Research ArticleA Comparative Study of Regularization Method in StructureLoad Identification

Bingrong Miao , Feng Zhou, Chuanying Jiang, Xiangyu Chen, and Shuwang Yang

State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu 610031, China

Correspondence should be addressed to Bingrong Miao; brmiao@home.swjtu.edu.cn

Received 13 June 2018; Accepted 12 September 2018; Published 15 October 2018

Academic Editor: Alvaro Cunha

Copyright © 2018 BingrongMiao et al.(is is an open access article distributed under the Creative CommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

(is study aims to correlate the vibration data with quantitative indicators of structural health by comparing and validating thefeasibility of identifying unknown excitation forces using output vibration responses. First, numerical analysis was performed toinvestigate the accuracy, convergence, and robustness of the load identification results for different noise levels, sensors numbers,and initial estimates of structural parameters. (en, the laboratory beam structure experiments were conducted. (e results showthat using the two identificationmethods Tikhonov (L-curve) and TSVD (GCV-curve) can successfully and accurately identify thedifferent excitation forces of the external hammer. (e TSVD based on GCV method has more advantages than the Tikhonovbased on L-curve method. (e proposed two kinds of load identification procedure based on vibration response can be applied tothe safety performance evaluation of the railway track structure in future inverse problems research.

1. Introduction

In the past two decades, a large number of researchers havecarried out many effective research studies on inverseproblems such as structural load identification and damageidentification [1–6], and many of the results have beenapplied to the structural health monitoring technology ofstructural integrity and safety assessment [7–10]. Most of theload identification techniques use these regularizationmethods, such as the Tikhonov method and the truncatedsingular value decomposition (TSVD) method. However,there are few comparative studies on the accuracy, noiseresistance, and robustness of these two methods in theidentification of engineering structure loads.

(ite and (ompson considered the influence of regu-larizing the matrix inversion, which discussed Tikhonovregularization with the ordinary cross-validation method forselecting the regularization parameter, and they also studiedan iterative inversion technique of the method [11]. Jac-quelin et al. analyzed the deconvolution problem and uti-lized general regularization techniques to solve the ill-posedfeature [12]. Choi et al. used the Tikhonov regularizationmethod to improve the stability of the load identification

results [13]. Wang et al. proposed an improved iterationregularization method for solving linear inverse problemsand chose Morozov’s discrepancy principle to verify theposterior regularization parameter [14]. Similarly, Sun et al.combined a new improved regularization operator withL-curve method to overcome the ill condition of load re-construction and obtained the stable and approximate so-lutions of inverse problems [15]. With the development oftechniques, some new regularization methods have beenproposed. Aucejo et al. proposed a method able to proceedwithout defining any regularization parameter beforehand,and the method is called multiplicative regularizationmethod [16]. Gao and Yu presented a new method based onquadratic programming theory to determine the regulari-zation parameter [17]. Considering the sparsity of force inthe time domain, Qiao et al. developed a general sparseregularization method based on minimizing l1-norm of thecoefficient vector of basic functions [18]. Different loads ona mechanical system also have different regularizationmethods. For random dynamic loads, Jia et al. proposeda weighted regularization approach based on the properorthogonal decomposition (POD), which the regularizationparameter was selected by the GCV method [19]. (en, in

HindawiShock and VibrationVolume 2018, Article ID 9204865, 8 pageshttps://doi.org/10.1155/2018/9204865

mailto:brmiao@home.swjtu.edu.cnhttp://orcid.org/0000-0001-9909-3873https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1155/2018/9204865

order to reconstruct the impact force, Gunawan adopted theWiener filter method and introduced a regularizationtechnique so that the method can provide an optimumsolution which balances the aspects of the minimum noiseand the minimum residual [20], and Boukria et al. developedan experimental method to identify the impact force ona reinforced concrete slab [21]. In addition, Qiao et al.imposed a novel impact force reconstruction method basedon the primal-dual interior point method (PDIPM), whichreplaced l2-norm by l1-norm [22]. Gunawan extended theLevenberg–Marquardt (LM) method and a variant of theGauss–Newton methods, in order to solve the ill-posedpulse-type impact-force inverse problem in conjunctionwith a trust region strategy [23]. A novel method wasproposed to identify loads with the combination of theTikhonov regularization and singular value decomposition(SVD) based on the condition [24].

In this paper, the TSVD of regularization parameterselection method based on generalized cross-validation(GCV) method and Tikhonov regularization based onL-curve method are compared and analyzed. (e identifi-cation accuracy and applicability of the two methods havebeen summarized through the simulation of the single-inputsingle-output model (SISO) and multiple-input multiple-output model (MIMO), and the identification characteristicsof sine wave load and triangle wave load have been dem-onstrated. Finally, the results of simulation analysis areverified by experiments.

2. Construction of Forward Problem

(is paper establishes the forward problem of load identi-fication based on the green kernel function. According to thecharacteristics of the pulse function, the integral of theproduct of the pulse function with any other function isequal to the function value of the pulse moment. (erefore,any dynamic load of a structure can be expressed as a su-perposition of the unit pulse signal in the time domain:

f(t) � t

0f(t)δ(t− τ) dτ, (1)

where f(t) is the external load and δ is the Dirac function.It is assumed that the Green function from the loading

point to the responding measurement point is z(t), which isthe response of the system under the unit pulse load δ(t).According to the principle of superposition, the relationbetween excitation and response of system can be written asconvolution form (Duhamel integral):

z(t) � t

0f(τ)H(t− τ) dτ, (2)

where H(t− τ) denotes the Green kernel function of thestructure impulse response, z(t) is the measured responseswhich can be displacement, velocity, acceleration, strain, andstress. For systems under zero initial conditions, the integral

form of Equation (2) in time domain is discretized intolinear equations as follows:

z1

z2

⋮

zm

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

�

h1

h2 h1

⋮ ⋮ ⋱

hm hm−1 · · · h1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

f0

f1

⋮

fm−1

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

Δt, (3)

where Δ(t) is the sampling interval and m is the samplingpoints.

(en, it is easily to written as

Z � HF. (4)

Equation (4) is represented as a linear discrete equationfor a SISO system. When a certain number of impact loadsare applied to the different locations simultaneously, thetotal responses of the system can be expressed as the su-perposition of the dynamic responses at each knownmeasurement point:

Zj � M

i�1H

ijFi, (5)

where Hij is the Green function matrix between the points ofapplication of force and sensor location.(erefore, Equation(5) is transformed into a matrix form as follows:

Z1

Z2

⋮

ZN

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

�

H11 H21 · · · H

M1

H12 H22 · · · H

M2

⋮ ⋮ ⋮ ⋮

H1N H2N · · · H

MN

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

F1

F2

⋮

FM

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

⎫⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎭

, (6)

whereM and N are the number of points of application andthe points of measurement, respectively, and it is needed toensure that N≥M. Equation (6) expresses the linear discreteequation model for MIMO system. In fact, the solution F ofthe linear system is very sensitive to data errors. And anattempt to solve directly the equation will likely lead to anerroneous solution. (en regularization is used to acquirea meaningful solution.

3. Regularized Method

3.1. Tikhonov Regularization. (e Tikhonov regularizationmethod can be expressed as an optimization problem asfollows [25]:

minf∈Rn

‖Hf− z‖22 + α2‖f‖

22, (7)

where ‖ · ‖22 is the square of the 2-norm for vector α(α> 0) isthe regularization parameter.

(e more exact representation of the objective functionin Equation (7) can be expressed as

fT

HTH + α2I f− 2zTHf + zTz. (8)

According to the concept that the gradient of the ob-jective function is equal to zero, the least squares solution ofEquation (8) is written as

2 Shock and Vibration

HTH + α2I f � HTz. (9)

Deriving from Equation (9), the identified load can beobtained by

F � HH

H + λI −1

HH

Z, (10)

where I is the unit matrix and H indicates Hermitiantranspose.

3.2. /e TSVD Method. (e method for dealing with ill-conditioned matrices is to derive new problems by using therank deficient coefficient matrix of rank deficit. (e rankdeficient matrix H (which can be deduced by SVD) is ap-proximated by k, the nearest matrix ofHk, which is extendedby the decomposition of the truncated singular value of therank K in Equation (6). Hk can be written as

Hk �

k

i�1uiσiv

Ti , k≤ n, (11)

where σi is the singular values of the matrix H, and thevectors ui, vi are the left and right singular vectors of thematrix H. (e truncated singular value decomposition(TSVD) method is used to solve the following equation ofthe problem [26]:

min Hkf− z����

����2. (12)

(e solution to Equation (12) can be written as

fk �

n

i�1

uTi z

σivi. (13)

(e solution of fk is the only solution to the regulari-zation of the numerical zero space in the matrix H withoutcomponents. In order to avoid excessive amplification of theidentified results' disturbance error, the right end of theresulting generalized solution formula is truncated by usingthe method. In addition, the parts of the previous k cor-responding to greater singular values are reserved, and thesmaller singular values will be filtered.

4. Methods for Selecting the TikhonovRegularization Parameter

4.1. L-Curve Criterion. ‖HF−Z‖ and ‖F‖ are both functionsof regularization parameter λ. (e minimum between theregularization solution and the residual norm has beenweighed through the L-curve method. When the regulari-zation parameter changes, it refers to the change of thedemonstration number and the norm of the residual norm.(e solution norm and the residual norm are well balancedat the inflection point of the L-curve (the maximum cur-vature of the L-curve), and the regularization parameter isthe optimal regularization parameter. (erefore, the rea-sonable value of the point obtained by the L-curve criterion

is located at the inflection point of the curve. If the re-lationships ρ(λ) � ‖HF−Z‖, η(λ) � ‖F‖ are assumed, thecurvature of the L-curve is calculated as follows:

K(λ) �ρ′η″ − ρ″η′

ρ′2 + η′2 3/2. (14)

Unfortunately, since optimal regularization parametervalue search range is too large, the computational cost will behigh or the L-curve graph is not satisfied.

4.2. Generalized Cross-Validation (GCV) Method. (e basicidea of generalized cross-validation (GCV) means if thearbitrary element zi of the z is ignored, the correspondingregular resolution will be predicted better, and the regula-rization parameter is chosen as the orthogonal trans-formation of z. (erefore, choosing the regularizationparameter is required to minimize the GCV function.

G �Hfreg − z

�����

�����2

2

trace Im −HHI( ( 2.

(15)

Equation (14) can also be expressed in discrete form.

G(k) �

Ni�1 u

Ti z/ σ2i + k2( (

2

Ni�1 1/ σ2i + k2( (

2 , (16)

where HI denotes the regularized solution produced byfreg multiplied with z (i.e., freg � HIz) and k is the reg-ularization parameter. A balance between the disturbanceerror and the regularization error is actually searched byusing the GCV method, and the inflection point of thecurve is produced.

5. Numerical Example and Test Validation

As shown in Figure 1, the example model is a beam with oneend fixed to simulate a steel cantilever beam. (e materialproperties of the beam are as follows: Young’s modulus is2.1× 1011 Pa, density is 7.8×103Kg/m3, and Poisson’s ratio is0.3. (e size of the beam is 1× 0.05× 0.005m. (e beam isdivided into ten elements, marked elements 1 to 10. (en,the number of finite element model nodes is expressed fromthe mark number of one to eleven. Numerical simulation isused to verify the efficiency of two proposed force identi-fication method as well as compared to the identified results.Furthermore, the identified results of different loads appliedon the same structure are analyzed.

For the purpose of studying the effect of noise onidentified results, the noise is added to the response data.

z � zcal + lp · std zcal( · nnoise, (17)

where z is the response of load identification and zcal rep-resents the response obtained by simulation. std(zcal) de-notes the standard deviation of the computed response. lpand nnoise are the noise level and white noise sequence withthe same length as zcal, which is a mean value of 0 anda variance of 1. And the relative error between the real loadand the identified load is defined as follows:

Shock and Vibration 3

RE �fidentify −freal

�����

�����

freal����

����× 100%. (18)

5.1. Load Identification of Single-Input Single-Output System(SISO System). A sinusoidal load f(t) � 10 sin(100πt)(0≤ t≤ 0.12) is applied at node 11. (e direction of the loadis perpendicular to the Z axis of the beam, and a samplingperiod is Δt � 0.0005 s. (e identified load is obtained byusing the response of node 10. At the noise levels 1%, 2%,and 5%, the two identificationmethods mentioned above areused to identify the loads. Figure 2 shows the results of thetwo methods when the noise level is 1%. (e overall relativeerror of the two methods are shown in Table 1 at 0%,1%, 2%,and 5% noise levels. Among them, the noise level is 0%,which means no noise.

As shown in Figure 2, the two methods can accuratelyidentify the load applied to the system. Obviously, it can befound clearly that the identification accuracy of Figure 2(a)is less than that of Figure 2(b) in the initial period of time. Itrefers that the two methods of the identified results showthe analogous situation in the absence of noise in Table 2. Itcan be found that the identification error of Tikhonovbased on the L-curve method is greater than the TSVDmethod based on GCV in the case of noise. In addition, thelatter method also has weak sensitivity to noise and hasbetter adaptability. In fact, periodic synthetic signal givesnaturally better results than the ones obtained fromnonperiodic signal for inverse analysis, because SVD(TSVD) is based on Fourier series.

5.2. Load Identification of Multiple-Input Multiple-OutputSystem (MIMO System). To study the differences betweenthe two methods, a sinusoidal load in the Z direction anda triangular load in the Y direction are applied at the nodes11 and 7, respectively, and the responses of the nodes 10 and6 are measured. Among them, the sinusoidal load f1(t) andthe triangular load f2(t) are described as follows:

f1(t) �10 sin 2πt/td( , 0≤ t≤ 4td,

0, t> 4td,

f2(t) �

5t/tf , 0≤ t≤ tf ,

−5t/tf + 10, tf ≤ t≤ 3tf ,

5t/tf − 20, 3tf ≤ t≤ 5tf ,

−5t/tf + 30, 5tf ≤ t≤ 7tf,

5t/tf − 40, 7tf ≤ t≤ 8tf ,

0, t> 8tf ,

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(19)

where td is a period of sinusoidal load and tf is a quarterperiod of triangle load. In this case, the sampling time in-terval Δt is 0.001 s, 0.0005 s, and 0.00025 s, respectively.Clearly, Figures 3 and 4 show the identification result of theabove two kinds of loads in the different sampling time. Dueto the great influence of high levels of noise on load iden-tification results of MIMO system, the main features of theload to the maximum amplitude are concerned. And at thenoise level 1%, the identified results of the two methods areshown in Figures 5 and 6.

According to Figures 3 and 4, it can be found that thesampling time interval has a great influence on the identi-fication of triangle wave load, while the sine wave load is not.It means that the sampling time required for the triangularwave load is shorter than the sinusoidal load. In addition, theTSVD method has a more accurate effect than the Tikhonovmethod, and the former method is not sensitive to noiseshown in Figures 5 and 6. Simultaneously, it can be foundthat the actual noise has a great influence on the identifi-cation results.

6. Experimental Validation

(e cantilever beam structure applied for force identificationis set up as shown in Figure 7.(ematerial is structural steel,the length is 1.196m, the width is 0.06m, and the thickness is0.0055m. In this experimental study, the beam is clamped atone end and the excitation point is selected to be near theend of the beam. For the material properties of the steelbeam, modulus of elasticity is 2.1× 1011N/m2, density is7.8×103Kg/m3, and Poisson ratio is 0.3. Four points of thebeam are attached by four accelerometers, which are IEPEvoltage output piezoelectric acceleration sensor. A shaker isused to generate the sinusoidal force. (e measured signalscontaining the force and acceleration are synchronouslyrecorded by DH9522 data acquisition system. For thepurpose of modal frequency, the finite element model of thestructure is established and calculated according to theactual information of the geometry, material, and boundaryconditions. A modal impact hammer is used to imposeimpact forces acting on the cantilever beam. (e naturalfrequencies of simulation and experiment are compared asshown in Table 2, and the finite element model is modifiedaccording to the experimental model. And the mentionedmethods of identification are verified by experiments.

From the natural frequency of Table 2, the relative errorof the natural frequency of the test and simulation is

11 3

45

67

89

1011

2

z

f1

f1

f2

x y

23

45

67

89

10

1

LoadFinite element unitNode number

1

Figure 1: (e finite element model of cantilever beam.

4 Shock and Vibration

relatively small and is less than 5%. It means that theestablished finite element model is more reliable and can beused for the identification of sinusoidal loads directly.

(e dynamic response of the structure is measured byexperiment, and the Green kernel function between theloading point and the measuring point is calculated by thefinite element method. (erefore, the load identificationequation based on the Green kernel function has beenestablished. As shown in Figure 8, the identified results of theTSVD based on the GCV method and the Tikhonov basedthe on L-curve method are compared and analyzed. (econclusion of the identified result shows that the identifi-cation load reconstructed by the TSVD based on the GCVmethod has higher accuracy than the Tikhonov based on theL-curve method, and it is in suitable agreement with theactual load. And using the Tikhonov regularization method

leads to obtain worse correlation error of identification nearthe initial moment.

7. Conclusions

(is study proposes a comparative research method for theinverse problem of load identification including the TSVD(GCV) method and Tikhonov based on the L-curve method.At the same time, numerical methods and experimentsare used to verify the recognition effectiveness and ro-bustness of these two approaches based on vibration re-sponse to identify the external excitation force. For these twokinds of regularization methods under different noise levels,

0–15

–10

–5

0

5

10

15

0.02 0.04 0.06t (s)

True loadIdentified load

F (N

)

0.08 0.1 0.12

(a)

True loadIdentified load

–15

–10

–5

0

5

10

15

F (N

)

0 0.02 0.04 0.06t (s)

0.08 0.1 0.12

(b)

Figure 2: Comparison of results using (a) Tikhonov regularization based on L-curve and (b) TSVD regularization based on GCV.

Table 1: Results of load identification of SISO system.

Noise level(%)

TSVD+GCV(RE/%)

Tikhonov + L-curve(RE/%)

0 7.14 7.361 8.20 7.352 11.35 8.865 24.44 16.78

15

10

5

0

–5

F (N

)

–10

–150 0.05 0.1 0.15

t (s)0.2 0.25

f1Step 0.001s

Step 0.0005sStep 0.00025s

Figure 3: (e sinusoidal load identification results with no noiseand different sampling time.

Table 2: Comparison of experimental and simulated natural fre-quencies of test specimens.

Modalorders

Naturalfrequency

value of test (Hz)

Natural frequencyvalue of simulation (Hz)

Error(%)

1 20.42 20.271 0.72 55.66 56.933 2.23 184.40 186.49 1.124 268.93 280.64 4.17

Shock and Vibration 5

the equations based on the established input-output re-lations for unknown external excitation forces are iterativelyestimated. Some detailed conclusions are briefly listed below:

(1) In the absence of noise, both the two mentionedmethods of the above identification methodscan accurately identify the load imposed on thesystem.

(2) By virtue of the proposed method, the noise shouldbe considered to the simulation and experiment. (eresults of numerical simulations and experimentalare indicated that the identification accuracy of the

TSVD based on the GCV method is much higherthan that of the Tikhonov based on the L-curvemethod. In other words, the former method is lesssensitive to noise.

(3) (e difference between the two methods is mainlydue to the fact that the L-curve generated by theexperiment is unobservable, and the optimalpoint on the L-curve is difficult to localize withaccuracy.

(4) (e results of different loads in different samplingtime can get the conclusion that the identified effect

0 0.05 0.1 0.15 0.2 0.25

15

10

5

4.5

4

3.5

3

2.5

20.01 0.015 0.02 0.025 0.03 0.035 0.04

5

F (N

)

t (s)

0

–5

f2Step 0.001s

Step 0.0005sStep 0.00025s

Figure 4: (e triangle load identification results with no noise and different sampling time.

–15

–10

–5

0

5

10

15

F (N

)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14t (s)

True loadTikhonov (L-curve)TSVD (GCV)

Figure 5: (e sinusoidal load identification results with 1 percent.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14t (s)

–6

–4

–2

0

2

4

6

F (N

)

True loadTikhonov (L-curve)TSVD (GCV)

Figure 6: (e triangular load identification results with 1 percent.

6 Shock and Vibration

of triangle wave load changes obviously with thechange of sampling time. (erefore, the samplingtime of linear load such as triangle wave is short; inother words, the sampling frequency could be large.

Data Availability

(e data used to support the findings of this study are in-cluded within the article. (e data are published on figsharewebsite. (https://figshare.com/s/5edc2999caa536c9ef42).

Conflicts of Interest

(e authors declare that they have no conflicts of interest.

Acknowledgments

(e authors thank the National Natural Science Foundationof China (51375405 and 51775456) and the Self-Developed

Research Project of the State Key Laboratory of TractionPower (2016TPL T10).

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Exciter

Beam

Acceleration sensor

Signal generator

Signal acquisitionsystem

Computer

(a)

Sensor

Cantileverbeam

Shaker

Computer

Poweramplifier

Forcetransducer

Spectralanalyzer

(b)

Figure 7: Experimental setup: (a) site map; (b) sketch map.

0 0.02 0.04 0.06 0.08 0.1 0.12t (s)

–4

–3

–2

–1

0

1

2

3

4

5

F (N

)

True loadTikhonov (L-curve)TSVD (GCV)

Figure 8: (e identified results in the experimental.

Shock and Vibration 7

https://figshare.com/s/5edc2999caa536c9ef42

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