Research Article  Open Access
Yongshuo Zong, Jinling Chen, Siyi Tao, Cheng Wang, Jianbing Xiahou, "Moving Window Differential Evolution Independent Component AnalysisBased Operational Modal Analysis for Slow Linear TimeVarying Structures", Scientific Programming, vol. 2020, Article ID 8879086, 9 pages, 2020. https://doi.org/10.1155/2020/8879086
Moving Window Differential Evolution Independent Component AnalysisBased Operational Modal Analysis for Slow Linear TimeVarying Structures
Abstract
In order to identify timevarying transient modal parameters only from nonstationary vibration response measurement signals for slow linear timevarying (SLTV) structures which are weakly damped, a moving window differential evolution (DE) independent component analysis (ICA) based operational modal analysis (OMA) method is proposed in this paper. Firstly, in order to overcome the problems in traditional ICAbased OMA, such as easy to go into local optima and difficulttoidentify highorder modal parameters, we combine DE with ICA and propose a differential evolution independent component analysis (DEICA) based OMA method for linear time invariant (LTI) structures. Secondly, we combine the moving widow technique with DEICA and propose a moving window differential evolution independent component analysis (MWDEICA) based OMA method for SLTV structures. The MWDEICAbased OMA method has high global searching ability, robustness, and complexity of time and space. The modal identification results in a threedegreeoffreedom structure with slow timevarying mass show that this MWDEICAbased OMA method can identify transient timevarying modal parameters effectively only from nonstationary vibration response measurement signals and has better performances than moving window traditional ICAbased OMA.
1. Introduction
It is hoped that the engineering structure will have a high degree of selfadaptive or selfcontrol ability, and it can automatically change its shape and performance to adapt to changes in environmental disturbances and new usage requirements as the environment or use functions change. Operational Modal Analysis (OMA) can estimate modal parameters without input data, which is not easy to obtain in largescale engineering structures [1]. The modal parameters of a linear timevarying structure can reflect the overall dynamics of the structure [2].
Blind Source Separation (BSS) is a technique that recovers source signals from mixed signals without prior knowledge [3]. To cite a few, independent component analysis (ICA) [4, 5], secondorder blind identification (SOBI) [6], complexity pursuit (CP) [7, 8], and sparse component analysis (SCA) [9] are typical BSS methods. The ICA method requires that the source signals be independent of each other and the number of sensors is not less than the number of source signals [10]. Kerschen [11] is the first to recognize the mapping between the modal shapes and the columns of the mixing matrix in BSS and, then, apply the ICA method to OMA. Some methods have been proposed to improve the ICA’s restrictions [12, 13]. However, because its optimization method is easy to go into local optima, traditional ICAbased OMA has low robustness, and it is difficult to identify highorder modal parameters using that [14]. Proposed by Strorn and Price [15, 16] firstly, differential evolution (DE) has much higher robustness and stronger global searching ability compared to the classical optimization method, such as the Gradient descent [17], stochastic gradient descent [18], and quasiNewton iteration method [19].
However, because the mixing matrix of BSS is a constant matrix, BSSbased algorithms can only be used for time invariant structures. In reality, most of the structural parameters change with time and are timevarying structures. The main research on the timevary structure is based on the frozenin coefficient method which is considering the timevarying system as time invariant over a short time. Ramnath defines the slow linearly timevarying (SLTV) structures as the change of the system coefficient is much slower than the change of the system solution [20]. The three main approaches of SLTV structures are timefrequency analysis [21], forgetting factor [22], and moving window [23, 24]. However, the timefrequency analysis method is not suitable for online identification. The performance of forgetting factor methods is closely related to prior knowledge, which is very troublesome.
Based on the moving window [24] and independent component analysis [15], this paper presents a moving window differential evolution independent component analysis (MWDEICA) based OMA method for weakly damped SLTV structures.
The primary contributions of this paper can be summarized as follows:(1)In this paper, we propose a new DEICAbased OMA method to identify modal parameters only from stationary random response signals for LTI structures. Using DE algorithm to replace the traditional linear regression optimization algorithm to search the separation matrix, the DEICAbased OMA method has high global searching ability, robustness, and complexity of time and space. Compared with the traditional ICA method based on gradient optimization algorithm, OMA based on DEICA can identify higherorder modal parameters and has high recognition accuracy.(2)A sliding window algorithm based on differential evolution independent analysis of sliding window is proposed by combining sliding window with the DEICA method. The instantaneous modal natural frequency and mode shape of the linear timevarying structure are identified by using the linear tracking characteristics of sliding window. This method can effectively identify the transient timevarying modal parameters only from the nonstationary random response measurement signals, which is better than the traditional moving window method. The modal identification method has better performance.(3)We design a threedegreeoffreedom structure with slowly timevarying mass simulation to verify the validity and recognition accuracy of the MWDEICAbased OMA algorithm.
The remainder of this paper is organized as follows. In section 2, we describe MWDEICA in detail. The simulation verification is demonstrated in section 3. Finally, we conclude this paper in section 4.
2. Theoretical Inference of Algorithm
2.1. Differential Evolution Independent Component AnalysisBased OMA for the LTI Structure
The gradientbased optimization algorithm used by the traditional ICA is easy to fall into the local optimum. Therefore, there are problems such as low robustness and inability to recognize highorder modes when performing modal analysis on a continuum structure. Differential evolution algorithm has the ability to solve global optimization problems. The DEICAbased OMA method uses DE as an optimization algorithm which replaces the traditional linear regression method, and the core is to optimize the separation matrix . Therefore, DEICAbased OMA can identify highorder modes and has higher precision. The specific steps are as follows:(1)The number of modes m to be extracted, the number of structural finite element nodes N, the number of response data collection points T, and the response matrix of the structure are determined, and then, . and .(2)The steps of the differential evolution algorithm are adjusted. Since needs to be extracted one by one, each individual in the population is a vector, which is stored in the population matrix P. and , (3) is zerocentered by and starts loop. The kurtosis is used as a function, which is . Each offspring of results is also zerocentered. Convergence condition of the algorithm: the maximum number of iterations (4000 times) is exceeded or . The optimal solution is , and let .(4); return to step (2) until .
The flow diagram is shown in Figure 1.
2.2. Nonstationary Response Signals Decomposition in the Modal Coordinate and MWDEICABased OMA for SLTV
Based on the “timefreezing” theory [24], the frozenin coefficient method, the “short time invariant,” and “quasistationary” assumptions, the nonstationary random response signals of the SLTV structures can be approximated as the stationary random response time series of LTI structures in short time interval. Figure 2 shows the length selection of moving window.
The choice of moving window length is fixed (the length is ), and the displacement responses can be decomposed by the modal expansion:where is the modal shapes matrix and is the modal coordinate response matrix. When the order of modal natural frequencies is not equal, the modal coordinate response vectors are independent of each other.
From equation (1), we can see that and represent, respectively, the LTV structure’s statistical average modal shapes and modal response over the window time . When time reaches , the predicted modal shape vector approximates to . Similarly, after using the singledegreeoffreedom (DOF) technique, the nature frequency of order at time can also be replaced by .
In view of equation (1), there is a onetoone relationship between the modal response matrix of weakly damped structures and the linear transformational matrix in MWICA and a onetoone mapping between the independent components of MWICA and the modal response matrix in equation (1). Through ICA decomposition, the uniqueness, certainty, and existence of the MWDEICAbased OMA method can be proved and shown in Figure 3.
2.3. Theoretical Analysis and Comparison between Traditional ICA and DEICA
As an optimization solution method, DE has much higher robustness and stronger global searching ability compared to the classical optimization method such as gradient descent, random gradient descent, and Newton iteration. Table 1 shows the characteristics of ICA and DEICA.

2.4. Application Scopes of the Proposed Method
The application scope of the OMA method based on moving window and DEICA is as follows:(1)The system should be a slow linear timevarying (SLTV) structure which is weakly damped.(2)Because of the limitation of blind source separation and independent component analysis, the number of vibration response sensors should be greater than or equal to the identification order of identifiable operational modal parameters.(3)Because of the limitation of sampling theorem, the sampling frequency of the vibration response signal should be greater than or equal to 2 times of the natural frequency of the identifiable modal natural frequency.(4)Every order of modal shape is with different amplitude. In case of the ICA method, the energy of separation matrix is not unique, and independent component also loses amplitude information. Unlike the principal component analysis method [25], the 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 the ICA method should be normalized.(5)The order of modal parameter identified by ICA is uncertain. The modal identified by the 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 firstorder 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. Simulation Identification
3.1. Simulation Dataset of a ThreeDegreeofFreedom Structure with Slow TimeVarying Mass
To confirm MWICA’s availability and identification accuracy, we have designed a simulation of an LTV threeDOF spring oscillator system in MATLAB/Simulink. This model is shown in Figure 4. The initial conditions for the three modal displacements and velocities are zero. The stiffness matrix is set to , , and , and the damping matrix is set to , , and . The external force is white noise excitation with zero mean and unit variance. The mass matrix is set to and , and is a timevarying parameter as follows:
This kinetic equation becomes
In the Simulink model, we use the Runge–Kutta algorithm to obtain responses of 40 Hz sampling frequency and 2000 s recorded time [23]. In the time domain, 2.0% level Gauss measurement noise disturbances are added to displacement response signals.
3.2. Simulation Parameters Setting
According to the third theoretical frequency 12.2951 Hz at time1987.225 s in Table 2, the sampling frequency is set to 40 Hz in simulation of the SLTV structure. The window length is set to in this simulation, and then, the frequency resolution of FFT .

This objective function of ICA algorithm in this paper is the measurement of maximization nonGaussian (kurtosis). The optimized method of class ICA algorithm is the quasiNewton iteration method. The ICA method is used multiple times simultaneously, and the maximum iterations time is 4000. The modal parameters can be identified most of the time in the case of 60 runs.
The parameter setting of DE: the individual number is 80; maximum iterations time is 4000; and proportionality factor and cross probability are random transformation each loop repeats.
3.3. Modal Assurance Criterion
To determine the accuracy of the modal shapes, the modal assurance criterion (MAC) is introduced to verify the operational modal identification results. The MAC is given by [26]where is the identified modal shape, is the theoretical modal shape, and and are the transpositions of and , respectively. The MAC values range between 0 (no coincidence) and 1 (complete coincidence). However, MAC only reveals information about the direction and shape of the modal results and does not contain any information about the modal amplitudes. The MAC value is always 1 if the two vectors are proportional.
3.4. Transient Operational Modal Parameters Identification Results
The two ICA algorithms were repeated 60 times, respectively. For the modal natural frequency, Figure 5 shows the theoretical transient natural frequencies and the identified natural frequencies by class MWICA and MWDEICA for the SLTV structure.
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Figure 5 shows that certain modal parameters are not identified by class MWICA and MWDEICA at any time. In MWICA, the percentage of modal parameters of unidentified windows is more than 0.2% in the average of 60 trials.
We choose two moments t = 50.25 s and t = 1350.00 s to compare the theoretical modal shapes with the identified modal shapes. The recognition value of the frequency is similar in each simulation, while the modal shape recognition result fluctuates. We select the best value in 60 times.
3.5. Results Analysis
(1)Figures 5 and 6 and Tables 3 and 4 show that using only nonstationary random response signals, the MWICAbased and MWDEICAbased OMA method can well identify the transient natural frequency and modal shape of the SLTV structure.(2)Figures 5, and 6 show that, in the case of adding 2.0% level Gauss measurement noise disturbances to displacement response signals, compared with the MWDEICAbased OMA method, the MVDEICA method can identify more modal parameters of windows, and the class MWICAbased OMA method has low robustness and it is difficult to identify modal parameters at some moment because its optimization method is easy to go into local optima [16]. For MWDEICAbased OMA, the recognition results are unstable.(3)Table 5 shows that the MWDEICAbased OMA method has much higher time and space requirements than the MWICAbased OMA method.
4. Conclusions
In order to solve the problem that the traditional ICA method is easy to fall into local optimum and difficult to identify highorder modal parameters, combining moving window technology and differential evolution independent component analysis, this paper proposes a transient operational modal parameters identification method only from nonstationary vibration response measurement signal s for slow linear timevarying structures which is weakly damped. Theoretical comparison and simulation identification results show that MWDEICAbased OMA can effectively identify timevarying modal parameters, while the parameters of some sample points are not recognized in MWICA.
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Due to moving window technology and differential evolution algorithm, we are still unable to identify modal parameters in some time points. How to reduce the complexity of the algorithm and use MVDEICA to identify higherorder modals is the future work. It is also the future work to solve the problem and experimentally verify on more complex threedimensional engineering structures.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was financially supported by CERNET Innovation Project (grant no. NGII20160410).
References
 E. Reynders, “System identification methods for (operational) modal analysis: review and comparison,” Archives of Computational Methods in Engineering, vol. 19, no. 1, pp. 51–124, 2012. View at: Publisher Site  Google Scholar
 K. Liu and M. R. Kujath, “Adaptation of the concept of modal analysis to timevarying structures,” Mechanical Systems and Signal Processing, vol. 13, no. 3, pp. 413–422, 1999. View at: Publisher Site  Google Scholar
 W. Guan, L. L. Dong, J. M. Zhou, Y. Han, and J. Zhou, “Datadriven methods for operational modal parameters identification: a comparison and application,” Measurement, vol. 132, pp. 238–251, 2019. View at: Publisher Site  Google Scholar
 J. Hang and Z. Yi, “Operational modal identification technique based on independent component analysis,” in Proceedings of the 2011 International Conference on Electric Technology and Civil Engineering (ICETCE), pp. 2430–2433, Lushan, China, April 2011. View at: Publisher Site  Google Scholar
 S. Chauhan, R. Martell, R. J. Allemang, and D. L. Brown, “Application of independent component analysis and blind source separation techniques to operational modal analysis,” in Proceedings of the 25th IMAC, Orlando, FL, USA, February 2007. View at: Google Scholar
 C. Rainieri, “Perspectives of secondorder blind identification for operational modal analysis of civil structures,” Shock and Vibration, vol. 2014, Article ID 845106, 2014. View at: Publisher Site  Google Scholar
 M. A. Eitner, B. G. Miller, J. Sirohi, and C. E. Tinney, “Operational modal analysis of a thinwalled rocket nozzle using phasebased image processing and complexity pursuit,” in Rotating Machinery, Optical Methods & Scanning LDV Methods, vol. 6, pp. 19–29, Springer, Cham, Switzerland, 2019. View at: Publisher Site  Google Scholar
 J. Antoni, R. Castiglione, and L. Garibaldi, “Interpretation and generalization of complexity pursuit for the blind separation of modal contributions,” Mechanical Systems and Signal Processing, vol. 85, pp. 773–788, 2017. View at: Publisher Site  Google Scholar
 H. Zhou, K. Yu, Y. Chen, R. Zhao, and Y. Wu, “Outputonly modal estimation using sparse component analysis and densitybased clustering algorithm,” Measurement, vol. 126, pp. 120–133, 2018. View at: Publisher Site  Google Scholar
 M. D. Spiridonakos, E. N. Chatzi, and B. Sudret, “Polynomial chaos expansion models for the monitoring of structures under operational variability,” ASCEASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering, vol. 2, no. 3, Article ID B4016003, 2016. View at: Publisher Site  Google Scholar
 G. Kerschen, F. Poncelet, and J.C. Golinval, “Physical interpretation of independent component analysis in structural dynamics,” Mechanical Systems and Signal Processing, vol. 21, no. 4, pp. 1561–1575, 2007. View at: Publisher Site  Google Scholar
 C. Wang, J. Wang, X. Lai et al., “Operation modal analysis following fast independent component analysis,” International Journal of Applied Electromagnetics and Mechanics, vol. 52, no. 12, pp. 103–111, 2016. View at: Publisher Site  Google Scholar
 C. Wang, J. Wang, B. Zhong et al., “Negentropy and gradient iteration based fast independent component analysis for multiple random fault sources blind identification and separation,” International Journal of Applied Electromagnetics and Mechanics, vol. 52, no. 12, pp. 711–719, 2016. View at: Publisher Site  Google Scholar
 J. Wang, C. Wang, T. Zhang, and B. Zhong, “Comparison of different independent component analysis algorithms for outputonly modal analysis,” Shock and Vibration, vol. 2016, pp. 1–25, 2016. View at: Publisher Site  Google Scholar
 R. Storn and K. Price, “Differential evolutiona simple and efficient heuristic for global optimization over continuous spaces,” Journal of Global Optimization, vol. 11, no. 4, pp. 341–359, 1997. View at: Publisher Site  Google Scholar
 P. Civicioglu and E. Besdok, “A conceptual comparison of the cuckoosearch, particle swarm optimization, differential evolution and artificial bee colony algorithms,” Artificial Intelligence Review, vol. 39, no. 4, pp. 315–346, 2013. View at: Publisher Site  Google Scholar
 C. Burges, T. Shaked, E. Renshaw et al., “Learning to rank using gradient descent,” in Proceedings of the 22nd International Conference on Machine Learning (ICML05), pp. 89–96, Bonn, Germany, August 2005. View at: Publisher Site  Google Scholar
 M. Zinkevich, M. Weimer, L. Li, and A. J. Smola, “Parallelized stochastic gradient descent,” Advances in Neural Information Processing Systems, vol. 23, no. 23, pp. 2595–2603, 2010. View at: Google Scholar
 D. F. Shanno, “Conditioning of QuasiNewton methods for function minimization,” Mathematics of Computation, vol. 24, no. 111, p. 647, 1970. View at: Publisher Site  Google Scholar
 R. V. Ramnath, Multiple Scales Theory and Aerospace Applications, Multiple Scales Theory and Aerospace Applications, American Institute of Aeronautics and Astronautics, Reston, VA, USA, 2015.
 A. Klepka and T. Uh, “Identification of modal parameters of nonstationary systems with the use of wavelet based adaptive filtering,” Mechanical Systems and Signal Processing, vol. 47, no. 12, pp. 21–34, 2014. View at: Publisher Site  Google Scholar
 W. Guan, C. Wang, D. S. Chen, X. Luo, and F. F. Su, “Recursive principal component analysis with forgetting factor for operational modal analysis of linear timevarying system,” International Journal of Applied Electromagnetics and Mechanics, vol. 52, no. 34, pp. 999–1006, 2016. View at: Publisher Site  Google Scholar
 C. Wang, J. Wang, and T. Zhang, “Operational modal analysis for slow linear timevarying structures based on moving window second order blind identification,” Signal Processing, vol. 133, pp. 169–186, 2017. View at: Publisher Site  Google Scholar
 C. Wang, H. Huang, J. Chen, W. Wei, and T. Wang, “An online and realtime adaptive operational modal parameter identification method based on fog computing in internet of things,” International Journal of Distributed Sensor Networks, vol. 16, no. 1, pp. 1–12, 2020. View at: Publisher Site  Google Scholar
 C. Wang, J. Gou, J. Bai, and Y. Guirong, “Modal parameter identification with principal component analysis,” Journal of Xi’an Jiaotong University, vol. 47, no. 11, pp. 97–104, 2013. View at: Publisher Site  Google Scholar
 J. Zou, K. Yu, and B. Yang, “Methods of timevarying structural parameter identification,” Advances in Mechanics, vol. 30, no. 3, pp. 370–377, 2000. View at: Google Scholar
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Copyright © 2020 Yongshuo Zong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.