Comparison of Different Independent Component Analysis Algorithms for Output-Only Modal Analysis

From the principle of independent component analysis (ICA) and the uncertainty of amplitude, order, and number of source signals, this paper expounds the root reasons for modal energy uncertainty, identified order uncertainty, andmodal missing in output-only modal analysis based on ICAmethods. Aiming at the problem of lack of comparison and evaluation of different ICA algorithms for output-only modal analysis, this paper studies the different objective functions and optimization methods of ICA for output-only modal parameter identification. Simulation results on simply supported beam verify the effectiveness, robustness, and convergence rate of five different ICA algorithms for output-only modal parameters identification and show that maximization negentropy with quasi-Newton iterative of ICA method is more suitable for modal parameter identification.


Introduction
Identification of modal parameters including modal frequencies, modal shapes, and damping ratio is an important research for structure dynamic characteristics.Modal analysis is a kind of "inverse problem" analysis method in structural dynamics field, which is different from traditional methods including finite element analysis (FEA) method and theoretical modeling method.On the basis of actual dynamic test on engineering structure, modal analysis deals with the dynamic problems through combining the experiment with the theory.
In output-only modal analysis, modal parameters are identified only according to the output response data when the input excitation and system are unknown.Comparing with experimental modal analysis, output-only modal analysis has numerous positive features.It is effective to solve the modal identification problem when the input excitation of engineering structure is difficult to measure.What is more, there is no need to consume plenty of materials in the laboratory to simulate the engineering structure.In addition, because the response data are tested in the working condition, the identified modal parameters can well reflect the actual situation of engineering structure and boundary conditions.Blind source separation (BSS) is a technique that recovers unknown independent source signals only from observed signals.At present, there are some BSS methods developed for output-only modal analysis [1].Kerschen et al. [2] explained the one-to-one relationship between vibration modes of mechanical systems and modes computed through a BSS technique called independent component analysis (ICA) and then applied BSS technique to the free and random responses of mechanical systems [3,4].Zhou and Chelidze [5] proposed a method for linear normal mode identification based on BSS and verified its effectiveness in a vibration system.Antoni [6] advocated the use of Fourier-domain separation techniques based on the short-time Fourier transformation.In recent years, experts and scholars continued to study BSS of operation modal analysis [7,8].Abazarsa et al. [9] adopted a second-order blind identification (SOBI) technique to nonclassically damped systems.Antoni et al. [10] showed that several separation criteria purposely dedicated to operational modal analysis (OMA) can be deduced from general physical considerations.Sadhu et al. [11][12][13] took use of multirank parallel factor decomposition, time-series analysis-based BSS, and a decentralized BSS algorithm for ambient modal identification.Hazra et al. [14] applied hybrid time-frequency BSS to full-scale structure data for a variety of conditions including the presence of supplemental damping devices.At the same time, it also can be applied in earthquake-excited structures [15], building vibration [16], and civil engineering structures [17].There are some methods to deal with decentralized modal identification with limited sensors by taking use of sparse component analysis [18,19].When the mechanical system is complex, such as nonlinear system, a convolutive BSS towards modal identification was employed [20].
In recent years, output-only modal analysis based on BSS has been further developed [21,22] and widely applied in practice.Modal identification issue was incorporated into the BSS formulation and transformed into a time-frequency framework [23].Furthermore, independent component analysis (ICA) combined with wavelet transform has been used to identify damage for time-varying systems [24].Therefore, the study of output-only modal analysis based on ICA is of great significance [7][8][9][10][11][12][13].
A generic approach to ICA consists of preprocessing the data, defining measures, and optimizing an objective function [25].The classical measure method of maximization non-Gaussianity includes maximization, kurtosis, and negentropy.And the three common approaches in ICA algorithms are minimum mutual information, Infomax, and maximum likelihood estimation [25,26].Based on the theoretically described method as mentioned, a comparison of different ICA algorithms for output-only modal analysis is made.This paper mainly focuses on comparison and evaluation of different ICA algorithms for output-only modal analysis and then finds out which method is of the best accuracy and robustness.
The remainder of this paper is organized as follows.In Section 2, from the principle of ICA and the uncertainty of amplitude, order, and number of source signals, we expound the root causes for modal energy uncertainty, order uncertainty, and modal missing in output-only modal analysis based on ICA methods.Different objective functions and optimization methods of ICA algorithms for output-only modal analysis are compared and evaluated in Section 3. The simulation verifies effectiveness, robustness, and convergence rate of different ICA algorithms for output-only modal analysis and judges the most suitable method for output-only modal analysis in Section 4. At the end of this paper, the conclusions and further research directions of the study are summarized in Section 5.

Modal Decomposition of Dynamic Response and Output-Only Modal Analysis of Multi-Degree-of-Freedom System.
Dynamic characteristic of the system is an intrinsic property, which reflects the linear relationship between the input and output of the system.Mechanical system is usually thought of as a proportional viscous damping system.Based on structural dynamic theory, for -degree-of-freedom vibration system, the equation of motion is where F() ∈ R × is excitation vector of additional load and Ẍ(), Ẋ(), and X() ∈ R × are on behalf of acceleration vector, velocity vector, and displacement vector, respectively.M, C, and K ∈ R × are mass matrix, damping matrix, and stiffness matrix of the vibration system, respectively.
For general engineering of small damping, the displacement response can be expressed in modal coordinates as follows: where Φ ∈ R × is modal matrix made up of order modal shape ⃗   ∈ R  and ⃗   and ⃗   are normalized and orthogonal mutually, while Q() ∈ R × is the vector matrix composed of order modal response ⃗   () ∈ R  , while ⃗   () and ⃗   () are mutually independent.
Classical modal parameter identification is to extract a series of mode shape   , mode frequency   , and modal damping ratio   from external excitation F() and response signal X() of the system.While in output-only modal analysis, modal parameters are identified only according to the output response signal X() when the input excitation F() and system are unknown.[26] published the paper on the signal processing in 1991, ICA became an effective solution to BSS.ICA is a method for finding underlying source signals from observed mixing signals.In order to estimate the source signals, ICA assumes that the source signals are statistically independent and the distributions of the components are non-Gaussian.Finally, to keep ICA model simpler, the linear system defined by the mixing matrix is invertible.

ICA for BSS. After Jutten and Herault
The linear model of ICA without measurement noise is where ()]  ∈ R × represents source signal.The goal of ICA algorithm is to isolate the source signals and the information about mixing matrix only from observation signals.In order to separate the statistically independent source signals from observation signals, it is necessary to find a separate matrix, shown in Figure 1.Its equation can be expressed as follows: where It is impossible to determine specific values of mixing matrix A and source signals S() simultaneously without any prior knowledge.But it is easy to find that there are some fuzzy factors and uncertainties from the model of ICA.It can be summarized as three uncertainties.
Figure 1: The essence of ICA method for BSS.

The Energy of ICA Is
Uncertain.Suppose D is a diagonal matrix; then, the linear instantaneous mixture modal can be expressed as So, the amplitude of separated signals is inconsistent with the amplitude of source signals.

The Order of ICA Is
Uncertain.Suppose P is a permutation matrix; the linear instantaneous mixture modal can be expressed as where PS() is a new source signals matrix after reordering and AP −1 is a new mixing matrix.

The Number of Independent Components Is Uncertain.
If the contribution of independent components is not enough, it is hard to identify and separate them.So, without any prior knowledge, determination of the number of independent components or sources by ICA algorithm is difficult.

ICA Based Output-Only Modal Analysis. Comparing (2)
with (3), the dynamic response modal coordinates decomposition of multi-degree-of-freedom system under the effect of free vibration or random excitation can be regarded as a special case of ICA [2][3][4].Therefore, as shown in Figure 2, there are a one-to-one relationship between vibration modal shapes of dynamic mechanical systems and separating matrix and a one-to-one relationship between modal responses and independent components.
On account of three assumptions of ICA method, the identified modal parameters are of the following characteristics.
(1) Every order of modal shape is with different amplitude.In case of ICA method, the energy of separation matrix is not unique, and independent component also loses amplitude information.Unlike principal  component analysis method [27], ICA method cannot get the contribution ratio information of each modal.Modal shape is a relative quantity rather than an absolute value.So, in order to compare the modal shape with the real modal shape, the separation matrix and modal shape identified by ICA method should be normalized.
(2) The order of modal parameter identified by ICA is uncertain.Modal identified by ICA method is not in accordance with the order from small to large.In fact, the first separated output source and vector are the ones whose independence is the strongest rather than the first-order modal parameter.Therefore, in order to compare the natural frequencies with the real natural frequencies, the modal parameters identified by ICA need to be reordered by the modal frequencies.
(3) If the contribution of independent components is not enough, it is hard to identify and separate them and to determine the number of modal parameters.So, in ICA based output-only modal analysis, if the modal parameter is with small contribution to independence, ICA may barely identify it, which would cause modal parameter missing.

Different Objective Functions and Optimization Methods of ICA for
Output-Only Modal Analysis If ⃗ () is Gaussian random variables, kurt( ⃗ ()) = 0.However, for a non-Gaussian variable, its kurtosis usually is not equal to zero.
When kurt( ⃗ ()) > 0, ⃗ () represents a super-Gaussian variable, and when kurt( ⃗ ()) < 0, ⃗ () represents a sub-Gaussian variable, and the greater the non-Gaussianity of ⃗ (), the greater the absolute value of kurtosis.Therefore, ICA problem can be converted into calculating the maximum of the absolute value or square of kurtosis of source signal component.

Maximization Negentropy.
Another non-Gaussian meas ure is negentropy.According to information theory, Gaussian variables have the maximum entropy of all the random variables with the same variance.
The negentropy of random variable ⃗ () is defined as where ⃗  Gauss () and ⃗ () are Gaussian random variables with the same covariance matrix and ( ⃗ is differential entropy.The negentropy value is always nonnegative and ( ⃗ ()) = 0 if and only if ⃗ () is the Gaussian distribution.To overcome the disadvantage of dyscalculia of negentropy, an approximate representation with high-order cumulant can be expressed as where ⃗ ]() is a Gaussian variable with zero-mean and unit variance and ⃗ () is a variable with zero-mean and unit variance, while (⋅) and {⋅} are nonquadratic function and expectation, respectively.

Minimization of Mutual Information. For 𝑚 random variables ⃗
(),  = 1, 2, . . ., , their mutual information is defined as for invertible linear transformation Y() = WX(), so When ⃗   () and ⃗   () are not related to each other and the variance of ⃗   () and ⃗   () is a unit matrix, then From ( 12), det W must be a constant, because According to (13), it suggests the basic relationship of maximization negentropy and mutual information.So the ICA problem of maximizing non-Gaussian transforms into the problem of minimizing mutual information.

Maximum Likelihood Estimation.
Maximum likelihood estimation can be interpreted as follows: when likelihood function reaches the maximum, its parameter corresponds to the separated matrix.The goal is to find the inverse matrix of mixing matrix; namely, W = A −1 .The likelihood function is defined as density function of variable density, with mixed matrix A as parameter.
Assume that p (X()) is estimation of probability density function   (X()) of observed signals X().The probability density function of source signals   (S()) is known, according to the relationship of two probability density functions; therefore, p (X()) satisfies the following equation: For the given model, the likelihood function of observed signals X() is function of parameter A and can be defined as When the parameter becomes separation matrix W = A −1 , the log-likelihood function is where  represents the number of source signals and  represents the sample number of observed signals with independent and identical distribution.
3.1.5.Infomax.If we consider the flow of information in a neural network, in order to realize the effective information transmission, it is necessary to maximize the mutual information between the input and output.Supposing that X() is the input of a neural network, then the output can be expressed as where   (⋅) is a nonlinear scalar function, ⃗   represents weight vector of neural network, and ⃗ () is additive measurement noise.
So, the output entropy is In the absence of measurement noise, maximization mutual information between input and output equals maximization output entropy.

Comparison of Different Objective Functions of ICA for
Output-Only Modal Analysis.Five objective functions have been introduced as mentioned above, and all the objective functions are interrelated.In fact, all the estimation theories are the same, and the only difference is the concrete forms.So there are some differences between the objective functions [28]: (a) Some objective functions such as maximization kurtosis and negentropy can estimate a component at once, but others estimate all the components at once.(b) Some objective functions such as maximization negentropy adopt the nonpolynomial function, but others adopt polynomial function which is in connection with cumulant.(c) Among the objective functions, the constraint of estimating independent component is irrelevant.However, when mutual information is the minimum, the independent components may not be irrelevant from each other.
(d) In the objective function of maximum likelihood estimation, the density of independent component is certain with prior knowledge, and it does not meet the model of ICA.However, maximization negentropy adopts approximation algorithm, without need of any prior knowledge.
(e) An important evaluation index is the robustness to outliers.It means that single and wrong observed values will not influence estimating independent components.
Above all, the advantages and disadvantages are summed up in Table 1.

Different Optimization Methods of ICA for Output-Only
Modal Analysis.Gradient descent [29], stochastic gradient descent [30], and quasi-Newton method [31] are three commonly used optimization methods.

Gradient Descent.
Gradient descent is an optimization algorithm, the essence of which is to make the change of steps and finally attain the minimal error.We minimize or maximize the objective function ( ⃗ ) = {( ⃗ , ⃗ ())} iteratively by starting from an initial point ⃗ (0), then compute the gradient of ( ⃗ ) at this point, and move in the direction of the negative gradient or with the steepest descent from a suitable distance.After that, we repeat the same procedure at the new point and so on.Finally, we get the update rule as follows: where {( ⃗ , ⃗ ())}/ ⃗  represents the gradient of ( ⃗ ) and () is step of moving in the direction of the negative gradient.

Stochastic Gradient Descent.
Stochastic gradient descent is a method combining random procedure with optimization method.Its basic idea is not to calculate the exact value of gradient directly but unbiasedly estimate the gradient as the direction of the negative gradient.
If the objective function of ICA is ( ⃗ ) = {( ⃗ , ⃗ ())}, we would like to minimize or maximize it.Firstly, we also need to initialize point ⃗ (0) and compute the gradient.Unlike gradient descent, what stochastic gradient descent method And then we move in the direction of the negative gradient or the steepest descent by a suitable distance and repeat the same procedure until we find the optimum point.Therefore, the update rule is as follows: 3.2.3.Quasi-Newton.In the optimal aspect, Newton's method is with fast convergence speed and high precision while it has the shortcoming of expensive computing, because it needs to calculate the second derivative and inverse of Hessian matrix.In order to keep its merits and to overcome the shortcoming, quasi-Newton method appeared.Quasi-Newton method calculates the second derivative and inverse of a quadratic function instead of Hessian matrix.
For the objective function of ICA is ( ⃗ ), if we want to minimize or maximize it, its update rule is where   =  2 ( ⃗ )/ ⃗   ⃗   is a Hessian matrix.However, it is quite difficult to calculate the inverse of Hessian matrix.And quasi-Newton method takes the measure by replacing the Hessian matrix with a quadratic function, so the update rule becomes where (⋅) represents a quadratic function.

Comparison of Different Optimization Methods of ICA
for Output-Only Modal Analysis.Gradient descent, stochastic gradient descent, and quasi-Newton are optimization algorithms, but each method is with respective advantages and disadvantages.Gradient descent is relatively easy but it is with low convergence rate and easy to be trapped in local minima.Stochastic gradient descent is simpler than gradient descent, but with the shortcoming of reaching the convergence precision more hardly.Quasi-Newton method is with fast convergence speed and high precision but with the more complex computation than the methods mentioned above.Table 2 shows comparison of three optimization methods with different evaluation factors.

Simulation Verification and Results Analysis
After weighing the advantages and disadvantages of different objective functions and optimization methods, we think the following five methods are more suitable to outputonly modal parameters identification.The methods include maximization of kurtosis with gradient descent method, maximization negentropy with quasi-Newton method, minimum mutual information with gradient descent method, maximum likelihood estimation with stochastic gradient descent method, and Infomax with gradient descent method.

Establishment of Simulation Model and Parameters Setting of Data
(1) Parameters setting of simulation model is a simply supported beam without damping, the length of which is 1 meter, elasticity modulus is 205 Gpa, Poison's ratio of material is 0.3, and density of material is 7850.
(2) This beam is divided into 1000 equal pieces and the number of measured stations is 1001.
( (5) The real modal shapes and natural frequencies are all calculated by FEA.
To reflect the accuracy of modal shapes in experiment, the modal assurance criterion (MAC) is introduced into [5].And the formula is as follows: where  MAC varies between 0 (no coincidence) and 1 (complete coincidence).

Results of Simulation Verification without Measurement Noise and Results
Analysis.With the purpose of verifying the effect of five methods mentioned above for modal parameter identification and judging which method is more suitable for operation modal analysis, the results of simulation verification without measurement noise are as follows.

Maximization Kurtosis with Gradient Descent Method.
According to the process of modal parameters identified by maximization kurtosis with gradient descent method, Figure 3 shows fast Fourier transform (FFT) of each independent component.The abscissa of main peak in each subfigure of Figure 3 corresponds to each natural frequency identified by maximization kurtosis with gradient descent method.And Table 3 shows the errors of natural frequencies identified by maximization kurtosis with gradient descent method.
According to the identified frequency value, Figure 4 shows the modal shapes which compared the real modal shapes calculated by FEA.
Table 4 shows the convergence times and MAC value of modal shapes.

Maximization Negentropy with Quasi-Newton Method.
According to the process of modal parameters identified by maximization negentropy with quasi-Newton method, Figure 5 shows FFT of each independent component.
The abscissa of main peak in each subfigure in Figure 5 is each natural frequency identified by maximization negentropy with quasi-Newton method.Table 5 shows error of natural frequencies identified by maximization negentropy with quasi-Newton method.According to the identified frequency value, Figure 6 shows the modal shapes which compared the real modal shapes calculated by FEA.
Table 6 shows the convergence times and MAC value of modal shapes.

Minimum Mutual Information with Gradient Descent
Method.According to the process of modal parameters identified by minimum mutual information with gradient descent method, Figure 7 shows FFT of each independent component.
The abscissa of main peak in each subfigure in Figure 7 is each natural frequency identified by minimum mutual information with gradient descent method.Table 7 shows error of natural frequencies identified by minimum mutual information with gradient descent method.
According to the identified frequency value, Figure 8 shows the modal shapes which compared the real modal shapes calculated by FEA.Table 8 shows the identified order and MAC value of modal shapes.

Maximum Likelihood Estimation with Stochastic Gradient Descent Method.
According to the process of modal parameters identification by maximum likelihood estimation with stochastic gradient descent method, Figure 9 shows FFT of each independent component.
The abscissa of main peak in each subfigure of Figure 9 is each natural frequency identified by maximum likelihood estimation with stochastic gradient descent method.Table 9 shows errors of natural frequencies identified by maximum likelihood estimation with stochastic gradient descent method.
According to the identified frequency value, Figure 10 shows the modal shapes which compared the real modal shapes calculated by FEA.
Table 10 shows the identified order and MAC value of modal shapes.

Infomax with Gradient Descent Method.
According to the process of modal parameters identification by Infomax with gradient descent method, Figure 11 shows FFT of each independent component.The abscissa of main peak in each subfigure of Figure 11 is each natural frequency identified by Infomax with gradient descent method.Table 11 shows errors of natural frequencies identified by Infomax with gradient descent method.
According to the identified frequency value, Figure 12 shows the modal shapes which compared the real modal shapes calculated by FEA.
Table 12 shows the identified orders and MAC values of modal shapes.

Results Analysis.
Comparing the results of output-only modal parameters identification with five objective functions of ICA algorithms, results analysis can be gained from the following respects: (1) From Figures 3-12, the effectiveness of five ICA algorithms for output-only modal parameter identification is verified.
(2) In Figures 3 and 5, natural frequencies identified are with no second peak.But in Figures 7, 9, and 11, natural frequencies identified are with second peak.A big reason for the difference is that the process of maximization kurtosis and negentropy for outputonly modal analysis is in a serial manner, and other methods are in parallel manner.
(       conclude that the effect of modal parameters identification is the best when the objective function is maximization non-Gaussian and the values of MACs are close to 1.

Results of Simulation Verification with Measurement Noise and Results
Analysis.From the results above, it is verified that the effect of modal parameters identified by the method is better than others when the objective function is non-Gaussian.However, a maximization non-Gaussian measure contains maximization kurtosis and negentropy.In order to verify the robustness of maximization kurtosis and negentropy, the response data are added with 1% measurement noise.13 shows errors of natural frequencies identified by maximization kurtosis with gradient descent method.

Maximization Kurtosis with
According to the identified frequency value, Figure 14 shows the modal shapes, which compared the real modal shapes calculated by FEA.
Table 14 shows the convergence times and MAC values of modal shapes when observed signals are with measurement noise.

Maximization Negentropy with Quasi-Newton Method.
Figure 15 shows FFT of each independent component when observed signals are with measurement noise.The abscissa of main peak in each subfigure of Figure 15 is each natural frequency identified by maximization negentropy with quasi-Newton method.And Table 15 shows errors of natural frequencies identified by maximization negentropy with quasi-Newton method.
According to the identified frequency values, Figure 16 shows the modal shapes, which compared the real modal shapes calculated by FEA.
Table 16 shows the convergence times and MAC values of modal shapes when observed signals are with measurement noise.

Results
Analysis.From Figures 3-12 above, we can see that when observed signals are with 1% measurement Gauss noise, maximization negentropy is more robust to measurement noise than maximization kurtosis.However, minimum mutual information and Infomax method as well as maximum likelihood estimation are with no advantage even when there is no measurement noise.

Conclusions
This paper verified the effectiveness of different objective functions and different optimization methods of ICA algorithm for output-only modal parameters identification.The methods are different from each other in content, but they are equivalent in essence.The simulation verification results show that modal parameters identification is the best when the objective function is maximization non-Gaussian and the values of MACs are close to 1, and maximization negentropy is more robust to measurement noise than kurtosis.

Figure 2 :
Figure 2: One-to-one corresponding relations between dynamic response modal coordinates decomposition and ICA.
FFT of the 6th independent component

Figure 3 :
Figure 3: FFT of each independent component identified by maximization kurtosis with gradient descent method.

Figure 4 : 23 (
Figure 4: Modal shape comparison between FEA and separating vector by maximization kurtosis with gradient descent method.

Figure 5 :
Figure 5: FFT of each independent component identified by maximization negentropy with quasi-Newton method.

Figure 6 :
Figure 6: Modal shape comparison between FEA and separating vector by maximization negentropy with quasi-Newton.

Figure 7 :
Figure 7: FFT of each independent component identified by minimum mutual information with gradient descent method.

Figure 8 :
Figure 8: Modal shape comparison between FEA and separating vector by minimum mutual information with gradient descent method.

( 4 )
From Figures 3-12, by comparing the modal shapes and natural frequencies as shown in Tables 3FFT of the 6th independent component

Figure 9 :
Figure 9: FFT of each independent component identified by maximum likelihood estimation with stochastic gradient descent method.

Figure 10 :
Figure 10: Modal shape comparison between FEA and separating vector by maximum likelihood estimation with stochastic gradient descent method.

Figure 11 :
Figure 11: FFT of each independent component identified by Infomax with gradient descent method.

Figure 12 :
Figure 12: Modal shape comparison between FEA and separating vector by Infomax with gradient descent method.
Gradient Descent Method.When observed signals are with measurement noise, Figure 13 shows FFT of each independent component.The abscissa of main peak in each subfigure of Figure 13 corresponds to each natural frequency identified by maximization kurtosis with gradient descent method.And Table (f) The 6th-order modal shape

Figure 14 : 9 (
Figure 14: Modal shape comparison between FEA and separating vector by maximization kurtosis with gradient descent method with measurement noise.

Figure 15 :
Figure 15: FFT of each independent component identified by maximization negentropy with quasi-Newton iteration method with measurement noise.

Figure 16 :
Figure 16: Modal shape comparison between FEA and separating vector by maximization negentropy with quasi-Newton iteration method with measurement noise.

Table 1 :
Comparison of five objective functions.

Table 2 :
Comparison of three optimization methods.
) Response data are obtained by the resonance method excited by sine signals.And the sine signals are composed of the frequencies that are 205 Hz, 91.3 Hz, 366 Hz, 572 Hz, 824 Hz, 1121 Hz, and 22 Hz with powers that are 60, 30, 30, 30, 30, 30, and 30, respectively.

Table 3 :
Error of natural frequencies identified by maximization kurtosis with gradient descent method.

Table 4 :
The convergence times and MAC value of modal shapes.

Table 5 :
Error of natural frequencies identified by maximization negentropy with quasi-Newton method.

Table 6 :
The convergence times and MAC value of modal shapes.

Table 7 :
Error of natural frequencies identified by minimum mutual information with gradient descent method.

Table 8 :
The identified order and MAC value of modal shapes.

Table 9 :
Error of natural frequencies identified by maximum likelihood estimation with stochastic gradient descent method.

Table 10 :
Identified order and MAC value of modal shapes.

Table 11 :
Error of natural frequencies identified by Infomax with gradient descent method.

Table 12 :
The identified order and MAC value of modal shapes.

Table 13 :
Error of natural frequencies identified by maximization kurtosis with gradient descent method.

Table 14 :
The convergence times and MAC value of modal shapes when observed signals are with measurement noise.

Table 15 :
Error of natural frequencies identified by maximization negentropy with quasi-Newton iteration method with measurement noise.

Table 16 :
The convergence times and MAC value of modal shapes when observed signals are with measurement noise.