Impact detection and location for a plate structure using least squares support vector machines

Structural Health Monitoring 2014, Vol 13(1) 5–18 Ó The Author(s) 2013 Reprints and permissions: sagepub.co.uk/journalsPermissions.nav DOI: 10.1177/1475921713495083 shm.sagepub.com

Qingsong Xu

Abstract Impact force magnitude detection and site location for clamped plates have direct relevance to the maintenance of aircraft and spacecraft structures. This article presents the impact identification of an aluminum plate structure based on least squares support vector machines. The structure is equipped with an array of surface-bounded piezoelectric sensors, which receive strain response signals excited by hammer impacts on different sites with varying magnitudes. To identify the magnitude and location of the impact, two features are extracted from each sensor signal by Hilbert transform, and intelligent models are established using least squares support vector machines. The regression models are then validated through experimental studies, which reveal that the least squares support vector machine–based approach achieves more reliable identification result and better detection accuracy than conventional artificial neural networks. In addition, an empirical index is presented to compare the system efficiency in terms of detection accuracy as well as the numbers of sensors and data sets. Results show that the proposed least squares support vector machine–based impact identification system affords a higher efficiency than the existing ones because it accomplishes a moderate detection accuracy along with greatly reduced hardware cost and workload. Keywords Impact identification, plate structures, artificial neural networks, support vector machines, structural health monitoring

Introduction Structural health monitoring (SHM) deals with damage identification for aerospace, civil, and mechanical infrastructures, and SHM has gained considerable attentions toward diverse engineering applications.1,2 In aircraft structures, plate and composite materials have been applied more and more widely owing to their merits in terms of high strength, light weight, and flexibility in design.3 Although the structures have been designed to tolerate certain degree of failures of the material, maintenance is required to guarantee the structural integrity for safety reason. During the maintenance, the visible damage can be easily detected and repaired. On the contrary, the invisible and undetected damage and fatigue induced by low-velocity impacts pose great challenges to maintenance workers. Such unwanted impact may be caused by the bird strikes during flight, runway stones during takeoff, and tool drops in maintenance processes.4 This kind of undetected, hidden damage is known as barely visible impact damage (BVID) in aerospace applications.5 Inability to identify the BVIDs

may result in catastrophe. In order to reduce the chance of catastrophe, it is necessary to detect and locate the low-velocity impacts on the structures. For such purpose, various damage identification strategies have been developed in the literature.6–8 Generally, the damage identification of plate structures can be classified into active and passive approaches. In active detection, the monitored structures are actuated or excited and the caused responses are recorded.9,10 Both impedance-based11 and guided wave methods12,13 have been presented, and a review of guided Lamb waves for damage identification was conducted by Su et al.14 In contrast, the passive detection approaches do

Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Macao, China Corresponding author: Qingsong Xu, Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Av. Padre Toma´s Pereira S.J., Taipa, Macao, China. Email: [email protected]

6 not require any actuation, whereas only sensors are used to monitor the responses induced by damages.3,15,16 Due to its simplicity in terms of operation, the latter gains intensive attentions from both industry and academia in the past two decades. In general, two groups of algorithms have been applied in passive detection. The first method is developed on the basis of system modeling.15 Its major shortcoming lies in that it is not easy to establish accurate system model for complex structures. The modeling error will be interpreted as significant identification errors even for simple structures such as plates. The second is based on data-driven techniques that only depend on the input–output data information, whereas no system model information is required. For instance, artificial intelligence (AI) techniques such as fuzzy logic17 and case-based reasoning18 have been applied to SHM. In particular, the machine learning methods are mostly employed to formulate a classification or a regression problem.16,19 Most of the existing works take into account the impact identification as a nonlinear regression problem. Although it can also be solved as a classification problem,20 subsequent regression modeling is usually required to improve the identification accuracy as revealed in LeClerc et al.21 Hence, the regression approach is employed in the present research. As typical machine learning techniques, artificial neural networks (ANNs) have been popularly adopted in previous works. For example, two backpropagation (BP) ANNs were trained to identify the location and magnitude of impact on an isotropic plate in Jones et al.22 However, the identifications of location and magnitude were carried out separately, which complicated the practical implementation procedure. Later, the impact location and quantification of a composite plate were reported in Worden and Staszewski23 by resorting to two ANNs, which were trained by the data acquired from the same setup of experiment. In the aforementioned work, an optimal sensor placement was determined using the genetic algorithm. More recently, the same ANNs have been extended to the impact identification for a real aircraft composite panel.21 It was reported that the ANN-based impact location strategy developed for simple plate structures is applicable to larger structures with more degree of complexity in terms of material and geometry. Hence, the investigation on impact identification for simple structures does make sense for practical applications. Nevertheless, concerning ANN method, it has been recognized that there is no universal method to determine an optimal ANN structure in terms of the number of hidden layers and number of neurons in each layer.4 Moreover, ANN possesses the shortcomings of overfitting and sinking into local optima. Alternatively, support vector machines (SVMs) pave a promising way to estimate nonlinear system models accurately.24,25 Based on

Structural Health Monitoring 13(1) statistical learning theory and structural risk minimization principle,26 SVM is capable of modeling nonlinear systems by transforming the regression problem into a convex quadratic programming (QP) problem and solving it with a QP solver.27 As compared with ANN, SVM exhibits the major advantages of global optimization and higher generalization capability.28 Furthermore, least squares support vector machines (LSSVMs) utilize equality constraints instead of inequality constraints as in the ordinary SVM. Hence, LSSVM simplifies the regression to a problem that can be easily solved from a set of linear equations.29,30 The potential of LSSVM in impact location for a composite plate has been demonstrated by some previous works.31 However, the location accuracy in Xie31 was guaranteed by a large sensor array. It is unclear whether the impact detection result is contributed dominantly by the large-scale sensor array or the LSSVM technique. Moreover, it is unknown how the LSSVM performs in comparison with some ANNs for the same impact identification problem. In this article, the identification of magnitude and location of impact on a clamped aluminum alloy plate structure is carried out by making use of only four surface-bounded piezoceramic discs. The nature of the work is to process the wave signal excited by the impact to find the wave arriving information for impact identification. Based on the acquired strain response data, both BP ANN and LSSVM regression models are trained to determine the force amplitude as well as the impact location. The major contribution of this article lies in the demonstration of the superiority of LSSVMs over some ANNs in terms of force amplitude quantification and impact site estimation through comparative experimental studies. Moreover, an empirical index is proposed to compare the detection efficiencies of various systems, which deliver different identification accuracy using different numbers of sensors and data sets. The remainder of this article is organized as follows. Section ‘‘Impact detection and location strategy’’ outlines the adopted impact detection and location strategy. The LSSVM-based regression models are developed in section ‘‘LSSVM modeling’’ for the determination of the impact amplitude and impact site. Afterward, a series of experimental studies are carried out in section ‘‘Experimental investigations on impact detection and location,’’ where a comparative investigation between ANN and LSSVM methods is performed and an empirical index is proposed to compare the detection efficiencies of different systems. Section ‘‘Conclusion’’ concludes this article and indicates the future work.

Impact detection and location strategy When a metallic or composite plate is impacted with some kinetic energy, propagating waves are induced,

Xu which can be detected by the surface-mounted piezoelectric sensors. The essence of impact identification is to acquire these induced propagating wave signals and to process the signals to find the site and magnitude of the impact. In the current research, impact experiments are carried out on an aluminum alloy (Al-6061) plate. The experimental setup is sketched in Figure 1. The plate structure has a dimension of 490 mm 3 390 mm 3 2.5 mm and is simply clamped on a table using four fixing screws. Four piezoelectric discs (material: Navy Type II) of 6.35 mm diameter and 0.191 mm thickness are bonded on the surface of the plate. Two-component conductive epoxy adhesive (CW2400 from ITW Chemtronics) consisting of hardener and adhesive is used to bond the transducers on to the plate surface. As illustrated in Figure 1, the four transducers are mounted on the xy plane with the coordinates of (170, 290 mm), (320, 290 mm), (320, 100 mm), and (170, 100 mm), respectively. The impacts are applied on the plate surface using an instrumented impulse force hammer (from PCB Piezotronics, Inc.). The magnitudes of the impact forces are kept small in order not to damage the structure. The analog voltage output signals of the piezoelectric sensors are acquired by an NI PCI-4472 board, which provides eight input channels with 24-bit resolution. In addition to the four sensor signals, the impact force signal of the hammer is also recorded. The data acquisitions of the five channels are triggered simultaneously once the amplitude of the impact force exceeds a predefined threshold value. With a sampling rate of 100 kHz, 10,000 data are recorded for each channel.

7 For instance, by applying an impact force signal as shown in Figure 2(a) to the structure, the time-varying strain data provided by piezoelectric sensor 2 are plotted in Figure 2(b). Owing to the dispersive nature of the waves, the signal with explicit amplitudes only occurs in a short time interval of less than 0.06 s. Nonetheless, the overall amount of strain data is too large to be used directly. Hence, feature extraction is conducted for the subsequent analysis. It has been shown that different features can be extracted from the original data.23 In this research, the time after impact of maximum response and the magnitude of maximum response are employed because these two features are the best among the known features.23 In order to extract the two features from each sensor output, the original sensor signals are processed to obtain their envelopes. In particular, the Hilbert transform is employed to obtain the envelope as follows. First, the Hilbert transform of each signal s(t) is computed as follows1,7

sh (t) =

1 p

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Next, the analytic signal is generated as a(t) = s(t) + ish (t)

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qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ s2 (t) + s2h (t)

Figure 1. Schematic diagram of experimental setup for impact magnitude detection and location of a plate structure. PC: personal computer.

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Figure 2. (a) An impact force signal, (b) the induced strain wave signal acquired by piezoelectric sensor 2, and (c) the strain signal envelope obtained by Hilbert transform.

Regarding the sensor signal as shown in Figure 2(b), its envelope is computed as depicted in Figure 2(c). The plot represents the time-varying magnitude of the induced propagating wave, which is acquired by the piezoelectric sensor 2. The maximum peak indicates that the propagating wave is passing the sensor. Based on the envelope of the wave signal, two features in terms of arrival time (A) and amplitude (B) of the maximum response are extracted from each channel. Hence, there are a total of eight features for the four sensor signals, which are employed for the identification of the impact magnitude and location as performed in the following section. Due to relatively small number of sensors and features, the signals are processed as conducted above. More details about the signal processing procedure can be found in Quek et al.1 and Coverley and Staszewski.7 It is worth mentioning that if the dimension of the data set is too high, a dimensionality reduction operation is more convenient for SHM applications.32

LSSVM modeling Using the aforementioned features extracted from the propagating wave signals, the problem of detection and location is to determine the force amplitude and site coordinates of the impact once an unknown impact occurs. To solve this issue, both SVM and LSSVM approaches can be employed. As compared with classical SVM, the LSSVM utilizes equality constraints instead of inequality constraints. Thus, the LSSVM has a relatively low complexity and is more computationally efficient than the classical SVM.33 Another advantage of LSSVM lies in that it has fewer parameters to tune.34 Hence, the LSSVM approach is employed in the present research. For more details about LSSVM algorithm, refer Suykens et al.27 Specifically, the impact identification is treated as a nonlinear regression problem owing to its detection accuracy. A single LSSVM model normally only produces one output. Therefore, three LSSVM models are

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employed to predict the maximum force amplitude (fmax ) and site coordinates (x and y) The regression problem is solved using the eight features from the four sensors as inputs and the force magnitude or site coordinates as outputs to train the LSSVM as outlined in the following.

LSSVM algorithm As is known, LSSVM maps the input data into a highdimensional feature space and constructs a linear regression function therein. The unknown force amplitude (fmax ) and coordinate (x and y) functions can be approximated by the mapping equation of LSSVM regression z(x) = wT u(x) + b

ð4Þ

fxi , zi gNi= 1 ,

with the given training data set where N represents the number of training data set, xi = fA1i , B1i , A2i , B2i , A3i , B3i , A4i , B4i g 2 R8 are the input data, and the output zi 2 R represent the force amplitude (fmax ) or coordinates (x and y). Additionally, the weight vector is w 2 ROh , the nonlinear mapping u( ) : R8 ! ROh denotes a map from the input space to a feature space of dimension Oh , and b is the bias. Because more details about the LSSVM algorithm can be found in the literature,22,33 only a brief description is presented here. First, the LSSVM approach formulates the regression as an optimization problem in the primal weight space min J (w, e) =

w, b, e

N 1 T 1 X w w+ g e2 2 2 i=1 i

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subject to the equality constraints zi = wT u(xi ) + b + ei ,

i = 1, 2, . . . , N

z(xt ) =

k x x i k2 K(x, xi ) = exp s2

z(xt ) =

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i=1

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k xt x i k 2 ai exp +b s2 i=1

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With the assigned regularization parameter g and kernel parameter s, the purpose of the training process is to determine the support values ai and the bias b.

Hyperparameter tuning It has been shown that LSSVM has two hyperparameters (g and s) to be adjusted. The high generalization ability of the LSSVM model relies on the appropriate tuning of these two parameters. To select the best hyperparameters, the approach of 10-fold cross validation is usually adopted. However, this method is very time-consuming. Alternatively, it has been reported that the Bayesian framework with three levels of inference is an effective way to infer the optimal hyperparameters of the LSSVM regressors.35 Given a set of training data, Bayesian inference is a robust framework to determine the distribution of the estimated model parameters based on the data sets. Afterward, the optimal model parameters can be predicted. The basic idea of the hyperparameter inference procedure using Bayesian framework arises from a modified version of the LSSVM regression problem min J (w, e) = mEW + zED

w, b, e

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subject to ei = zi wT u(xi ) + b ,

i = 1, 2, . . . , N

ð11Þ

with EW = ED =

ai K(xt , xi ) + b

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with s.0 denoting the width parameter (which specifies the kernel sample variance s2 ) and k k representing the Euclidean distance, the LSSVM model for the impact magnitude and location estimation becomes

ð6Þ

where g is the regularization parameter and ei is the residual error. Generally, the weight vector w cannot be solved directly since it is of high dimension. Thus, a dual formation is constructed and a kernel function is introduced to simplify the computation. Then, necessary calculations result in the following solution to the regression problem N X

By adopting the radial basis function (RBF) as kernel function

1 T w w 2

N N 2 1X 1X e2i = zi wT u(xi ) + b 2 i=1 2 i=1

ð12Þ ð13Þ

where m is the new regularization factor and z denotes the variance of the noise for the residual error ei . The hyperparameter g is related to m and z by g = z=m. It is noticeable that by substituting equations

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Figure 3. Experimental setup for the impact test of a plate structure.

(12) and (13) into equation (10), the same optimization problem is generated as described by equation (5). Given a training data set, the details about Bayesian inference algorithm for the calculation of the two optimal hyperparameters can be obtained from Vong et al.36 In the current research, the procedure is implemented by resorting to a toolbox37 running in MATLAB environment.

Experimental investigations on impact detection and location In this section, the LSSVM-based identification models are verified by conducting a series of experimental investigations. A photograph of the experimental setup is shown in Figure 3, and the setup has been described in section ‘‘Impact detection and location strategy’’ earlier. For a comparative study, a BP ANN is also constructed and trained due to its popularity. Generally, the experimental data are divided into two main sets. In a previous work,23 the randomly distributed impacts were used as the training set, and the impacts applied on a regular mesh were employed as the testing set. In the present research, both training and testing sets are obtained by randomly distributed impacts. A preliminary test revealed that such a data sampling scheme achieves better detection accuracy than the conventional method. In the current research, a location error in terms of area is defined by multiplying the mean x error by the mean y error. A percentage area error is then calculated by equation (14), which represents the percent of the area error with respect to the overall area (X 3Y ) of the test plate structure. In addition, the detection error of the maximum force amplitude is characterized by the mean absolute error and percentage mean error (see equation (15)) with respect to the experimental value

N model 1X fmax i fmax i 3100% N i=1 fmax i

ð15Þ

where N is the number of data sets. In addition, zmodel i and zi represent the ANN or LSSVM model output and experimental output for the ith variable (fmax , x, or y), respectively. For the convenience of regression modeling, the data sets are normalized before the training process. Specifically, both the input and output data are normalized into the range between 0 and 1 through the following transformation N (v) = v =

v vmin vmax vmin

ð16Þ

where N represents the normalization operation, and vmin and vmax denote the minimum and maximum values of the input or output data v, respectively. After training, the ANN or LSSVM output z is retransformed into the actual value z by the inverse normalization operation N 1 .

ANN detection and location results It is well known that ANN owns universal approximation capabilities. It has been shown that a three-layer perceptron ANN with one hidden layer is capable of representing any continuous function.38 Hence, the ANN of three-layer structure is employed in the current research. Additionally, the popular BP learning algorithm and Levenberg–Marquardt training algorithm are adopted. In view of the input and output variables, one ANN model can be developed to estimate the force amplitude and location of the impact using eight neurons in the input layer and three neurons in the output layer, respectively. Even so, in order to illustrate a fair comparison with LSSVM, which normally owns a single-output capability, three ANN models are established for the three outputs. Besides, the transfer functions for the hidden and output layers are selected as the tangent-sigmoid and linear functions, respectively. Concerning the selection of optimum number of neurons for the hidden layer, there is little guidance available. Hence, a preliminary search has been performed to determine an optimal number of neurons in hidden layer. By varying the number of hidden neurons between 1 and 40, results show that 20 hidden neurons

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produce the best performance in this study. The first 109 sets of impact data are used to train each of the three ANN models. During the training procedure, the data sets are divided into three subsets of 60%, 20%, and 20%, which are used for the purposes of training, validation, and testing, respectively. Two stop criteria are set for the training process, that is, the training is stopped when the average squared error between network outputs and target outputs is lower than 1026 or when the number of iterations exceeds 5000. Moreover, in order to remove the overfitting and improve the network generalization ability, an early stopping technique is applied by monitoring the validation error. Specifically, when the validation error increases for a number of five consecutive iterations, the training is stopped, and the weights and biases at the minimum of the validation error are returned. The training time of the ANN is about 0.8 s, which is executed on a personal computer (PC) equipped with 2.83-GHz Quad CPU and 8-GB RAM running Windows 7 (64-bit) operating system. It is known that the result of ANN can heavily rely on the initial values assigned to the weights.4 To examine the effects of different initial conditions on the ANN performance, the three ANN models are trained by multiple times. Moreover, in order to obtain more accurate results, the ANN models have been carried out 30 times to conduct a large-sample statistical analysis. Before each training process, the weight values of the ANN are initialized randomly. The histograms of the training results for the impact magnitude and location identification are shown in Figure 4(a) and (b), respectively. It is observed that majority of the training processes produce Ef and Exy

LSSVM detection and location results As for the LSSVM, the aforementioned 109 and 54 data sets are used for the purposes of training and testing, respectively. Considering the output ability of LSSVM (i.e. a single-output variable), three LSSVM models are trained to detect the maximum force amplitude fmax and location coordinates x and y, respectively. For example, given arbitrary initial values (g 0 = 1 and s0 = 1), the optimized hyperparameters for the LSSVM model of coordinate x are obtained by the Bayesian framework automatically. The optimal hyperparameters are g = 1:58 and s = 23:85. They are used to train the LSSVM to obtain the values of ai and b, which minimize the objection function of optimization.

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around 55% and 1.0%, respectively. This phenomenon is caused by the initial conditions in the ANN training. The result of the best trained ANN is shown in Figure 5. The produced mean error of the maximum force amplitude fmax is 8.2 N, which stands for a percentage mean error of 43.2%. Additionally, the resulted mean errors of the coordinates x and y are 29.1 and 26.9 mm, respectively, which correspond to a percentage area error of 0.41%. Once the ANN is trained, the remaining 54 sets of impact data are used for generalization testing, and the testing results are depicted in Figure 6. It is observed that the testing mean errors are 14.9 N and 74.0 and 70.5 mm for the force fmax and coordinates x and y, respectively, that is, 81.1% percentage mean force error and 2.73% percentage area error.

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Figure 5. Training results of ANN models for (a) maximum force amplitude, (b) coordinate x, and (c) coordinate y. (d) Comparison of experimental result and ANN training result. ANN: artificial neural network.

To illustrate the influences of the initial conditions on LSSVM performance, the training procedures of the three LSSVM models are conducted 30 times. The histograms of the results are depicted in Figure 7, which indicate that majority of the training results give Ef and Exy around 55% and 0.183%, respectively, indicating the effects of the different initial conditions. For each LSSVM, the training time is less than 0.1 s, which is also executed on the PC equipped with 2.83-GHz Quad CPU and 8-GB RAM running Windows 7 (64bit) operating system. Since we are more interested in the best performance of the algorithms, only the results of best trained LSSVM models are presented. The best trained LSSVM models produce the results of force magnitude and two coordinates as shown in Figure 8(a) to (c), respectively. Figure 8(d) describes the planar plot of the location results. It is found that the LSSVM delivers the mean errors of 3.8 N and 30.5 and 11.3 mm

for fmax and x and y coordinates, respectively, which are equivalent to 20.0% percentage mean force error and 0.18% percentage area error, respectively. Concerning the best trained LSSVM models, their generalization testing results are shown in Figure 9. It is observed that the mean error of the force amplitude fmax is 10.0 N, that is, 54.4% percentage mean error. In addition, the mean errors of coordinates x and y are 44.6 and 45.5 mm, respectively, which correspond to 1.06% percentage area error with respect to the overall plate size.

Discussion on identification results From the histograms of ANN and LSSVM training results as shown in Figures 4 and 7, respectively, it is observed that the BP ANN results span wider ranges of 43.2%–90.4% and 0.41%–2.36% for Ef and Exy , respectively. This indicates that the ANN identification results are very dependent on the starting conditions of

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Figure 6. Testing results of ANN models for (a) maximum force amplitude, (b) coordinate x, and (c) coordinate y and (d) comparison of experimental result and ANN location result. ANN: artificial neural network.

the weight values. On the contrary, the LSSVM results of Ef are distributed in only two small subranges of 19.98%–21.78% and 55.92%–56.67%, and the results of Exy are spread over a small range of 0.174%– 0.193%. It is noticeable that the histograms are constructed using 30 samples of ANN and LSSVM results. It appears that the 30 results of LSSVM models approximately follow a normal distribution as shown in Figure 7(b). In order to obtain a normal distribution for the ANN results, more samples are required, that is, more experiments are needed to be performed. From this point of view, the LSSVM model produces a normally distributed result more easily than ANN for the impact site location under the influences of random initial conditions. Hence, as compared with ANN, the LSSVM enables more reliable identification of the impact site because it is less sensitive to the starting points and produces more stable results.

Concerning the force magnitude identification, both ANN and LSSVM produce fmax errors around some particular values as shown in Figures 4(a) and 7(a), respectively. In addition, the errors are much larger than the impact site location errors. In order to have a better understanding of the model performances, the ANN and LSSVM models have been trained 1000 more times for the force detection. The histograms of the 1000 training results of ANN and LSSVM models are shown in Figure 10(a) and (b), respectively. The larger data set of the ANN model results shows that a regular distribution is produced. In addition, the best trained ANN model delivers a testing error of Ef = 58:4%. The wide span (27.55%–247.44%) of the training results reveals that the ANN model is significantly affected by the initial conditions. As for the LSSVM model, although it produces the best training error of less than 1%, the testing error is as large as 51.2%. Moreover,

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Figure 9. Testing results of LSSVM models for (a) maximum force amplitude, (b) coordinate x, and (c) coordinate y and (d) comparison of experimental result and LSSVM location result. LSSVM: least squares support vector machine.

majority of LSSVM model training results still stack around two particular values of 20% and 56%. This phenomenon is caused by the system issues of LSSVM model along with the Bayesian inference approach. Further inspection reveals that although the LSSVM model parameters are randomly initialized at each time of training, the optimal hyperparameters inferred by the Bayesian method fall into two sets of values, which lead to two groups of identification results consequently. On the one hand, the results exhibit the robustness of the Bayesian inference under the influence of different initial conditions. On the other hand, it indicates the limitation of the LSSVM model in terms of impact force magnitude identification using the present setup. In the future work, other approaches to tuning the LSSVM parameters will be exploited to enhance the impact detection accuracy.

For a clear comparison, the training and testing results of the best ANN and LSSVM models are tabulated in Table 1. Since the generalization capabilities of the models are reflected by the testing errors, these errors are adopted in the comparison study. It is evident that the LSSVM enhances the force amplitude detection accuracy by 12.3% in comparison with ANN method. Moreover, as compared with ANN location error, the LSSVM substantially improves the site location accuracy by 61.1% in terms of percentage area error. To facilitate the understanding of the identification accuracy, a scenario is assumed where an impact is exerted on the plate and the impact is located by the trained ANN and LSSVM. Subsequently, if a maintenance worker is required to search for the impact site accurately by using nondestructive evaluation (NDE) technique, the workload is reduced by 61.1% using

16

Structural Health Monitoring 13(1)

(a)

(b)

400

1000 600

900

350

400

No. of training results

No. of training results

800 300 250 200 150

200

700

0 54

600 500

30

400

20

300

10

56

58

100

200 50 0 0

0 0

10

20

30

100 50

100 Percent f

max

150 error (%)

200

0 0

250

10

20 30 40 Percent fmax error (%)

50

60

Figure 10. Histograms of 1000 training results of (a) ANN model and (b) LSSVM model for the force magnitude detection. ANN: artificial neural network; LSSVM: least squares support vector machine.

Table 1. Impact detection and location results of the best trained ANN and LSSVM models. Performance

Mean error, fmax (N) Mean error, x (mm) Mean error, y (mm) Percent fmax error (%) Percent area error (%)

ANN

LSSVM

Training

Testing

Training

Testing

5.2 29.1 26.9 27.5 0.41

10.7 74.0 70.5 58.4 2.73

0.1 30.5 11.3 0.6 0.18

9.4 44.6 45.5 51.2 1.06

ANN: artificial neural networks; LSSVM: least squares support vector machine.

LSSVM result as compared with the case of employing ANN result. Although both LSSVM and ANN are capable of estimating the location of the impact with a percentage area error less than 3%, neither of them can determine the force amplitude accurately since both detection errors exceed 50%. That is, although the extracted two features from each channel are effective in terms of the location of impact site, their effectiveness in the quantification of the impact force magnitude is not evident. It has been shown in the previous works23 that among the known characteristics parameters, the aforementioned features are able to obtain the best identification results. The reason why these features produce a relatively low accuracy for the present force magnitude detection may lie in two reasons. One possible reason is that the employed Hilbert transform is inadequate to extract features for accurate force magnitude identification. Hence, in the future work, more feasible signal processing algorithms will be explored to extract more appropriate characteristic parameters for improving

the impact detection performance. Another reason may come from the too small number of the training sets. The magnitude quantification accuracy may be enhanced by acquiring more data sets yet at the cost of more workload. Besides, other models will also be exploited aiming to improve the force detection accuracy. It is noticeable that the impact detection and location are conducted by ANN and LSSVM with a relatively small number of 109 training sets and 54 testing sets. For comparison, typical previous investigations on plate structures based upon regression approaches have been selected. The numbers of overall data sets and the location errors (see equation (14)) are described in Table 2. As compared with the existing works,4,23,31 this research produces a relatively large percentage area error of Exy = 1:06%. The reason mainly arises from the small number of sensors and sparse impact test points. It is further observed that the previous works generate a better Exy by resorting to either more sensors (NS ) or more impact data sets (ND ), that is, higher hardware

Xu

17

Table 2. Conditions and impact location results of several works. Reference

Number of sensors

Number of data sets

Plate size (mm2)

Mean error (mm2)

Area error Exy (%)

Efficiency h (%)

Worden and Staszewski23 Haywood et al.4 Xie31 This work

17 12 9 4

175 630 85 163

530 3 300 608 3 304 400 3 320 490 3 390

23.1 3 25.7 26 3 15 N/A 44.6 3 45.5

0.37 0.21 0.04 1.06

0.033 0.013 0.131 0.152

cost or more workload. Intuitively, the smaller the parameters NS , ND , and Exy , the better the overall performance because this represents the situation with fewer sensors, fewer impact tests, and better detection accuracy. Hence, an empirical index is presented in equation (17) to describe the efficiency of accuracy of the impact identification system. It can be deduced that the larger the index, the better the efficiency of accuracy h=

1 Exy 3100% NS ND

ð17Þ

The h values of the four works are shown in the last column of Table 2. It is seen that the index of this work is much higher than those of the existing ones, which reveals that the efficiency of impact identification system in this research is higher than the previous ones. The reason lies in that it achieves a better detection efficiency along with reduced cost of sensors (as well as data acquisition channels) and less workload in generating the impact data sets. In the future, the impact identification strategy based on online algorithms30 will be investigated to further improve the detection performance of the system once new testing data are added to the data sets. Even though there is plenty of room for performance improvement of the proposed LSSVM-based impact identification scheme, the achievement of satisfactory impact location results for the plate structure elaborated by the conducted investigations demonstrates the effectiveness of the proposed LSSVM-based strategy over conventional ANN-based one. Furthermore, it reveals a great potential for future research toward practical applications of the impact identification strategy on the structures of aircrafts, spacecrafts, and so on.

Conclusion In this research, the LSSVM approach has been employed to deal with the identification of impact force magnitude and impact site location. This issue has been solved by formulating three nonlinear regression models. Comparative experimental studies have been conducted using the LSSVM and BP ANN methods. It is found that adopting the same numbers of sensors and

impact data sets, the LSSVM produces more reliable result and better accuracy for both force magnitude and location identification. An empirical index reveals that the reported impact identification technique achieves better efficiency than the existing ones in consideration of the number of sensors, number of data sets, and the location detection accuracy simultaneously. The proposed technique can be extended to the impact identification of more complicated structures as well. Future work includes the extraction of better features by means of advanced signal processing algorithms and advanced tuning of the LSSVM parameters to accomplish more accurate detection and location of the impact. Declaration of conflicting interests The author declares that there is no conflict of interest.

Funding This work was supported by the Macao Science and Technology Development Fund under Grants 024/2011/A and 070/2012/A3 and the Research Committee of the University of Macau under Grants MYRG083(Y1-L2)FST12-XQS and MYRG078(Y1-L2)-FST13-XQS.

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