An Orthogonal Matching Pursuit based signal compression and reconstruction approach for electromechanical admittance based structural health monitoring

Mechanical Systems and Signal Processing 133 (2019) 106276

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

An Orthogonal Matching Pursuit based signal compression and reconstruction approach for electromechanical admittance based structural health monitoring Hedong Li, Demi Ai ⇑, Hongping Zhu, Hui Luo School of Civil Engineering and Mechanics, Huazhong University of Science and Technology, Wuhan, China

a r t i c l e

i n f o

Article history: Received 10 May 2018 Received in revised form 7 June 2019 Accepted 25 July 2019

Keywords: Signal compression Data reconstruction Electromechanical admittance Structural health monitoring Compressed sensing Sparsity Orthogonal Matching Pursuit

a b s t r a c t Signal compression and reconstruction were critical for damage detection in engineering structural health monitoring (SHM), on account of large amounts of sensor data collected and processed in signal acquisition system. A signal compression and recovery approach for damage detection using piezoelectric ceramic transducer (PZT) in structural monitoring system was proposed in this article. The basis of this approach was to first perform a linear projection of the transmitted data x in the monitoring system onto y by a random matrix and subsequently to feedback the data y to the receiving system. An algorithm of sparsity-adaptive Orthogonal Matching Pursuit (OMP) modified via optimal parameter analysis was explored to improve both the recovery effect and the compression ratio (CR) of compressed sensing (CS) in data processing stage. A statistical index was then introduced to identify the vector characteristics. The proposed method was sufficiently validated with the electromechanical admittance (EMA) data collected in an experiment for local damage detection on a simply-supported steel beam, and further applied to a longtime health monitoring of full-scaled shield tunnel segment structure. Qualitative and quantitative comparisons between the reconstructed and the original signals in structural damage detection under multiple conditions indicated that the proposed compression and recovery approach was of high accuracy and robustness to immune from the sensor conditions and temperature/environment impact, thus providing promising assistance to the impedance/admittance based SHM practice. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Application of piezoelectric ceramic transducer (PZT) based electromechanical impedance technique in structural health monitoring (SHM) has shown promising results due to its superior sensitivity to small local damages, which involved with aeronautic, mechanical and civil engineering domains in the past two decades. In the impedance technique, a PZT patch is often surface-bonded onto the surface of the target structure. The frequency-domain admittance (inverse of impedance) signals of PZT transducers are usually served as reference for damage detection. Since the impedance/admittance signal couples the electrical impedance of the PZT patch and the mechanical impedance of its host structure, any changes in structural

* Corresponding author at: School of Civil Engineering and Mechanics, Huazhong University of Science and Technology, 1037 Luoyu Road, 430074 Wuhan, PR China. E-mail address: [email protected] (D. Ai). https://doi.org/10.1016/j.ymssp.2019.106276 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.

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H. Li et al. / Mechanical Systems and Signal Processing 133 (2019) 106276

mass, stiffness or damping caused by damages thus reversely impact the mechanical impedance and reflect in the PZT signatures. G. Park et al. [1] utilized the impedance-based technique for damage identification on a pipeline system under normal operation conditions. C. K. Soh et al. [2] applied the PZT transducers to testing a loaded reinforced concrete bridge, and validated its sensitivity to local-region damages. The applicability of strain monitoring in the real time domain was validated and a novel method of converting piezoelectric signals from the frequency to the time domain was proposed [3]. D. Ai et al. [4–7] extended the mechanical impedance technique into the monitoring of steel structural corrosion, reinforced concrete stress/damages and proposed a sensor self-diagnosis process based on extracting the features of the real admittance signal to discriminate structural and sensor damages. Recently, L. M. Campeiro et al. [8] used a series of experiments on an aluminum bar to analyze the noise and vibration effects on impedance-based structural damage detection based on coherence function and basic damage indices. Impedance technique integrated with Lamb wave-based SHM strategy were also developed for long-term composite structural damage classification and health condition diagnosis of sensor network under varying pressure [9]. A non-dimensional hydration parameter based on electromechanical impedance technique was proposed to monitor the early-age hydration of concrete [10]. In recent years, the research with regard to the engineering applications of impedance technique has been constantly attracting extensive interests in civil engineering domains. However in practice, there are several challenges for the practical application of PZT sensors and its networks, including a substantial amount of sensors required to be installed on the critical regions of structure for continuously collecting signals in the whole-life cycle of the monitoring object, for higher accuracy of damage quantification when fast dealing with these monitoring signals. The cost of data transmission and storage is also considerable with the demand of high accuracy in structural damage identification. Regarding with these issues, efficient data compression algorithms are necessary for far-field transmission and storage of monitoring signals from sensors to command system due to the bulkiness of civil structures. Since conventional electromechanical impedance technique was unable to satisfy the demand for meticulous structural damage detection, inter-disciplines such as wireless sensing, signal processing and data driven theories were investigated. A data normalization technique using Kernel principal component analysis (PCA) was applied to structural damage detection under varying temperature and external loading conditions [11]. Impedance data collected from a composite wing specimen was utilized to examine the novel method. The results showed that the proposed technique might cause fewer false alarms than the conventional technique. J. Min et al. [12] proposed a neural network (NN) based pattern analysis method to identify damage-sensitive frequency ranges autonomously for impedance-based SHM technique and to output damage type and severity. P. Selva et al. [13] used the electromechanical impedance technique for damage detection and localization. Numerical experiments of carbon fiber reinforced plates were set up more than 100 damage scenarios for extracting damage metrics to evaluate the test model. The damage localization process was achieved using artificial neural networks (ANNs). Three optimization algorithms, including grid-search, particle swarm optimization and genetic algorithm combined with support vector machines (SVM) were proposed to effectively analyze sensor data for structural damage detection [14]. A benchmark model of a small-scale three-story frame aluminum structure within 17 different scenarios was utilized to verify the proposed method. The results revealed that the genetic algorithm based SVM performed better results than other methods. A critical issue to realize the real-time, far-field and in-situ transmission and analysis of PZT signals is data compression and recovery, due to the tremendous information produced during monitoring process, which has raised the attentions of researchers. H. Sohn et al. [15] implemented a pattern recognition strategy for structural vibration signal based SHM and applied the method to a bridge column and a surface-effect fast boat. S. Park et al. [16] combined PCA-data compression with K-means clustering algorithms to realize a wireless structural monitoring system. The PCA algorithm was utilized to compress the original impedance data into significant feature vectors contained the essential characteristics and eliminated the noise interference. The experimental results obtained from a bolt-jointed aluminum structure showed that root-mean-square deviation (RMSD) of the PCA-compressed data differed little from that of the original data, and the unsupervised pattern recognition method using two principal components as damage-sensitive features were found effective. Considering the limits of typical operation consisting of recording signals from all the transmit-receive pairs and subtracting pre-recorded baselines to detect changes caused by multiple damage sites, sparse reconstruction techniques of basis pursuit (BP) denoising and orthogonal matching pursuit were applied to achieve the decomposition of differential signals into a linear combination of location-based components [17]. Since the frequencies of decomposed sinusoid were arbitrary and signal of structural vibration would lose its sparsity in most cases, the continuous basis pursuit (CBP), semidefinite programming (SDP) and spectral iterative hard threshold (SIHT) were applied in a numerical 3-story frame experiment to achieve the recovery with higher precision and less measurements by Z. Duan et al. [18]. Y. Sun et al. [19,20] applied ultrasonic phased array technology using multi-sensor detection for structural monitoring. Compressive sensing theory was utilized to reduce the cost of data storage, transmission and processing. Compressed sensing was implemented to compress structural response data from a reinforced concrete structure and the damage detection process via the reconstructed data showed high accuracy and effective compression ratio [21]. Y. Bao et al. [22,23] investigated the data compression and data loss recovery based on compressive sampling, by which the original data x was projected linearly onto y by a random matrix and the data y was fed back to the base station. In addition, an adaptive sparse time–frequency analysis method was developed to identify the time-varying cable tension forces of bridges [24]. More recently, an electromechanical impedance sensitivity-based structural damage detecting method was integrated with sparse regularization technique to identify damage location and severity [25], and a hybrid method based on the redundant concatenated dictionary and weighted l1-norm regularization method was proposed for moving force identification [26]. Majority of these researches and discussions about compressed sensing for data compression in SHM were merely limited to the reconstruction

H. Li et al. / Mechanical Systems and Signal Processing 133 (2019) 106276

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effectiveness in consideration of basis types and measurement numbers, regardless of the sparsity level which may significantly influence the measurements and iterations. Thus, systematic study on the pursuit of optimal solution for signal/data compression and reconstruction in the impedance-based SHM domain is still necessary to overcome these limitations, with the aid of other greedy algorithms such as Orthogonal Matching Pursuit (OMP) algorithm. Unfortunately, this is still lacking in the literature. In this study, an innovative approach combined piezoelectric impedance technique with OMP algorithm based data compression and reconstruction process was proposed and deduced for structural damage evaluation. The OMP based method was improved via optimal parameter analysis for compressing and reconstructing impedance signals. An experiment of simply-supported steel beam was first conducted to validate the feasibility of the proposed approach. Six PZT sensors under different conditions were used to obtain impedance measurements, which were then transmitted to the data processing system. Parametric analysis was implemented and discussed to seek an optimal balance between compression ratio and total error. Both the RMSD and the covariance coefficient (CC) indices were computed to evaluate the effectiveness of the recovery signals for damage quantification. Besides, the approach was also applied to a long-time monitoring of a full-scaled shield tunnel segment in laboratory environment, in which the impedance measurements under different environmental conditions were sufficiently testified. 2. Principle of the electromechanical impedance technique As an intelligent material, PZT can be simultaneously served as sensor and actuator for structural damage detection based on its direct and converse piezoelectric effects that are electrical charges produced in response to an applied strain field and conversely mechanical strain produced in response to an applied electric field, which are illustrated in the impedance model. 2.1. One-dimensional impedance model C. Liang et al. [27] proposed a one-dimensional impedance model considering interaction between the PZT transducer and the host structure, where the patch was subjected to an electric field perpendicular to x-axis. Under the variable electric field, x-axial vibration would be produced on the patch regarded as a thin bar. One side of the patch was fixed, and the other side was connected to a structure as a member simplified as a single-degree-of-freedom system. The electromechanical admittance (EMA) in the PZT-structure system can be expressed as:

Y ¼ GðxÞ þ jBðxÞ ¼ jx

ba la ha




er33 

Z s ðxÞ 2 E d E Z s ðxÞ þ Z a ðxÞ 31 11

 ð1Þ

where G(x) and B(x) denote the conductance and susceptance of the admittance respectively; j is ð1Þ1=2 ; x denotes the angular frequency of excitation; ba, la and ha denote the width, length and thickness of the PZT patch respectively; er33 ¼ er33 ð1  djÞ is the complex electric permittivity of the PZT patch at a constant stress; d is the dielectric loss factor of the PZT patch; g is the structural mechanical loss factor; Zs(x) and Za(x) denote the mechanical impedance of the structure E

and the PZT patch respectively; d31 is the coupling PZT constant; E11 ¼ EE11 ð1 þ jgÞ is the complex Young’s modulus of the PZT patch at a constant electric field. 2.2. One-dimensional impedance model considering the bonding layer The aforementioned model ignores the interaction of the bonding layer between the PZT patch and the structure, through which PZT sensors can be stuck to the measured structure, such as epoxy resin adhesive. The force transmission between PZT sensors and the host structure is achieved via the bonding layer, indicating that the bonding layer participates in the above force interaction and directly influences the force transfer efficiency. Therefore, debonding or deterioration in the bonding layer can also change the impedance signatures [7]. As shown in Fig. 1, the bonding layer and the structure are respectively deemed as single-degree-of-freedom system for taking the adhesive between the PZT patch and the structural surface into account [28].

Y ¼ jx

ba la ha

2

er33 þ

E

d31 E11 Z a ðxÞ tanðjla Þ 2 E  d31 E11 nZ s ðxÞ þ Z a ðxÞ jla

! ð2Þ

qffiffiffiffiffiffiffiffiffiffiffiffiffi E where j ¼ x q=E11 is the wave number, q is the density of the patch; n modifies the structural dynamic interaction in the actuator-driven system for characterizing the bonding layer. Other parameters of PZT and structural properties are the same with that addressed in Eq. (1). Regardless of the difference in the PZT-structure interaction models, the EMA of PZT is uniquely determined by the mechanical impedance of the host structure as long as the parameters and performance of PZT and bonding layer maintain constant. Variation of structural mechanical impedance can be reflected via the changes in the EMA of PZT, so as to reflect structural defects, damages or other physical changes. Through the observations on the alternation of admittance plots, structural damages are determined. However, the storage and analysis of the data measurements are tremendous,

4

H. Li et al. / Mechanical Systems and Signal Processing 133 (2019) 106276 I = isin( t+ )

Electric Admittance: Y=Re(Y)+jIm(Y) KB

V = vsin( t )

x

Laptop

PZT

MS

MB

CB

Impedance Analyzer

KS

Bonding Layer

CS Structure

Fig. 1. Data measuring system in one-dimensional interaction model of PZT-structure considering the influence of bonding layer.

time-consuming and tedious, especially when the amount of PZT sensors is necessary enlarged for real-life structures. Therefore, data compression and reconstruction are prerequisite for the fast analysis on the measurements for accurate damage detection. As a prospective approach for data compression/reconstruction, the basic theory of compressed sensing used in this study is introduced in the following section. 3. Compressed sensing 3.1. Principle of compressed sensing Compressed sensing mainly contains sparse representation of the collected signal, design of measurement matrix and signal reconstruction. Prerequisite of compressed sensing is that the original signal presents a good sparsity itself or shows a sparse characteristic in some transform [29–31]. Generally, there is no sparse characteristic itself either in time domain or frequency domain for experimental signal collection. To find the sparse land of the original signal, transform such as wavelet basis, Fourier basis, cosine basis and so on, should be applied to the original signal. Introducing a n  n matrix W ¼ ½w1 ; w2 ;    ; wn , where wi is a column vector, discrete time signal X ðiÞ; i ¼ 1;    ; n in Rn can be transformed into a which is the corresponding basis coefficient matrix in the orthogonal basis W [32,33].

X ¼ Wa

ð3Þ m

Within a linear projection of X, a data vector Y 2 R of length m can be obtained.

Y ¼ UX

ð4Þ

where U is called measurement matrix. Substituting Eq. (3) into Eq. (4), Y is obtained as following:

Y ¼ UX ¼ UWa ¼ Ha

ð5Þ

where H is called sensing matrix. If the basis coefficients a in Eq. (5) have K (K  N) non-zero values, decay exponentially and approximate zero after sorting, the signal is called K-sparse in domain [30–33]. Equation (5) is known as a non-deterministic polynomial problem. The solution of the equation can be solved via the minimum l0-norm optimization problem [30,31], 

ab ¼ arg mink a k0 s:t: Ha ¼ Y

ð6Þ

Considering the reconstruction error, Eq. (6) can be converted to a minimum l1-norm optimization problem, 

ab ¼ arg mink a k1 s:t: k Ha  Y k2  e

ð7Þ

If the sensing matrix H satisfies a so-called restricted isometry property (RIP):

1  dk 

k Hmk k22 k mk k22

 1 þ dk

ð8Þ

for all K-sparse vectors vk, where dk is an isometry constant with dk 2 (0,1), the K-sparse signals can be accurately reconstructed via the optimization problem [30–33]. The exact recovery phenomenon would occur if the columns of the sensing matrix H are approximately orthogonal [34,35]. Since prerequisite of compressed sensing is that the original signal shows a sparse characteristic in some basis, the accuracy of the signal reconstruction can be ensured if the signal turns to own K-sparse characteristic when it is projected onto an appropriate orthogonal basis. Generally, Gaussian random matrix and Bernoulli random matrix are selected as common measurement matrix [29–31].

1 G 2 Rmn : Gði; jÞ ¼ pffiffiffiffiffi g ij ; g ij  Nð0; 1Þ M

ð9Þ

H. Li et al. / Mechanical Systems and Signal Processing 133 (2019) 106276

5

where G is called Gaussian random matrix which is a strongly random measurement matrix and has been validated to conform to RIP.

  1 1 1 B 2 Rmn : Bði; jÞ ¼ pffiffiffiffiffi bij ; bij  0:5 0:5 M

ð10Þ

where B is called Bernoulli random matrix. 3.2. Orthogonal Matching Pursuit The ultimate object of compressed sensing theory is to uttermost reconstruct the original data. One of the reconstruction approaches is convex optimization algorithm, such as BP algorithm based on the linear programming and the gradient projection sparse algorithm [34,38]. The major deficiencies in the above algorithms are that the computing speed is rather slow to hinder its application in processing a substantial amount of data, although the reconstruction precision is high [29,36]. Another reconstruction approach is OMP algorithm, in which the combination of local optimization is utilized to explore the nonzero coefficients to reconstruct the original data [29,36–39], which shows advantages such as (1) the less computational consumption for sparse recovery; (2) the fast convergence for calculating iterations; and (3) the convenient implement to module design in practice. To achieve an optimal accuracy of reconstruction, the OMP algorithm was improved via optimal parameter analysis for impedance signal based structural damage detection. 3.2.1. Procedures of Matching Pursuit Given a collection of vectors D = {xi } in a Hilbert space H, each vector is called an atom, which has the same length n with the signal f, D is referred to as a dictionary. And then normalization is implemented on these vectors, namelykxi k = 1. The atom optimally matched the signal is selected to construct a sparse approximation and to obtain the signal residual, forming an iterative procedure [36]. S. G. Mallat et al. [37] presented an iterative algorithm based on Matching Pursuit (MP) where orthogonal projection is expressed as:

PV f ¼

X

an xn

ð11Þ

n

where PV is the orthogonal projection operator onto V. In each iteration,

f ¼

k X

ai xni þ Rk f ¼ f k þ Rk f

ð12Þ

i¼1

where f k is the current approximation and Rk f is the current residual. Initializing R0 f ¼ f ; f 0 ¼ 0; k ¼ 1, the procedures of MP algorithm are summarized as follows [29,36–38]: (I) Calculate the inner products fhRk f ; xn ign . (II) Find nkþ1 such that

    R k f ; xn asupj Rk f ; xf kþ1

ð13Þ

where 0 < a  1. (III) Set

  f kþ1 ¼ f k þ Rk f ; xnkþ1 xnkþ1

ð14Þ

  Rkþ1 f ¼ Rk f  Rk f ; xnkþ1 xnkþ1

ð15Þ

  (IV) Set k = k + 1, and repeat steps (I) – (IV) until the criterion is converged, relying on that Rkþ1 f ; xnkþ1 ¼ 0 [36]. 3.2.2. Procedures of Orthogonal Matching Pursuit As a shortcoming of the MP algorithm, the approximation could be suboptimal in some cases even though the convergence criterion has been satisfied [36,39]. Full backward orthogonality of the residual during each iteration is maintained to improve convergence, thus improving the shortage of only guaranteeing the orthogonality between the residual and the best atom via MP algorithm [39,40]. Introducing the kth-order model for f 2 H,

f ¼

k X n¼1

akn xn þ Rk f ; with hRk f ; xn i ¼ 0; n ¼ 1; . . . k

ð16Þ

6

H. Li et al. / Mechanical Systems and Signal Processing 133 (2019) 106276

where the superscript k in akn denotes the different weight on different model order. The above kth-order model is updated to a model of (k + 1)th-order,

f ¼

kþ1 X

akþ1 n xn þ R kþ1 f ; with hRkþ1 f ; xn i ¼ 0; n ¼ 1;    ; k þ 1

ð17Þ

n¼1

An auxiliary model for the relationship between xkþ1 and all the previous xk ðk ¼ 1;    ; nÞ is required to perform the transform [36,41].

xkþ1 ¼

k X

k

bn xn þ ck ; with hck ; xn i ¼ 0; n ¼ 1; . . . ; k

ð18Þ

n¼1

According to Eqs. (11) and (12), it can be obtained that, k X

k

bn xn ¼ PVk xkþ1

ð19Þ

n¼1

ck ¼ PV?k xkþ1

ð20Þ

Thus the correct can be updated by a loop computation [36,41], as below: k

akþ1 ¼ akn  ak bn ; n ¼ 1;    ; k n

ð21Þ

2 akþ1 kþ1 ¼ ak ¼ hRk f ; xkþ1 i=hck ; xkþ1 i ¼ hRk f ; xkþ1 i=k ck k
 k P k ¼ hRk f ; xkþ1 i= k xkþ1 k2  bn hxn ; xkþ1 i

ð22Þ

n¼1

When the residual Rk+1f is satisfied, Rk f ¼ Rkþ1 f þ ak ck , and the convergence is obtained: 2

2

k Rkþ1 f k ¼ k Rk f k  jhRk ; xkþ1 ij2 =k ck k2

ð23Þ

Finally, based on the above OMP theory, the procedures of sparsity adaptive OMP algorithm improved via optimal parameter analysis can be carried out as follows: (1) Original admittance data is first performed normalization as a pre-processing step for better calculation performance. (2) The range of sparsity level [Ks: Ke] is selected via empirical reference and trialand-error tests. (3) The selected sparsity level K is input to the parameter analysis procedure, and corresponding measurement number, iterations and other parameters are set out to obtaining the linear measurements. (4) Making use of the OMP algorithm, sparse coefficient matrix, reconstructed data and total error are exported for parameter analysis. The above procedure is adaptive to obtain the optimal sparse level for subsequent damage detection based on reconstructed admittance data. The whole procedures are constructed step by step in Fig. 2. By introducing an additional orthogonalization step, the OMP algorithm can perform excellent results in its guaranty of convergence to the projection after finite iterations. The feasibility of OMP algorithm for data handling is validated in the following experimental studies. 4. Validation of the approach using monitoring data obtained from an simply-supported steel beam 4.1. Procedure of the simply-supported beam experiment An experiment of steel beam with 500 mm  35 mm  5 mm (length, width, thickness) rectangular section was implemented to validate the proposed approach in this article. The steel beam was simply-supported on both the ends within block rubbers on a desk. Six PZT patches were bonded to the beam as showed in Fig. 3, and the material parameters were listed in Table 1. PZT#1 and PZT#6 were 200 mm apart from the midspan, PZT#2 and PZT#5 were 100 mm apart from the midspan, and PZT#3 and PZT#4 were 50 mm from the midspan respectively. The experimental procedure was designed as four steps. To serve as a reference, all the sensors in case 1 (C1) were perfectly bonded to the steel beam except PZT#4, which was partially bonded as approximately three-quarter area of the patch was painted with high strength epoxy, while the steel beam was undamaged. PZT#5 was scratched with a knife as slight sensor damage while PZT#6 was damaged to several fissures as severe damage relatively (case 2, C2). Then, eight coins with dimeter of 19 mm, thickness of 1.67 mm, and mass of 3.20 g were placed in the middle of the beam span, as a kind of structural damage caused by added mass (case 3, C3). In the last step, a notch was cut on the mid-span (case 4, C4), the dimension of which was 35 mm  4 mm  2 mm (length, width, thickness). These experimental conditions are presented in Fig. 4. The whole experimental process was under approximately 20 °C, all the impedance measurements were collected in each case via a structural monitoring system consisted of a laptop and commercial-available Agilent 4294A analyzer within 1 V voltage excitation.

H. Li et al. / Mechanical Systems and Signal Processing 133 (2019) 106276

Fig. 2. Procedures of the improved OMP algorithm.

Impedance analyzer

Laptop

PZT

Tested steel beam

Fig. 3. Measuring system for steel beam with damaged sensors by Agilent 4294A impedance analyzer.

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H. Li et al. / Mechanical Systems and Signal Processing 133 (2019) 106276

Table 1 Main material characteristics of PZT patches. Label

Density (g/cm2)

dielectric constant

electromechanical coupling coefficient

piezoelectric coefficients (C/N)

Curie temperature (°C)

insulation resistance (mX)

PZT#1

7.86

1400 ± 810%

0.8

400  10@12

330

1000

Case 1: undamaged beam

PZT#1

25

PZT#2

100

partially bonded PZT#4 PZT#5

PZT#3

50

50

50

PZT#6

50

100

25

500 Case 2: undamaged beam

PZT#1

25

PZT#2

100

partially bonded PZT#4

PZT#3

50

50

50

scratched PZT#5

50

damaged PZT#6

100

25

500 Case 3: mass added in the middle of the span partially bonded PZT#4

Coins PZT#1

PZT#2

25

100

50

PZT#3

50

50

scratched

damaged

PZT#5

50

PZT#6

100

25

500 Case 4: notch cut on the midline of the span

PZT#1

PZT#2

partially bonded PZT#4

PZT#3

scratched PZT#5

damaged PZT#6

Notch

25

100

50

50

50

50

100

25

500

Fig. 4. Sensor placement and experiment conditions of the tested steel beam.

4.2. Impedance data compression and reconstruction using OMP algorithm 4.2.1. Data reconstruction results First of all, linear normalization processing was performed to all the original impedance data to increase the efficiency and accuracy for obtaining the optimum solution via the gradient descent method, as below: 0

x ¼ ðx  minðxÞÞ=ðmaxðxÞ  minðxÞÞ

ð24Þ

Comparisons between original discrete admittance signal X(t) and reconstructed signal Y(t) in different cases are conducted, where the sparsity level of all the above measurements ranges from 400 to 2100, and the number of observation alters correspondingly, over a frequency range of 80 to 440 kHz. It is noted that the sparsity level is estimated via firstly performing empirical parameters using several sparsity levels from a geometric progression, then selecting the level for minimizing the difference between the perturbed measurement vectorsY ¼ UX and the output one UX, namely kUX  Yk2 , until a proper range is determined. Fig. 5 shows partial comparisons when it comes to 1300 for the sparsity level K, where the

H. Li et al. / Mechanical Systems and Signal Processing 133 (2019) 106276

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Fig. 5. Comparisons of original admittance data with reconstructed data: (a) PZT#2 (perfectly bonded) in case 1, (b) PZT#4 (partially bonded) in case 1, (c) PZT#5 (undamaged) in case 1, (d) PZT#5 (scratched) in case 2, (e) PZT#6 (damaged) in case 2, (f) PZT#2 (undamaged) in case 3, (g) PZT#5 (scratched) in case 3, (h) PZT#6 (damaged) in case 3, (i) PZT#1 (undamaged) in case 4, (j) PZT#5 (scratched) in case 4, (k) PZT#6 (damaged) in case 4.

conductance variation tendencies along with frequency in four different cases are presented. Three kinds of sensor damage severities, including healthy, scratched and damaged states, which were conducted by a knife, were set up to consider selfdamage of sensors. Signals from PZT#1, PZT#2, PZT#3, PZT#5 and PZT#6 which were perfectly bonded in case 1 were utilized as baseline, while a partially bonded sensor PZT#4 was designated as a reference to provide the situation suffering from sensor bonding issues. Conductance signals of PZT#5 and PZT#6 in case 2 were collected after the treatment for PZT patches. In both case 3 and case 4, PZT#1, PZT#2, PZT#3 were regarded as healthy sensors, so-called regular monitoring situation, PZT#5 and PZT#6 represented damaged sensors as contrast, which were utilized to investigate the effectiveness of identification when encountering sensor faults. It can be concluded that the reconstructed data matched well with the corresponding original data within a certain allowable error, despite some big fluctuations along with the climbing section of the curve. For each case, the entire procedure of compressing and reconstructing the admittance signals of all the six PZTs consumed merely 5260 s, which is short in longterm practical monitoring process. The results also demonstrate that different kinds of sensor conditions would not affect the feasibility of the reconstruction algorithm to some degree. This is important in the SHM process, since the PZT transducer itself may also subject to damages. Maximum reconstruction of the original signal is actually imperative for ensuring the accurate results of structural and sensor diagnosis. 4.2.2. Compression ratio In order to achieve in quantifying compression effects, compression ratio is introduced here to measure the data compression performance:

CR ¼ Sorig =Scomp

ð25Þ

where Sorig and Scomp denote the original data size and compressed data size respectively. Compression ratios for different sparsity levels in various cases are shown in Fig. 6: for K = 400, the compression ratio can be up to around 10, while low to approximate 1.7 for K = 2100. Compression ratio curves are nonlinear but monotone decreasing along with sparsity level, while compression effect is also decreased. The variation tendency shows similar characteristics in different cases. 4.2.3. Influences of sparsity level on reconstruction accuracy Fig. 7 shows the corresponding total errors for different sparsity levels to discern the optimal sparsity for each PZT sensor data processing. As for the total error, there is a sharp decline at first, it then falls into a steady phase, but rises again towards the end. Note that majority of the optimal total errors ranges from 1.67% to 2.79%, except the partially bonded PZT#4 data, whose optimal total error is 6.84% in case 1, 7.24% in case 3 and 6.69% in case 4 respectively, namely that reconstructed

10

H. Li et al. / Mechanical Systems and Signal Processing 133 (2019) 106276

Fig. 6. Compression ratio for different sparsity levels: (a) case 1, (b) case 2, (c) case 3, (d) case 4.

admittance data with bonding defect is considered to cause distortions. Therefore, it is difficult to determine whether the reconstructed data from PZT#4 could be utilized to identify the structural damage. The total error can reach to the lowest level when the sparsity K is in the range of 1400 to 1600 for all the impedance data. 4.3. Damage detections with the reconstructed data Comparison for original normalized data of PZT#2 in different cases is shown in Fig. 8(a). It is found that PZT#2 has small changes on the processed signal under the mass addition case, while that under notch damage case shows an obvious left shift of the curve in Fig. 8(b), similarly in the reconstructed normalized data of PZT#2, both successfully indicate the occurrence of structural damages. Fig. 8(c) and (d) show that PZT#3 can also distinguish the healthy and the abnormal conditions since an obvious increase and left shift in the main resonant peaks are presented. Small fluctuations of increase and decrease in the resonance peaks in Fig. 8(e) indicate that the partially-debonded PZT#4 almost lost its ability to detect structural damage. Similar characters can be also seen in the reconstructed data in Fig. 8(f). Fig. 8(g) and (h) demonstrate that magnitude decreases in resonance peaks for PZT#5 after scratched, and fluctuations in resonance peaks without obvious shifts in the resonance frequency hardly indicate the structural crack damage, due to the deterioration of sensor validity [1,7]. It is clear that the reconstructed data well repeats the features of the original ones. These observations indicate that the original data from different conditions can be effectively replaced by the reconstructed ones. To more clearly discern the difference between the original and reconstructed admittance signatures, damage metrics are further computed and compared in the next section. 4.3.1. Damage detection using RMSD The RMSD of the real part of the admittance data is utilized as a damage indicator, expressed as [2]:

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Fig. 7. Total error for different sparsity levels: (a) PZT#3 (perfectly bonded) in case 1, (b) PZT#4 (partially bonded) in case 1, (c) PZT#5 (undamaged) in case 1, (d) PZT#5 (scratched) in case 2, (e) PZT#6 (damaged) in case 2, (f) PZT#3 (undamaged) in case 3, (g) PZT#4 (partially bonded) in case 3, (h) PZT#5 (scratched) in case 3, (i) PZT#6 (damaged) in case 3, (j) PZT#3 (undamaged) in case 4, (k) PZT#4 (partially bonded) in case 4, (l) PZT#5 (scratched) in case 4.

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u n

n

2 X 2 uX 1 RMSD ¼ t Gi  G0i = G0i i¼1

ð26Þ

i¼1

where G0i denotes the undamaged conductance signature at the i-th measurement point, G1i denotes the damaged conductance signature at the corresponding point. 4.3.2. Damage detection using CC CC index as an effective indicator in damage quantification is also introduced to assess the damage, expressed as [8]:

 CC ¼ Cov G0 ; G1 =r0 r1

ð27Þ

where r0 and r1 denote the standard deviation of the undamaged and damaged conductance data respectively. Both the baselines of the RMSD and CC values are calculated before the structural damage is introduced. Fig. 8(a) shows that the RMSD index for the original PZT#2 data between case 1 and case 3 is 0.071, while the reconstructed data denotes 0.076, where the error is 7.183%. The RMSD index for PZT#3 is more accurate since the error between the original and reconstructed data denotes only 1.499%. Furthermore, the differences of the RMSD index between the mass-added case and the notch-damaged case can be clearly distinguished. A distinct increase of RMSD index is found from the case 3 to the case 4 in Fig. 9(a) and (b). Fig. 9(c) also displays the original and reconstructed data based RMSD indices for PZT#5 and PZT#6. RMSD of PZT#6 in notch-damaged case is 0.442, almost 8 times more than that in the mass-added case, indicating the development of structural damages. However, there is little difference of RMSD for PZT#6 among the reference case, the massadded case and the notch-damaged case. In addition, Fig. 9(c) shows that the error of the reconstructed PZT#6 data in case 3 denotes 17.082%, which indicates the sensor may be damaged and could not offer effective feedback, whereas those of the scratched PZT#5 are all under 6%, which demonstrates that the sensor data was still available, as a result of merely a light deterioration. Fig. 10(a) shows that the CC index for the reconstructed data of PZT#2 and PZT#3 match well with the corresponding original data in different cases, as same as that of PZT#4 in Fig. 10(b), PZT#5 and PZT#6 in Fig. 10(c). Due to the negative relationship between CC index and structural damage severity, it can be seen that the CC index for the original/reconstructed PZT#2 data between the undamaged case 1 and case 3 is 1.000 and 0.994, which successfully denotes the mass additional damage in case 3, and then it decreases from case 3 to case 4 for PZT#2 (0.994 vs. 0.962) denotes the increase of structural damage severity (i.e. from mass addition to crack damage). Similarly, the indices for PZT#3 and PZT#4 respectively decrease

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Fig. 8. Comparisons among different cases for normalized admittance data: (a) original PZT#2, (b) reconstructed PZT#2, (c) original PZT#3, (d) reconstructed PZT#3, (e) original PZT#4, (f) reconstructed PZT#4, (g) original PZT#5, (h) reconstructed PZT#5.

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Fig. 9. Comparisons of data damage metrics using RMSD depending on original data and reconstructed data: (a) PZT#2 and PZT#3 using RMSD, (b) PZT#3 and PZT#4 using RMSD, (c) PZT#5 and PZT#6 using RMSD.

Fig. 10. Comparisons of data damage metrics using CC depending on original data and reconstructed data: (a) PZT#2 and PZT#3 using CC, (b) PZT#2 and PZT#4 using CC, (c) PZT#5 and PZT#6 using CC.

from 0.997 to 0.977 and from 0.996 to 0.978, which also effectively indicate the damage growth. Additionally, the CC values for PZT#5 and PZT#6 in case 2 (i.e., 0.984 and 0.734, respectively) can account for the sensor scratch and damaged condition respectively, and the decreased CC values from case 3 to case 4 (0.995 vs. 0.989 for PZT#5 and 0.997 vs. 0.772 for PZT#6) further indicate the increase of structural damage severity. The damage detection results by CC are primarily consistent with that by RMSD, in spite of their reverse nature in damage quantification. All the errors of the CC values between the original and reconstructed data are under 0.20% except that of PZT#4 is 0.964%, which could be possibly attributed to that the multiple resonance peaks in the admittance signal caused by that partial debonding decrease the reconstructed efficiency.

5. Application of the approach using monitoring data obtained from a full-scaled shield tunnel segment 5.1. Experimental procedure This section further validates the proposed approach with an application to a long-time monitoring of full-scaled shield tunnel-segment structure. As shown in Fig. 11, the segment structure with dimensions of 4000/3500 mm  500 mm  2432 mm (external/internal diameter, wall thickness, width) was implemented. The standard blocks, contiguous blocks and roof block were connected with each other via bolts along both the longitudinal and transverse direction. Epoxy resin was utilized to bond the PZT sensors to the structural surface. Transverse bolts, namely Bolt#1 and Bolt #2 in Fig. 12(a), were loosened as a damage situation for shield tunnel operation, where bolt #1 was loosened as case 2 and both the bolts were loosened as case 3. Three groups of tests were conducted under different temperature and humidity to compare the monitoring effects in different environments, where the first one was at 27.5 °C and the humidity was 84%, the second one was 17.5 °C and the similar humidity after a half year, and the third one was 27.7 °C and 77% humidity after about one year. The sensor arrangement of the 1st test is showed in Fig. 12(a), where PZT#1, PZT#3, PZT#7 were bonded on the edge of concrete bolt hole, PZT#5 and PZT#6 were bonded near the edge of the segment joining, PZT#4 and PZT#8 were bonded on the middle of the segment in the first group test. Different from other sensors bonded on the concrete surface, PZT#2 was bonded on the steel screw nut, which was 600 mm from the loosened bolt nut. PZT#1 till #8 was 600 mm, 600 mm, 0 mm, 300 mm, 300 mm, 350 mm, 600 mm and 650 mm from bolt #1, while 850 mm, 850 mm, 600 mm, 300 mm, 350 mm, 350 mm, 0 mm, 650 mm from bolt #2 respectively. Admittance data was collected by HP4294A impedance analyser in a realtime structural monitoring system. In both the second and the third group test, the sensors were installed and arranged as shown in Fig. 12(b), more PZT sensors were bonded on the concrete segment to obtain more information. PZT#4, PZT#5, PZT#6, PZT#7 and PZT#8 in the 1st test were replaced by PZT#5, PZT#7, PZT#9, PZT#11 and PZT#13 in 2nd and 3rd tests respectively. PZT#6, PZT#8,

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0m

m

14

250mm

175

250mm

3500mm

Laptop PZT located areas

Impedance analyzer

Prototype of tunnel segment structure Fig. 11. Structural monitoring system for the full-scale shield tunnel segment structure.

Fig. 12. Sensor placement and experiment conditions of the shield tunnel segment: (a) the first group test, (b) the second group test and the third group test.

PZT#10 and PZT#17 were bonded near the corresponding sensors to realize repeatability. PZT#12 was bonded near the concrete hole while PZT#16 was set on the bolt hole. The long-term temperature effect was considered when reconstructing the data.

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Fig. 13. Comparisons of original admittance data with reconstructed data in the first group test: (a) PZT#1 in case 1, (b) PZT#3 in case 1, (c) PZT#5 in case 2, (d) PZT#5 in case 3, (e) PZT#7 in case 1, (f) PZT#7 in case 2, (g) PZT#7 in case 3.

5.2. Impedance data compression and reconstruction using OMP algorithm 5.2.1. Data reconstruction results Fig. 13 shows the comparisons between original admittance data and reconstructed impedance data under different situations in the first test. The frequency band ranges within 40 Hz-320 kHz. Consuming time for compressing and reconstructing the admittance signal of one PZT in each case was approximate 200 s, which was much less than that used for steel beam (Section 4.2.1). The results again demonstrate that the proposed approach is excellent in the application of recovering monitoring data from the shield tunnel segment. Admittance data in Fig. 14 was collected in different temperature conditions comparing to the 1st test (17.5 °C vs. 27.5 °C). Fig. 14(a)–(g) is corresponding to that in Fig. 13. Noted that PZT#7 in the 2nd test was on the same position of PZT#5 in the 1st test, which was on the edge of the segment joining, while PZT#11 in 2nd test represented PZT#7 in the 1st test, which was bonded on the edge of Bolt#2 hole. Results of the 2nd test show high accuracy of reconstruction, which indicate that the proposed algorithm is immune from the temperature impact, since data reconstruction using OMP rises superior to up-anddown fluctuations or right-and-left shifts of the data to be processed in the whole algorithm. Comparisons between original and reconstructed data in the 3rd test are shown in Fig. 15, which are corresponding to those in the 1st test, explaining for the reconstruction effectiveness considering the time-varying effect on the PZT transducers. The results indicate that the algorithm is still reliable in long-time monitoring process, in spite of the admittance signals might be altered by temperature and humidity. The above illustrations sufficiently validate the robustness of the proposed approach regardless of environmental conditions. 5.2.2. Compression ratio Compression ratios for different sparsity levels are presented in Fig. 16. It is clear that compression ratio can be up to 16 when the sparsity comes to 100, while low to 1.7778 when the sparsity is 900. As for all the compression ratios, it comes from constant high level to a minimum after experiencing a sharp reduction. 5.2.3. Influences of sparsity level on reconstruction accuracy The optimal sparsity for each PZT is presented in Fig. 17, where the total error varies along with the corresponding sparsity level. The following total errors are all within 2%, except PZT#4 in case 2 of the 3rd test, possibly attributed to the multiple resonance peaks in admittance signals caused by the deterioration of bonding layer after one-year monitoring, as explained in the beam experiment. The effectiveness of reconstruction comes to an optimum, among which PZT#1, PZT#7 and PZT#13 turn out to be close to optimum value, when the sparsity K is around 600. 5.3. Damage quantification using statistical metrics 5.3.1. Damage detection using RMSD Fig. 18(a) and (b) show that the RMSD values for the original data of PZT#1-PZT#8 between the baseline case (i.e. Case 1/C1) and the damaged cases (i.e. one-bolt-loosened case: C2, two-bolts-loosened case: C3). The RMSD index for PZT#3 and PZT#7 in the first group test is more accurate than other PZTs since the error between the original and reconstructed data are below 0.839%. Besides, the effects of the RMSD index for the reconstructed data from other PZT sensors in the first group test are also reliable, whose total errors are all less than 3.626%. The difference of the RMSD values between case 2 and

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Fig. 14. Comparisons of original admittance data with reconstructed data in the second group test: (a) PZT#1 in case 1, (b) PZT#3 in case 1, (c) PZT#7 in case 2, (d) PZT#7 in case 3, (e) PZT#11 in case 1, (f) PZT#11 in case 2, (g) PZT#11 in case 3, (h) PZT#4 in case 1, (i) PZT#12 in case 2, (j) PZT#14 in case 3, (k) PZT#16 in case 2.

Fig. 15. Comparison of original admittance data with reconstructed data in the third group test: (a) PZT#1 in case 1, (b) PZT#3 in case 1, (c) PZT#7 in case 2, (d) PZT#11 in case 1, (e) PZT#11 in case 2, (f) PZT#4 in case 1, (g) PZT#12 in case 2, (h) PZT#16 in case 2.

case 3 can be distinguished, on account of a proportion larger than 5.575%. The RMSD values of PZT#2 in the first test are larger than other values, mainly because this patch was bonded on the screw nut and more sensitive to bolt loosen than that on the concrete surface. In addition, the RMSD value of PZT#1 in the first group test has an obvious decrease as high as 21.339% from case 2 to case 3, by which the damage degree can be distinguished clearly, and so is PZT#7. It is noted that PZT#1, PZT#3 and PZT#7 were on the edge of the bolt hole, whose RMSD values vary much more than those of PZT#4 and PZT#5. It is also noted that a large variation of the RMSD index for PZT#6 and PZT#8 is showed in Fig. 18(b), while PZT#6 was on the edge of the segment joining, 350 mm far from bolt#1, and PZT#8 was on the middle of the segment, 650 mm far from bolt#1, which indicates that the variations of the PZT sensors far away from the damage position can not offer the damage degree information. Hence that PZT sensors are recommended to be bonded on bolt nuts to monitor

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Fig. 16. Compression ratio for different sparsity levels: (a) case 1 of the first group test, (b) case 1 of the second group test, (c) case 1 of the third group test.

Fig. 17. Total error for different sparsity levels: (a) PZT#1 in case 1 of 1st group test, (b) PZT#3 in case 1 of 1st group test, (c) PZT#5 in case 2 of 1st group test, (d) PZT#7 in case 1 of 1st group test, (e) PZT#1 in case 1 of 2nd group test, (f) PZT#3 in case 1 of 2nd group test, (g) PZT#7 in case 2 of 2nd group test, (h) PZT#11 in case 1 of 2nd group test, (i) PZT#1 in case 1 of 3rd group test, (j) PZT#3 in case 1 of 3rd group test, (k) PZT#7 in case 2 of 3rd test, (l) PZT#11 in case 1 of 3rd group test.

health status aimed at bolt-loosened situation, furthermore, sensors should be arranged near the structural key positions directly, where damages probably occurred. The RMSD indices of the second group exhibit high sensitivity in distinguishing one-bolt-loosened case and two-boltsloosened case from the results of Fig. 18(c)–(f). Most of the aforementioned RMSD values have an increase more than 8% from case 2 to case 3, resulting in a similar situation as that in the steel beam experiment (Section 4). 5.3.2. Damage detection using CC Fig. 19 shows that the CC indices for the reconstructed data match well with the corresponding original data in different cases, since the total errors are not exceeding 0.871%. Smaller CC values for PZT#1–3 than the other PZTs in the 1st and 2nd tests denote that these regions (i.e. near the loosen bolt or on the bolt nut) are sensitive to bolt loosen damage, and so is PZT#7, these observations are coincident with the RMSD results. The CC values of PZT#8, PZT#9, PZT#10, PZT#14 in the 2nd test show a little decrease from case 2 to case 3 (Fig. 19d–f), which could be caused by the larger stress on the segment joints when another bolt was loosened. However, majority of the changes in CC values from one-bolt-loosened case to

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Fig. 18. Comparisons of data damage metrics using RMSD depending on original data and reconstructed data: (a) PZT#1-PZT#4 of 1st test using RMSD, (b) PZT#5-PZT#8 of 1st test using RMSD, (c) PZT#1-PZT#4 of 2nd test using RMSD, (d) PZT#5-PZT#8 of 2nd test using RMSD, (e) PZT#9-PZT#12 of 2nd test using RMSD, (f) PZT#13-PZT#16 of 2nd test using RMSD.

Fig. 19. Comparisons of data damage metrics using CC depending on original data and reconstructed data: (a) PZT#1-PZT#4 of 1st test using CC, (b) PZT#5-PZT#8 of 1st test using CC, (c) PZT#1-PZT#4 of 2nd test using CC, (d) PZT#5-PZT#8 of 2nd test using CC, (e) PZT#9-PZT#12 of 2nd test using CC, (f) PZT#13-PZT#16 of 2nd test using CC.

two-bolts-loosened case are below 0.462% much smaller than that in RMSD index (more than 8%). In this respect, RMSD index is more sensitive than CC index for evaluating bolt-loosen damages on the shield-tunnel segment structure. 6. Conclusions This article proposed an innovative data compression/reconstruction approach for EMA signals based SHM in frequency domain. In the approach, the original impedance/admittance signals of PZT transducer interacted with its host structure

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were first collected. Then a linear projection was implemented for the transmitted data in the system by a random matrix. A sparsity-adaptive OMP algorithm improved via optimal parametric analysis was performed to reconstruct the normalized admittance data. To validate the proposed method, experiment on a simply-supported steel beam bonded with multiple PZT transducers under different conditions was conducted to measure the original admittance signals. Both the reconstruction error and compression ratio were discussed to evaluate the influences of sparsity level K on the accuracy of data reconstruction. It was found that the total error between the original and reconstructed signals showed a sharp decline at first and then remained a steady phase, however an increase was presented to the end. Reconstructed data displayed uniform accordance with the original ones, regardless of the transducer states and structural damage conditions. Quantitative comparisons of the reconstructed and original signals for damage identification using RMSD and CC demonstrated that the reconstructed signals behaved a high accuracy in damage quantification. RMSD index was also found more effective than CC index in damage identification, even to mass additional damages. Furthermore, the proposed approach was applied to a long-time health monitoring of a full-size shield tunnel concrete structure in the laboratory environment. The compressed and reconstructed results again presented high accuracy in recovering the original signal characters, which indicated the proposed approach was also suitable for the SHM of prototype concrete structures under different environmental conditions. Specifically, the structural bolt-loosened damages can be well identified and damage degree can be quantified via the variations of RMSD with regard to the admittance of PZT transducers near the bolt holes, while those far away from the damage locus failed to offer accurate damage information, as they exceeded the monitoring scope. Admittance signals in these damage cases were effectively reconstructed when the processed data showed a sparse character on some basis, and the reconstruction error can be decreased to an acceptable level via parametric analysis on sparsity level. Therefore, it can be concluded that the proposed method showed high accuracy and strong robustness in the applications of impedance based metal and concrete SHM, which possessed the capability to immune from the sensor conditions and environmental impacts. A step further, the reduction of optimal reconstruction error using more efficient way is necessary to provide reliable information and reduce false alarms in structural damage detection. Besides, wireless sensing technology of PZT sensors is beneficial to make the data acquisition system more convenient, where the embedded module design of the proposed algorithm procedures is under developing to future engineering applications.

Acknowledgement The authors gratefully acknowledge the Natural Science Foundation of China (Grant Nos. 51808242, 51578260, 51578261) and the China Postdoctoral Science Foundation (Grant No. 2018M632861).

References [1] G. Park, H.H. Cudney, D.J. Inman, Feasibility of using impedance-based damage assessment for pipeline structures, Earthq. Eng. Struct. D 30 (2001) 1463–1474. [2] C.K. Soh, C.K. Tseng, S. Bhalla, A. Gupta, Performance of smart piezoceramic patches in health monitoring of a RC bridge, Smart Mater. Struct. 9 (2000) 533–542. [3] V.G.M. Annamadas, C.K. Soh, Load monitoring using a calibrated piezo diaphragm based impedance strain sensor and wireless sensor network in real time, Smart Mater. Struct. 26 (2017) 045036. [4] D. Ai, H. Zhu, H. Luo, J. Yang, An effective electromechanical impedance technique for steel structural health monitoring, Constr. Build. Mater. 73 (2014) 97–104. [5] D. Ai, H. Luo, C. Wang, H. Zhu, Monitoring of the load-induced RC beam structural tension/compression stress and damage using piezoelectric transducers, Eng. Struct. 154 (2018) 38–51. [6] D. Ai, H. Zhu, H. Luo, C. Wang, Mechanical impedance based embedded piezoelectric transducer for reinforced concrete structural impact damage detection: a comparative study, Constr. Build. Mater. 165 (2018) 472–483. [7] D. Ai, H. Luo, H. Zhu, Diagnosis and validation of damaged piezoelectric sensor in electromechanical impedance technique, J. Intel. Mat. Syst. Str. 28 (2017) 837–850. [8] L.M. Campeiro, R.Z. Silveira, F.G. Baptista, Impedance-based damage detection under noise and vibration effects, Struct. Health Monit. 17 (2018) 654– 667. [9] D. Gao, Z. Wu, L. Yang, et al, Integrated impedance and Lamb wave-based structural health monitoring strategy for long-term cycle-loaded composite structure, Struct. Health Monit. 17 (2018) 763–776. [10] T. Visalakshi, S. Bhalla, A. Gupta, Monitoring early hydration of reinforced concrete struc-tures using structural parameters identified by piezo sensors via electromechanical impedance technique, Mech. Syst. Signal Pr. 99 (2018) 129–141. [11] H.J. Lim, M.K. Kim, H. Sohn, C.Y. Park, Impedance based damage detection under varying temperature and loading conditions, NDT&E Int. 44 (2011) 740–750. [12] J.Y. Min, S. Park, C.B. Yun, C.G. Lee, C. Lee, Impedance-based structural health monitoring incorporating neural network technique for identification of damage type and severity, Eng. Struct. 39 (2012) 210–220. [13] P. Selva, O. Cherrier, V. Budinger, F. Lachaud, J. Morlier, Smart monitoring of aeronautical composite plates based on electromechanical impedance measurements and artificial neural networks, Eng. Struct. 56 (2013) 794–804. [14] G. Gui, H. Pan, Z. Lin, Y. Li, Z. Yuan, Data-driven support vector machine with optimization techniques for structural health monitoring and damage detection, KSCE J. Civ. Eng. 21 (2017) 523–534. [15] H. Sohn, C.R. Farrar, A statistical pattern recognition paradigm for vibration-based structural health monitoring, Proc Smart Engineering System Design, Neural Networks, Fuzzy Logic, Evolutionary Programming, Complex Systems, and Data Mining (St. Louis, Missouri, USA). (2000) 57–64. [16] S. Park, J.J. Lee, C.B. Yun, D.J. Inman, Electro-mechanical impedance-based wireless structural health monitoring using PCA-data compression and kmeans clustering algorithms, J. Intel. Mat. Syst. Str. 00 (2008) 1–12. [17] R.M. Levine, J.E. Michaels, Model-based imaging of damage with Lamb waves via sparse reconstruction, J. Acoust. Soc. Am. 133 (2013) 1525–1534. [18] Z. Duan, J. Kang, Compressed sensing techniques for arbitrary frequency-sparse signals in structural health monitoring, Proc SPIE 9061, Sensors and Smart Structures Technologies for Civil, Mechanical, and Aerospace Systems (San Diego, California, United States) 9061 (2014) 90612W.

20

H. Li et al. / Mechanical Systems and Signal Processing 133 (2019) 106276

[19] Y. Sun, F. Gu, S. Ji, L. Wang, Composite plate phased array structural health monitoring signal reconstruction based on orthogonal matching pursuit algorithm, J. Sensors 2017 (2017) 3157329. [20] Y. Sun, F. Gu, Compressive sensing of piezoelectric sensor response signal for phased array structural health monitoring, Int. J. Sens. Netw. 23 (2017) 258. [21] M. Jayawardhana, X. Zhu, R. Liyanapathirana, Compressive sensing for efficient health monitoring and effective damage detection of structures, Mech. Syst. Signal Pr. 84 (2017) 414–430. [22] Y. Bao, J.L. Beck, H. Li, Compressive sampling for accelerometer signals in structural health monitoring, Struct. Health Monit. 10 (2011) 235–246. [23] Y. Bao, H. Li, X. Sun, Y. Yu, J. Ou, Compressive sampling-based data loss recovery for wireless sensor networks used in civil structural health monitoring, Struct. Health Monit. 12 (2013) 78–95. [24] Y. Bao, Z. Shi, J.L. Beck, H. Li, T.Y. Hou, Identification of time-varying cable tension forces based on adaptive sparse time-frequency analysis of cable vibrations, Struct. Control Hlth. 24 (2017) 1–17. [25] X. Fan, J. Li, H. Hao, et al, Identification of minor structural damage based on electromechan-ical impedance sensitivity and sparse regularization, J. Aerospace Eng. 31 (2018) 04018061. [26] C. Pan, L. Yu, H. Liu, et al, Moving force identification based on redundant concatenated dictionary and weighted l1-norm regularization, Mech. Syst. Signal Pr. 98 (2018) 32–49. [27] C. Liang, F.P. Sun, C.A. Rogers, Coupled electromechanical analysis of piezoelectric ceramic actuator-driven systems: determination of the actuator power consumption and system energy transfer, Proc SPIE 1917, Smart Structures and Intelligent Systems (Albuquerque, NM, United States) 1993 (1917) 286–298. [28] Y.G. Xu, G.R. Liu, A modified electro-mechanical impedance model of piezoelectric actuator-sensors for debonding detection of composite patches, J. Intel. Mat. Syst. Str. 13 (2002) 389–396. [29] G. Davis, S. Mallat, M. Avellaneda, Adaptive greedy approximations, Constr. Approx. 13 (1997) 57–98. [30] E.J. Candes, Compressive sampling, Proc International Congress of Mathematicsians, (Madrid Spain) (2006) 1433–1452. [31] D.L. Donoho, Compressed sensing, IEEE T. Inform. Theory 52 (2006) 1289–1306. [32] E.J. Candes, J. Romberg, T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information, IEEE T. Inform. Theory 52 (2006) 489–509. [33] E.J. Candes, M.B. Wakin, An introduction to compressive sampling, IEEE Signal Process. Mag. 25 (2008) 21–30. [34] E.J. Candes, T. Tao, Decoding by linear programming, IEEE T. Inform. Theory 51 (2005) 4203–4215. [35] E.J. Candes, J. Romberg, T. Tao, Stable signal recovery from incomplete and inaccurate measurements, Commun. Pur Appl. Anal. 59 (2006) 1207–1223. [36] Y. C. Pati, R. Rezaiifar, P. S. Krishnaprasad, Orthogonal Matching Pursuit: Recursive function approximation with applications to wavelet decomposition, in: Proc 27th Ann, Asilomar Conference on Signals, Systems and Computers (Pacific Grove, California, United States). (1993) pp. 40–44. [37] S.G. Mallat, Z. Zhang, Matching pursuits with time-frequency dictionaries, IEEE T. Inform. Theory 41 (1993) 187–200. [38] S.S. Chen, D.L. Donoho, M.A. Saunders, Atomic decomposition by basis pursuit, SIAM Rev. Soc. Ind. Appl. Math. 43 (2005) 129–159. [39] J.A. Tropp, A.C. Gilbert, Signal recovery from random measurements via orthogonal matching pursuit, IEEE Trans. Inf. Theory 53 (2007) 4655–4666. [40] D. Needell, R. Vershynin, Signal recovery from incomplete and inaccurate measurements via regularized orthogonal matching pursuit, IEEE J. Sel. Top. Signal Process. 4 (2010) 310–316. [41] V.K. Singh, A.K. Rai, M. Kumar, Sparse data recovery using optimized orthogonal matching pursuit for WSNs, Proc. 8th Int. Conf. ANT 109 (2017) 210– 216.