 6-14 Singular spectrum in rSSA step……….. 186 **
**Figure 6-15 Singular spectrum in SSI-COV step, for different combinations of **

**the number of SV in rSSA step and the moving window length L’.... 187 ****Figure 6-16 Frequency tracking by rSSA-SSI-COV. Noise correlated with **

**output. Comparison of the 4 cases……….. 188 **
**Figure 6-17 Compring the recursive frequency tracking by a) rSSA-SSI-COV, **

**system order 16 and b) RSSI-COV, order 30……….. 188 **
**Figure 7-1 ** 3-story steel frame with extra stiffener and lock-up system in the 1^{st}

**story………. 189 **
**Figure 7-2 ** **Singular spectrum obtained by SSI-COV………... 189 **

**Figure 7-3 ** **Stabilization diagram for pole discrimination……… ** 189
**Figure 7-4 ** Three dimensional mode shapes before and after removing the

**brace……… 190 **
**Figure 7-5 ** Additionally identified three dimensional mode shapes after

**removing the brace……….. 191 **
**Figure 7-6 ** Recursive identification of modal frequencies for white noise

**excitation, a) AB order 6, b) AB order 16……….. ** 191
**Figure 7-6 ** Recursive identification of modal frequencies for white noise

**excitation, c) NB order 6, d) NB order 20……….. ** 192
**Figure 7-7 ** Recursive identification of modal frequencies for El Centro

**earthquake, with no instantaneous stiffener release……… 192 **
**Figure 7-8 ** Short Time Fourier Transform for steel frame subjected to El

**Centro earthquake, with no instantaneous stiffener release………… 193 **
**Figure 7-9 ** Recursive identification of modal frequencies for El Centro

**earthquake. Siffener released at 14 and 29 seconds……… 193 **
**Figure 7-9 ** Recursive identification of modal frequencies for El Centro

**earthquake. Siffener released at 14 and 29 seconds……… 194 **
**Figure 7-10 Short Time Fourier Transform for steel frame subjected to for El **

**Centro earthquake, instantaneous stiffener release at 14 and 29 s... 194 **
**Figure 7-11 Singular spectrum for different time segments………... 194 **
**Figure 7-12 Three dimensional mode shapes identified with system order equals **

**to 6, case where the brace is removed at 14 seconds……….. 195 **
**Figure 7-13 Three dimensional torsion and coupled mode shapes identified with **

system order 16, case where the brace is removed at 29 seconds,
**comparing with the corresponding offline identified modes……….. 196 **
**Figure 7-14 Recursive identification of modal frequencies for TCU082 **

**earthquake. Siffener released at 38 seconds……… 197 **

xviii

**Figure 7-15 Recursive identification of modal frequencies for TCU082 **
**earthquake. Siffener released at 52 seconds……… 197 **
**Figure 7-16 Dimensions and the design detail of 1-story 2-bay RC frame……… 197 **
**Figure 7-17 Installation and instrumentation of the RC frame………... 198 **
**Figure 7-18 Arrangement of the series RCF6 shaking table test……… 198 **
**Figure 7-19 Frequency traced by RSSI-COV for the RCF6 frame subjected to **

**series of TCU082 earthquake……….. 198 **
**Figure 7-20 Frequency traced by RSSI-COV for the RCF6 frame subjected to **

**30 gal white noise excitation at different damage state……….. 199 **
**Figure 7-21 Singular spectrum obtained from the data points between 25 and 40 **

**seconds……… 199 **
**Figure 7-22 Frequency traced by RSSI-COV for the RCF6 frame subjected to **

**series of TCU082 earthquake. System order increased to 4………... 200 **
**Figure 7-23 The first 35 seconds of 600 gal TCU082 earthquake……….. 200 **
**Figure 7-24 Kalman filter prediction of the structure response under WN1 **

**excitation………. 201 **
**Figure 7-25 Kalman filter prediction of the structure response under WN2 **

**excitation………. 201 **
**Figure 7-26 Mahalanobis and Euclidean norm of the Kalman filter prediction **

**error………. 202 **
**Figure 7-27 The relationship between outlier analysis (Mahalanobis norm and **

Euclidean norm) with respect to the identified system natural
**frequency………. 202 **
**Figure 7-28 (a) Bridge configuration and sensors location (b) Field setup of **

**the bridge (c) After concluded the scouring experiment………. 203 **
**Figure 7-29 Singular spectrum for different choices of singular values in rSSA... 203 **
**Figure 7-30 Variation of bridge modal frequencies traced by a) rSSA-SSI-COV **

**and b) RSSI-COV……… 204 **

**Figure 7-31 Application of stability criterion to the time-frequency plot of **
**bridge modal frequencies, test conducted in 2011/01/24……… 205 **
**Figure 7-32 Scouring depth for 3 piers, test conducted in 2011/01/19………….. ** 205
**Figure 7-33 Singular spectrum for different choices of singular values in rSSA... 205 **
**Figure 7-34 Evolution of bridge modal frequencies traced by rSSA-SSI-COV **

**with applied stability criterion, test conducted in 2011/01/24……… 206 **
**Figure 7-35 Singular spectrums: a) rSSA, b) RSSI-COV for the subspace order **

**with 25 SV………... 206 **
**Figure 7-36 Evolution of bridge modal frequencies traced by rSSA-SSI-COV **

**with applied stability criterion, test conducted in 2011/01/26……… 207 **
**Figure 7-37 Singular spectrum for different choices of singular values in rSSA... 207 **
**Figure 7-38 Evolution of bridge modal frequencies traced by both a) RSSI-COV **

and b) rSSA-SSI-COV, applying stability criterion, test conducted
**in 2011/03/29 measured by accelerometers……… 208 **
**Figure 7-39 Zoom in the evolution of bridge modal frequencies between 4500 **

and 5500 seconds, test conducted in 2011/03/29 measured by

**accelerometers……… ** 209

**Figure 7-40 Identified 1**^{st} mode shapes from two time instants. Test in
**2011/01/19………... 209 **
**Figure 7-41 Examples of identified 2**^{nd} mode shapes from two time instants.

**Test in 2011/01/19……….. ** 210

**Figure 7-42 1**^{st} mode shape slope ratio for a) smoothed mode shapes.

**2011/01/19 test ………... 210 **
**Figure 7-42 1**^{st} mode shape slope ratio for b) non-smoothed mode shapes.

**2011/01/19 test……… 211 **
**Figure 7-43 1**^{st} mode shape slope ratio for a) smoothed and b) non-smoothed

**mode shapes. 2011/01/19 test……….. 211 **

xx

**Figure 7-44 Examples of identified mode shapes from two time instants. Test in **
**2011/01/26 ……….. 212 **
**Figure 7-45 1**^{st} mode shape slope ratio for a) smoothed mode shapes.

**2011/01/26 test……… 212 **
**Figure 7-45 1**^{st} mode shape slope ratio for b) non-smoothed mode shapes.

**2011/01/26 test ………... 213 **
**Figure 7-46 2**^{nd} mode shape slope ratio for a) smoothed and b) non-smoothed

**mode shapes. 2011/01/26 test……….. 213 **
**Figure 7-47 1**^{st} mode shape slope ratio for a) smoothed and b) non-smoothed

**mode shapes. 2011/03/29 test……….. 214 **
**Figure 7-48 2**^{nd} mode shape slope ratio for b) non-smoothed mode shapes.

**2011/03/29 test ………... 215 **
**Figure 7-49 3**^{rd} mode shape slope ratio for a) non-smoothed, R=0.90; b)

non-smoothed, R=0.70; and c) smoothed, R=0.70 mode shapes.

**2011/03/29 test ………... 216 **
**Figure 7-50 Outlier analysis for a) Euclidean norm and, b) Mahalanobis norm… 217 **
**Figure 7-51 Error mean for a) Euclidean norm and, b) Mahalanobis norm……... 217 **
**Figure 7-52 Error standard deviation for a) Euclidean norm and, b) Mahalanobis **

**norm.………...…… 218 **
**Figure 7-53 Error RMS mean per sensor………...……. 219 **
**Figure 7-54 Error RMS standard deviation per sensor………...… 220 **

**Chapter 1 ** **Introduction **

**1.1 ** **Background **

Structural health monitoring and damage detection in civil infrastructures is an issue that has attracted much attention in the last decades, a dense research work was carried out trying to prevent disasters caused by the agedness, deterioration and damage in structures. In recent years, there are painful example like the sudden collapse of I-35W Mississippi River Bridge on August 1, 2007, in the United States, with 13 dead and 144 injured as the victims; the collapse of Kao-Ping Bridge (高屏大橋) in Taiwan on August 27, 2000, with 30 injured, and the collapse of Ho-Feng Bridge (后豐大橋) in Taiwan on September 14, 2008, with 6 dead; the last two has occurd during the Typhoon struck and caused by the bridge pier scouring.

The raise in the safety concern on the civil infrastructures and the need of strategies and methods able to detect damage from the large scale civil structural systems and hence to make early warnings, is the reason that explains the intensive research activity in this challenging field over the last years.

The vibration-based damage detection is a global monitoring and assessment method [14], which has as its hypothesis that the global dynamic behavior of the structure is a function of the physical properties of the structure (mass, damping, and stiffness) whose changes will be reflected in the vibration signals, collect through sensors like: displacement transducers, velocity sensors, accelerometers…, etc. The statistical pattern recognition from the vibration signals is fundamental for the health

2

monitoring process [16]. Based on that point out in [44], it consists in a four-part process: (1) operational evaluation, (2) data acquisition, fusion and cleansing, (3) feature extraction and information condensation, and (4) statistical model development for feature discrimination.

The identification of damage can be grouped into two branches: the model-based and non-model-based [11, 14]. The construction of the mathematical model of dynamic systems from experimental data is the so-called system identification, different model-based identification approaches are available in classical literatures [24, 29, 43].

During the past few years, the subspace identification algorithms had been successfully applied on structural system identification. The subspace method can be classified into the Subspace Identification (SI) algorithm which uses both input and output data, and the Stochastic Subspace Identification (SSI) algorithm which is an output-only identification algorithm. The developments of these methods are based on concepts from linear algebra, system theory and statistics. There are two essential numerical tool for the subspace methods in linear algebra: Singular Value Decomposition (SVD) and the QR decomposition. Classical algorithms to perform such matrix decomposition tools are completely described in [18].

For large scale civil structures such as bridges, the input excitation to the structural system is unknown, the output-only Stochastic Subspace Identification (SSI) is suitable for the identification and monitoring of these structures excited by ambient vibrations.

There are several varieties of SSI technique such as Covariance-driven (SSI-COV), Data driven (SSI-DATA), or combined with other methods like Expectation

Maximization technique (SSI-EM) [39, 42] and the Empirical Mode Decomposition (EMD) based stochastic subspace identification [52].

SSI-DATA algorithms were fully enhanced by Van Overschee and De Moor [47].

The core of output-only identification through SSI-DATA is the orthogonal projection carried out by LQ decomposition [47, 50], followed by the SVD used to extract the system subspace. There are variants of the Data-driven algorithm which correspond to a different choice of weighting matrices before factorizing the projection matrix. The well-known SSI-DATA algorithms include CVA, N4SID, MOESP and IV-4SID [27, 46, 48]. Application of the SSI-DATA algorithm to investigate the dynamic characteristics of a cable-stayed bridge had been studied in [49]. In [9] the algorithm was also applied to the identification of a Steel-Quake benchmark structure. In [55] the method is applied to identify the modal parameters of The Heritage Court Tower in Vancouver, Canada, and the Beichuan Bridge located in China, which has its arch made by concrete filled steel tube. The reference-based SSI algorithms were also developed in [36, 37, 38] and applied in the identification of a steel transmitter mast and a prestressed concrete bridge (the Z24-bridge in Switzerland).

As opposed to SSI-DATA, the SSI-COV algorithm avoids the computation of orthogonal projection, instead, it is replaced by converting raw time histories in an assemble of block covariances which is called Toeplitz matrix. The SSI-COV algorithm appears early as the Modified Instrumental Variable methdod, with applications in laboratory tests, such as the identification of a vertical steel clamped-free beam and modal analysis of a carrying bogie [7]. Other application can be found in the identification of offshore structures and rotating machinery [1] and an aircraft [2].

4

Different family of SSI-COV exist, a very famous identification algorithm is a combined approach of the Natural Excitation Technique (NExT) [23] and the Eigensystem Realization Algorithm (ERA) [25] to find modal parameters from ambient response. It is called as the Natural Excitation Technique and Eigensystem Realization Algorithm (NExT-ERA) [10] which has been applied to the identification and damage detection of a 4-story 2-bay steel IASC-ASCE Benchmark structure. The same algorithm is also applied to the identification of a cable stayed bridge in [53], where alternative form to construct the stabilization diagram [37] was proposed. Another similar approach exists: it is a combination of the Random Decrement method (RD) [4, 33] and ERA using Data Correlations (ERA/DC) [26]. This RD-ERA/DC algorithm is applied to the modal identification of Tsing Ma Bridge, located in Hong Kong [40].

Improvements were achieved by substituting random decrement functions by their cross-correlation in the assembling of the Hankel matrices.

Different from the off-line analysis, the output-only system identification and damage detection through on-line recursive algorithms has received considerable attention recently, it is suitable for long-term continuous monitoring systems and development of early warning systems. In the past few years, several recursive subspace identification algorithms have been proposed to update in an recursive fashion the main decomposition tools of the SSI algorithm: the LQ decomposition and SVD.

The updating of the LQ decomposition is done by means of Givens rotations [18], the SVD updating problem is circumvented by considering the similarities between recursive SSI and adaptive signal processing techniques for direction of arrival estimation [56], and only the column space of extended observability matrix is updated

[19, 20, 34]. The recursive stochastic realization by the classical Covariance-driven SSI algorithm (RSSI-COV) is proposed in [17], and the application for in-flight flutter monitoring is discussed in the paper. However, recursive Data-driven subspace algorithm is the most widely used method in the literature. Application for in-flight modal analysis of airplanes can be found in [13]. In [28] the RSSI-DATA is applied to the system identification of Donghai Bridge located in China. Damage detection example of the mentioned 4-story 2-bay steel IASC-ASCE Benchmark structure can be found in [12], and finally, application to the health monitoring and damage detection of a single pier subjected to scour, and to the 1-story 2-bay reinforced concrete frame can be found in [58].

Although the literature of SSI algorithms were reviewed, there is another useful
output-only subspace tool called the Singular Spectrum Analysis (SSA), which is a
novel non-parametric technique and it was firstly applied to extract tendencies and
harmonic components in meteorological and geophysical time series [3]. Except the
extraction of tendency, SSA can be applied to eliminate noise effect, or to detect the
*singularities, e.g., to extract structural residual deformation [32]. *

The conjunction of SSA to SSI-COV will be the main contribution of this thesis.

Although it is simply the addition of a pre-processing tool and no more, this action allows the determination of the best system order from the connection in-series of two SVD decomposition engine, and has greatly enhanced the identification quality and stability. Moreover, the recursive Singular Spectrum Analysis algorithm will be proposed in this thesis, which in conjunction with RSSI-COV method offers a very stable and accurate online tracking capacity.

6

**1.2 Research Objectives **

The objective of this research is to, first, enhance the Covariance-driven Stochastic Subspace Identification method (SSI-COV) to the named “Singular Spectrum Analysis–

Covariance driven Stochastic Subspace identification method” (SSA-SSI-COV), validated both by numerical simulation and the application in system identification of Canton Tower, a benchmark problem for structural health monitoring of high-rise slender structures.

Second, develop the recursive Singular Spectrum Analysis method (rSSA), and in conjunction with the recursive Covariance-driven Stochastic Subspace Identification method to construct the named “recursive Singular Spectrum Analysis – Covariance driven Stochastic Subspace Identification method” (rSSA-SSI-COV), through a moving window approach. The method will be validated firstly by numerical simulation and later by application in the damage detection and health monitoring of laboratory experiments.

The organization of this thesis is briefly described as follows:

**Chapter 2: The basic methodology of subspace identification algorithm is recalled **
through, firstly, the introduction of the dynamic model of a linear system, followed by
the formulation of SSI-COV and SSI-DATA method, and finally, the Singular Spectrum
Analysis (SSA) procedure will be described.

In system identification algorithms, it is important to distinguish the structure modes from the spurious modes because the order of the real system is always unknown. The alternatives to build the stabilization diagram will be introduced and compared one to

another. A comparison benefit-drawback and implementation issues will be discussed through a numerical simulation example and the identification of a laboratory test.

**Chapter 3: A comprehensive numerical study and comparison between different SSI **
algorithms is carried out. Measurement noise effect and the addition of a noise which
violates the SSI assumption is discussed. Identification of the simulated nonlinear
signals, signals with time-varying frequency, signals with closely-spaced frequencies
mixed with white noise is done to understand the performance of SSI algorithms under
different scenarios of assumption violation and the mechanism to overcome this
difficulties. The SSA-SSI-COV algorithm is introduced in this chapter to solve the
identification problem of closely-spaced frequencies with added white noise.

**Chapter 4: Application of SSI algorithms in system identification of the Canton Tower **
is discussed. The order determination procedure through the SSA-SSI-COV algorithm
will be described. Comparison between different SSI approaches is made in this chapter.

The procedure called decimation is although studied and applied to increase the convergence speed of the stabilization diagram.

**Chapter 5: the derivation of Covariance-driven Recursive Stochastic Subspace **
Identification algorithm (RSSI-COV) can be found in this chapter. The Projection
Approximation Subspace Tracking algorithm (PAST) and its Instrumental Variable
extensions (EIV-PAST) is also described and implemented to RSSI-COV. To consider
the noise contaminated data, a recursive pre-processing technique called recursive
singular spectrum analysis technique (rSSA) is derived to enhance the accuracy and
stability in the online tracking capability.

8

**Chapter 6: the RSSI-COV method and the proposed rSSA-SSI-COV algorithm through **
a moving window approach are validated in this chapter by means of numerical
simulation of a 6 DOF system, cases with sudden reduction and slow decreasing in
system stiffness are studied. The effects of the selected RSSI model parameters in the
online modal analysis, and the influence of time-varying frequencies in the selection of
system order are also discussed.

**Chapter 7: the RSSI-COV method and the proposed rSSA-SSI-COV algorithm through **
a moving window approach are applied to the monitoring and damage detection of, first,
shaking table test of a 3-story steel structure with instantaneous stiffness reduction.

Second, the shaking table test of a 1-story 2-bay reinforced concrete frame subjected to earthquake excitations with increasing intensity. Finally, application to the monitoring of a three pier and four span steel bridge under continuous scour is carried out.

**Chapter 8: Summaries and suggestions for the use of the proposed algorithms will be **
given here. The potential research topics are indicated at the end.

**Chapter 2 **

**Stochastic Subspace Identification Methods **

**2.1 ** **Introduction **

In output-only characterization, the ambient response of a structure is recorded
*during ambient influence (i.e. without artificial excitation) by means of highly-sensitive *
velocity or acceleration sensing transducers. The Stochastic Subspace Identification
(SSI) technique is a well known multivariate identification technique for output-only
measurements. It was proved by several researchers to be numerically stable, robust to
noise perturbation and suitable for conducting non-stationarity of the ambient
excitations although its stationary assumption is violated [5, 37, 53].

The SSI-DATA algorithm was fully enhaced by Van Overschee and De Moor [47], while SSI-COV algorithm has as its antecedent the Eigensystem Realization Algorithm [25] for the free response of a structure, which are applied along with the Natural Excitation Technique (NExT) or Random Decrement (RD) functions. This chapter will begin with the introduction of the dynamic model of structures, followed by the stochastic properties and the system realization methods of each subspace algorithm.

**2.2 ** **Models of vibrating structures **

**2.2.1 ** **Continuous-time state-space model **

The Finite Element model of a linear time-invariant dynamic system can be expressed as:

### ( )

^{t}

^{q}### ( )

^{t}

^{q}### ( )

^{t}

^{F}^{(t)}

^{u}^{(t)}**q****C** **K** **L**

**M**&& + ** _{2}**& + = =

_{(2.1) }

10

**where M, C**_{2}** and K**∈ℜ^{n}^{×}* ^{n}* are the mass, damping and stiffness matrix.

### ( )

^{t}^{∈}

^{ℜ}

^{n}**q*** is the displacement vector at continuous time t. *

### ( )

^{t}* q&* is the velocity vector.

### ( )

^{t}* q&*& is the acceleration vector with the same dimension as the displacement vector.

*(t)*∈ℜ*n*

* F* is the excitation vector.

*m*
*n*×

ℜ

∈

**L** is the input location matrix.

*(t)*∈ℜ*m*

**u***is the vector describing m inputs as a function of time t. *

**n is the number of DOFs and m is the number of inputs. **

The above second order differential equation can be rearranged into a first order differential equation known as the state-space model, which consist of two equations [24]:

**The state equation: **

### ( )

^{t}

^{x}### ( )

^{t}

^{u}### ( )

^{t}* x*& =

**A**

**+**

_{c}**B**

**(2.2)**

_{c}where

### ( ) ( )

### ( )

^{}

^{∈}

^{ℜ}

^{2}

^{×}

^{1}

= ^{n}

*t*
*t* *t*

**q****x****q**

& * is the state vector at continuous time t and therefore *

### ( ) ( ) ( )

^{}

=
*t*
*t* *t*

**q****x****q**

&

&

&

& **. A**c is the so-called system matrix since it contains all the information

**related to the system (M, C**2**, K in the equation of motion), and B**c is the definition of
**input matrix in the state equation. A**_{c}** and B**_{c} are arranged as follows:

*n*
*n 2*
2
1

1

×

−

− ∈ℜ

−

= −

**2**

**c** **M** **K** **M** **C**

**I**

**A** **0** , _{−} ∈ℜ ^{n×}^{m}

= ^{2}

**L**
**M**

**B**_{c}**0**** _{1}** (2.3)

The state equation which is a first order differential equation has the following

solution [57]:

### ( )

^{t}^{=}

^{e}^{(}

^{t}^{−}

^{t}^{)}

### ( )

^{t}^{+}

### ∫

_{t}

^{t}

^{e}^{( )}

^{t}^{−}

### ( )

^{d}0 0

0 ^{τ} *^{u}*τ τ

**x**

**x**^{A}^{c}^{A}^{c}**B**** _{c}** (2.4)

where the 1^{st}** term is the free vibration solution given an initial condition x(t*** _{0}*), and the
2

^{nd}term is a typical convolution integral. Through an eigen-analysis of the system

**matrix A**c, the state equation can be decoupled through a coordinate transformation using the obtained complex eigenvectors.

**A**** _{c}**=

**ΨΛ**

_{c}**Ψ**

^{−}

^{1},

^{x}### ( )

*=*

^{t}

^{Ψ}

^{η}### ( )

*(2.5)*

^{t}where^{η}

### ( )

*is the generalized coordinate.*

^{t}**Λ**

**c**∈ℜ

^{2}

^{n 2}^{×}

*is a diagonal matrix containing complex eigenvalues λ*

^{n}*in the diagonal which appear in conjugate pairs,*

_{i}**Ψ**∈ℜ

^{2}

^{n 2}^{×}

*are the complex eigenvectors. From the eigen-analysis*

^{n}**A**

_{c}**Ψ**=

**ΨΛ**

**, one may find that they have the following structure:**

_{c}

= _{*}
**Λ**

**Λ**_{c}**Λ** = ,

= _{*}^{*}_{*}
**Λ**
**Θ**
**ΘΛ**

**Θ**

**Ψ** **Θ** _{(2.6) }

**In fact, it can be easily verified that Λ are the same eigenvalues and Θ the **
*same eigenvectors, i.e., mode shapes, than those obtained by conducting eigen-analysis *
directly in the unforced equation of motion (2.1), but they cannot be used to decouple
the equation of motion unless it is a proportionally damped system.

Then, the decoupled state equation can be written as follows:

*^{η}*&

### ( )

*=*

^{t}**Λ**

**c**

^{η}### ( )

*+*

^{t}**Ψ**

^{−}

^{1}

**B**

**c**

^{u}### ( )

*(2.7)*

^{t}Furthermore, to relate the obtained complex eigenvalues to a physical
interpretation, a Taylor Expansion is required to decouple the free vibration term *e*^{A}^{c}* ^{t}*
in (2.4), which is a matrix exponential:

12

### ( )

^{1}

1

1 3

3 2 2 3

3 2 2

3 ...

... 2 3 2

−

−

−

=

=

⇒

+ + + +

= + +

+ +

=

**Ψ**
**Ψ**

**Ψ**
**Ψ**

**Λ** **Ψ**
**Λ** **Λ**

**I**
**A** **Ψ**

**A** **A**
**I**

**c**
**c**

**c**

**Λ**
**A**

**c**
**c**

**c**
**c**

**c**
**c**
**A**

*t*
*t*

*t*
*t*

*e* *i*

*diag*
*e*

*e*

*!*
*t*

*!*
*t* *t*

*!*
*t*

*!*
*t* *t*
*e*

λ

(2.8)

*where diag(·) is the diagonal operator. Therefore, considering only this free vibration *
term in (2.4) and having in mind that the complex eigenvalue has its real and imaginary
part: λ* _{i}* =α

*+*

_{i}*j*β

_{i}*, solution to the i-th mode free vibration is:*

### ( )

*t*

*e*

^{(}

^{j}^{)(}

^{t}

^{t}^{0}

^{)}

_{i}### ( )

*t*0

*e*

^{(}

^{t}

^{t}^{0}

^{)}

### [

cos

_{i}### (

*t*

*t*0

### )

*j*sin

_{i}### (

*t*

*t*0

### ) ]

_{i}### ( )

*t*0

*i*

*i*
*i*

*i* η β β η

η = ^{α}^{+} ^{β} ^{−} = ^{α} ^{−} − + − (2.9)

where the coordinate transformation shown in (2.5) has been applied to decouple the
free vibration solution. Comparing (2.9) with the well-known free vibration solution of
*a SDOF system, the so-called i-th effective modal frequency * ω_{i}^{′} and effective damping
ratio ζ_{i}^{′} can be realized:

*i*
*i*
*i*

*i* α β λ

ω′= ^{2}+ ^{2} = ,

*i*
*i*

*i*
*i*

*i*

*i* λα

β α

ζ α =^{−}

− +

′=

2

2 (2.10)

The effective modal frequency and damping ratio are exactly those obtained by
normal mode approach if it is classical or proportional damping. In the case of
non-proportional damping, ω_{i}^{′} will be slightly different than the normal natural
frequency, and ζ_{i}^{′} can be called as the i-th effective attenuation rate [57].

One can note that ω_{i}^{′} is actually the amplitude of the complex system pole, and

ζ*i*^{′} is related to the phase. Hence, when a structural system is changed due to damage,
the migration of system poles will be directly reflected by the computed effective modal
frequency and damping ratio, which the term “effective” will be omitted hereafter.

**The observation equation: **

*If only subsets of the n DOF can be measured, and considering that measurements *
*are taken at l locations and the sensors can be either accelerometers, velocity or *
displacement transducers, the observation equation can be defined as:

### ( )

^{t}

^{q}### ( )

^{t}

^{q}### ( )

^{t}

^{q}### ( )

^{t}* y* =

**C**

**&& +**

_{a}**C**

**& +**

_{v}**C**

**(2.11)**

_{d}where ^{y}

### ( )

^{t}^{∈}

^{ℜ}

^{l}

**represents the l outputs. C**

_{a}**, C**

_{v}**and C**

**∈ℜ**

_{d}

^{l}^{×}

*are the output location matrices corresponding to acceleration, velocity and displacement respectively.*

^{n}* To relate the output y(t) to the system state x(t), the equation of motion (2.1) can be *
used to eliminate

^{q&}^{&}

### ( )

*, and by arranging and grouping location matrices, the observation equation become:*

^{t}### ( )

*=*

^{t}**C**

**c**

^{x}

### ( )

*+*

^{t}**D**

**c**

^{u}

### ( )

^{t}* y* (2.12)

where ^{C}^{c}^{=}

### (

^{C}

^{d}^{−}

^{C}

^{a}

^{M}^{−}

^{1}

^{K}

^{C}

^{v}^{−}

^{C}

^{a}

^{M}^{−}

^{1}

^{C}

^{2}### )

^{∈}

^{ℜ}

^{l 2}^{×}

*is the output matrix, and*

^{n}*l×**m*

− ∈ℜ

=**C** **M** **L**

**D**_{c}_{a}^{1} is the direct transmission matrix.

**Although the eigenvectors of system matrix A contains mode shapes information **
as that shown in (2.6), however, there is no knowledge about the location of each DOF
* when the matrix A is identified, moreover, usually the number of modes, i.e., order of *
the system extracted from measurement data is different than the number of sensors,
thus, the system eigenvectors should be mapped to the sensor locations through the

**output location matrix C**

**:**

_{c}**Ψ**
**C**

**V**** _{c}** =

**(2.13)**

_{c}**where V**** _{c}** are the observed mode shapes.

14

**2.2.2 ** **Discrete-time state-space model **

Since all data is sampled in discrete time, the above continuous time state-space model can be converted into a discrete time state-space model. By gathering together the state and observation equation:

**k****k****k**

**k****k****1****k**

**u****x****y**

**u****x****x**

**D**
**C**

**B**
**A**

+

=

+

+ =

(2.14)

where ^{x}^{k}^{=}^{x}

### ( )

^{k}^{∆}

^{t}^{=}

### [

^{q}

^{T}

^{k}

^{q}^{&}

^{T}

^{k}### ]

*is the discrete state vector containing the sampled*

^{T}**displacements and velocities. u****k**** and y*** k* are sampled input excitation and output

**measurement. A is the system matrix, B is the input matrix, C is the observation matrix**

**and D, the direct transmission matrix, all in discrete-time. The relationships between**these matrices in discrete-time and continuous time are the following [24]:

*e* ^{∆}*t*

= ^{A}^{c}

**A** , ^{B}^{=}^{A}^{c}^{−}^{1}

### (

^{e}

^{A}

^{c}^{∆}

^{t}^{−}

^{Ι}### )

^{B}

^{c}^{ , }

^{C}^{=}

^{C}

^{c}^{ , }

^{D}^{=}

^{D}

^{c}^{(2.15) }

A basic assumption behind these relationships is that, the external perturbation is
*constant within a sampling period, i.e., * **u***k* =**u**

### ( )

*∆*

^{k}*for the period of time*

^{t}### (

^{k}### )

^{t}*t*

*k*∆ ≤τ < +1∆ **. It is provided that the inverse of system matrix A**c exists.

*The eigenvalues µ**i*** of the discrete-time system matrix A can be, therefore, related **
to the continuous-time eigenvalues by

### ( )

*e* ^{t}_{i}*t*^{i}

*i*

*i* ⇔ = ∆

= ^{∆} λ µ

µ ^{λ} ^{ln} (2.16)

Then, frequencies and damping ratios can be computed as mentioned before. Both the observation matrix and complex eigenvectors are not affected by the discretization in time, the above-mentioned equations can be used without any change.