Data-driven Stochastic Subspace Identification Research Articles

A mechanics-informed neural network method for structural modal identification

Modal identification is one of the core topics within the realm of structural health monitoring (SHM). In this study, we summarize four modal mechanical properties and propose a mechanics-informed neural network (MINN) method for structural modal identification. The proposed MINN method incorporates the sparsity of the data in the time–frequency domain and cross-correlation minimization in the time domain into the neural network to obtain modal parameters, which uses sparsity constraint and cross-correlation minimization constraint to obtain the accurate modal responses and mode shapes. Subsequently, modal frequencies and damping ratios can be derived from the modal responses. The proposed MINN method is verified by numerical simulations and two actual suspension bridges. Compared with traditional methods, the proposed MINN method has two major advantages. Firstly, the proposed MINN method presents explicit mathematical equations to distinguish the modes and the spurious modes, which obviates the necessity for priori information such as model order or time-consuming manual intervention to distinguish the modes and the spurious modes. Therefore, it can be implemented adaptively to determine the modal order and obtain the modal parameters. Secondly, the proposed MINN method can obtain a greater number of accurate modal parameters than traditional methods and achieves an increase of 102.6%, 43.4%, and 31.5% in the number of accurate results when compared to covariance-driven stochastic subspace identification (SSI-COV), data-driven stochastic subspace identification (SSI-DATA) and the natural excitation technique and the eigensystem realization algorithm (NExT-ERA), respectively. Therefore, the proposed MINN method provides an adaptively modal identification method that has clear modal mechanical properties to distinguish the modes and the spurious modes and can obtain a greater number of accurate results.

Intelligent damage diagnosis method for offshore platforms based on enhanced stabilization diagrams and convolutional neural network

Considering that it is difficult to evaluate the damage state of offshore platform structures under environmental excitation by stochastic subspace identification (SSI) stability diagrams alone, an intelligent damage diagnosis method based on enhanced stability diagrams and convolutional neural network (CNN) is proposed. The data-driven SSI algorithm and covariance-driven SSI algorithm are utilized to identify stability diagrams of monitoring data, and the stability diagrams of the two algorithms are superimposed together for image enhancement. Further, the enhanced stability diagrams are used as input samples for CNN training to distinguish the damage state of the structure. In the meanwhile, the whale optimization algorithm is employed to optimize the hyper parameters of CNN to ulteriorly improve the recognition performance. The final test accuracy of CNN is 97.20%, and is 13.09% higher than before hyper parameter optimization, which indicates that the damage diagnosis method based on enhanced stability diagrams and CNN is reasonable and effective, and is expected to be applied to real-time damage diagnosis of offshore platform structures.

Long-Term Seismic Monitoring of a Passively-Controlled Steel Building on Performance Assessment under Strong Earthquake

Analysis of recorded seismic response of an eight-story passively-controlled steel building located in Sendai is reported in this paper. Vibration monitoring system has been instrumented to test the effectiveness of dampers and actual performance of passively-controlled structures under earthquake since it was constructed in 2003. A data-driven stochastic subspace identification methodology with an alternative strategy to remove spurious modes is developed and implemented to the recorded actual earthquake response to extract dynamic properties of the passively-controlled eight-story steel building from the recorded floor acceleration data. Thus, the inherent damping characteristic of the building is identified under various earthquakes. Moreover, the variation of the estimated natural frequencies, mode shapes, and damping ratios for all the earthquake recordings are illustrated. To further investigate serviceability of the passively-controlled steel building during an earthquake, probabilistic model updating is developed to estimate the model parameters and infer the response of the structure based on the identified modes. Besides, seismic performance assessment of the passively-controlled steel building from the estimated dynamic characteristics and model parameters during service period is also discussed.

Dynamic displacement estimation and modal analysis of long-span bridges integrating multi-GNSS and acceleration measurements

Compared with acceleration-based modal analysis, displacement can provide a more reliable and robust identification result for output-only modal analysis of long-span bridges. However, the estimated displacements from acceleration records are frequently unavailable due to unrealistic drifts. Aiming at obtaining more accurate and stable results for determining the modal parameters, this study develops a multi-rate weighted data fusion approach for estimating displacement responses in dynamic monitoring of structures based on global navigation satellite system (GNSS) and acceleration measurements. The approach initially derives the local estimations from displacement and acceleration sensors via a Kalman filter algorithm with colored measurement noise, and later uses a weighted fusion criterion of scalar linear minimum variance to fuse the results of local estimations. Then, structural modal pamameters are identified by employing data-driven stochastic subspace identification (SSI) algorithm. The proposed approach is validated in a four degree-of-freedom numerical model and then applied to a long-span bridge in engineering practice. The results illustrate that the proposed approach can reduce the error of GNSS-obtained displacement and expand recognizable frequency range by introducing dynamic displacement component from acceleration measurement.

A vision-based method for estimating shallow cable tension via vibrational properties

Shallow cable is the component of cable-supported glass roof systems, which has been widely used in modern building constructions. In this research, a vision-based method is proposed for estimating tension values of shallow cables subjected to harmonic excitations. To validate the accuracy of the proposed low-cost, easy-to-operate, and multipoint synchronous measurement method, vision-based results are compared with the data obtained from contact sensors, and the error is within 2%. In addition, the data-driven stochastic subspace identification (Data-SSI) is employed to identify the first-order modal shapes of the shallow cable by vision-based data, and the results have a precise match with the computed shape of the numerical analysis. To investigate the practicality of the proposed method, the uncertainty evaluation for shallow cable tensions obtained by vision-based data is conducted based on the Monte Carlo method (MCM). These results demonstrate that the method proposed in this research is effective for capturing vibrational properties and estimating shallow cable tension.

Multi-stage semi-automated methodology for modal parameters estimation adopting parametric system identification algorithms

In recent years, a new research direction in structural condition assessment has been focusing on developing automated or semi-automated procedures to identify a structure’s modal parameters from its response measurements. This is because long-term structural monitoring systems rely on the implementation of system identification methodologies that often involve the intervention of an expert user with an acquired experience in the field.This paper aims to offer a semi-automated methodology for extracting the modal parameters independently of the chosen parametric system identification technique with minimum user involvement in the parameter selection process. Here, the framework is applied to two different parametric system identification algorithms: Data-Driven Stochastic Subspace Identification (DD-SSI) and Output Only Observer Kalman Filter (O/O OKID). The procedure can be represented as a multi-stage strategy where unsupervised tools and three clustering options are offered to the user to reach a reliable estimate of the modal parameters. The proposed procedure is validated with an application in the operational modal analysis of an existing hospital structure located in Italy. The results demonstrated excellent accuracy and robust performance of the methodology, even in the presence of closely spaced modes. The proposed procedure helps to improve the data analysis process in continuous monitoring, where usually, the algorithm’s parameters need to be constantly updated by the user.

Modal analysis of structures based on distributed measurement of dynamic strains with optical fibers

Dynamic properties of structures, in terms of frequencies and mode shapes, represent inherent characteristics and health states of structural systems. A method for extraction of modal parameters of structures based on distributed measurements of dynamic strains is described in this article. AT-BOTDA (Amplitude Transfer-Brillouin Optical Time Domain Analysis) optical fiber sensor based on Brillouin scattering is employed for distributed measurements of dynamic strains in a laboratory beam. SSI-DATA (Data-driven Stochastic Subspace Identification) is selected in the formulation of the method, based on its capability for efficient analysis of large amounts of data. The analysis approach involves pre-processing of the distributed data based on basic principles of dynamics, determination of the sensing range, building the strain state space model, extracting modes and improving the results with selection of the sensing points. Laboratory experiments on a beam indicate that distributed vibration strains collected by AT-BOTDA are sufficient for convergence, resulting in an accurate determination of the first two mode shapes.

Monitoring-based evaluation of dynamic characteristics of a long span suspension bridge under typhoons

The Xihoumen Bridge, with a main span of 1650 m and two side spans of 578 m and 485 m, respectively, is a long-span suspension bridge spanning the Jintang and Cezi islands in Zhejiang Province, China. In order to establish a reference baseline for vibration mitigation and condition assessment of the bridge under typhoons, the wind field properties, vibration energy distribution, and modal parameters of the bridge are identified through in-situ monitoring. Dynamic responses at different locations on main span and wind speeds at 1/2 main span of the bridge under 10 typhoons and under normal wind were collected for this study. Vibration energy distribution is estimated using wavelet packet transform. Modal parameters are identified by the peak-picking (PP) and data-driven stochastic subspace identification (SSI-DATA) methods. The results indicate that in the low-frequency range, the fluctuating wind energy of typhoons changes more significantly than that of normal wind. More than 80% of the vibration energy is concentrated in the frequency band (0–1.5625 Hz) under typhoons, while nearly 90% of the vibration energy under normal winds is evenly distributed in three frequency bands (0–1.5625 Hz, 1.5625–3.125 Hz, and 3.125–4.6875 Hz). When using typhoon-induced dynamic responses as input, the SSI-DATA method performs better than the PP method in the identification of modal parameters. The results obtained in this study can be used as a baseline for future structural condition assessment of the bridge after typhoon events.

Uncertainty quantification in data-driven stochastic subspace identification

A crucial aspect in system identification is the assessment of the accuracy of the identified system matrices. Stochastic Subspace Identification (SSI) is a widely used approach for the identification of linear systems from output-only data because it combines a high computational robustness and efficiency with a high estimation accuracy. Practical approaches for estimating the (co)variance of system matrices that are identified using SSI exist for the case where the system matrices are obtained from the shift-invariant structure of the extended observability matrix. However, in data-driven SSI, the system matrices are often obtained in a different way, using identified state sequences. This case is treated in the present work, for three common types of weighting. First, it is shown that the estimated system matrices depend entirely on sample output correlation estimates, the covariance of which can be straightforwardly estimated. Subsequently, a linear sensitivity analysis of the data-driven SSI algorithm is performed, such that the covariance of the identified system matrices can be also computed. A memory efficient implementation is obtained by computing the related Jacobian only implicitly. An extensive numerical validation, covering a range of parameter choices, demonstrates the accuracy of the estimated variance of the identified system description. Finally, the practical use of the method in the context of operational modal analysis is demonstrated in an experimental case study.

Dynamic monitoring and data analysis of a long-span arch bridge based on high-rate GNSS-RTK measurement combining CF-CEEMD method

The purpose of this article is to develop a combined data analysis method of Chebyshev filter (CF) and complementary ensemble empirical mode decomposition (CEEMD) for weakening the influence of the background noise of global navigation satellite system (GNSS) sensors. To test the effect of noise reduction using the proposed CF-CEEMD method, a nonlinear signal with additive noise is first introduced. Then, the GNSS measured signal of a long-span arch bridge is analyzed using CF-CEEMD. Moreover, the dynamic characteristic parameters (i.e. natural frequencies, mode shapes and damping ratios) of the bridge are extracted from the de-noised signal employing the data-driven stochastic subspace identification (DD-SSI) algorithm. Meanwhile, the finite element (FE) model of the bridge is established to predict the natural frequencies and the mode shapes via modal analysis. Finally, the results depict that GNSS-RTK technique is applicable to monitor the dynamic response of long-span bridges with reasonable accuracy via CF-CEEMD analysis. Furthermore, the natural frequencies and mode shapes derived experimentally via DD-SSI analysis have good agreement with the predicted values based on FE model.

Modal Parameters Identification of A Real Offshore Platform From the Response Excited by Natural Ice Loading

This paper investigates the possibility of utilizing response from natural ice loading for modal parameter identification of real offshore platforms. The test platform is the JZ20-2MUQ jacket platform located in the Liaodong Bay, China. A field experiment is carried out in winter season, as the platform is excited by floating ices. The feasibility is demonstrated by the acceleration response of two different segments. By the SSI-data method, the modal frequencies and damping ratios of four structural modes can be successfully identified from both segments. The estimated information from both segments is almost identical, which demonstrates that the modal identification is trustworthy. Furthermore, by taking the Jacket platform as a benchmark, the numerical performance of five popular time-domain EMA methods is systematically compared from different viewpoints. The comparisons are categorized as: (1) stochastic methods versus deterministic methods; (2) high-order methods versus low-order methods; (3) data-driven versus covariance-driven stochastic subspace identification methods.

Modal parameter identification of a multiple-span post-tensioned concrete bridge using hybrid vibration testing data

The paper describes and evaluates application of output-only system identification to an eleven-span post-tensioned concrete bridge using hybrid excitation. A linear chirp sweeping force, induced by two light-weight electro-dynamic shakers, augmented environmental sources to excite the bridge during the hybrid testing exercise. To obtain the modal characteristics of the structure, two output-only time domain system identification methods were employed, namely auto-regressive (AR) time series model and eigensystem realization algorithm with observer/Kalman identification (ERA-OKID), with the traditional data-driven stochastic subspace identification method (SSI-data) providing a comparative benchmark. The accuracy and efficiency of both system identification algorithms when used on hybrid testing data are investigated and compared to the results from purely ambient vibration testing data. The study demonstrates that using both output-only identification algorithms the collected vibration responses induced by the proposed hybrid vibration testing methodology can be used for extracting modal parameters with enhanced accuracy and reliability (i.e. more identified modes) for the large-scale post-tensioned concrete bridge due to the increase in the excitation strength and better coverage of the relevant frequency bands. Compared to the classical SSI-data, the AR and ERA-OKID techniques were able to identify more modes at reduced computational cost when applied to voluminous data from multi-channel measurements.

A comparative study on in-flight modal identification of an aircraft using time- and frequency-domain techniques

In this study, in-flight modal identification analyses are made based on vibration data collected during a flight test of an aircraft, by using two different output-only identification techniques: frequency domain decomposition and data-driven stochastic subspace identification. The purpose of this study was to evaluate and compare the efficacy of the two methods in modal parameter estimation and to validate their capability in dealing with some challenging tasks such as time tracking of modal parameters and estimating modal damping ratios. In addition, the effects of different environmental conditions and maneuvers are investigated by separating the flight-test data, such as static engine start, taxi, takeoff, cruise, roll, climb, descend, and yaw maneuvers. It is demonstrated that the selection of operational conditions and maneuvers plays a crucial role in identifying the modal parameters of the aircraft.

Customised active monitoring system for structural control and maintenance optimisation

The paper illustrates the development and the application of an active monitoring system, and analyzes the investigated dynamic behaviour of the structure where the system is applied. The system is installed on a 250 m suspended arch steel bridge that has been instrumented with sensors of different type. This work focuses on the employed methodologies and obtained results related to the dynamic monitoring of the bridge. The use of the Operational Modal Analysis techniques with data-driven stochastic subspace identification algorithms allows the extraction of the structural dynamic characteristics: natural frequencies, damping ratios, and mode shapes. These data are in overall accordance with those calculated through the computational model utilized by the active monitoring system. Besides, the definition of a range of scattering of modal parameters, under specific conditions, has been obtained. The bridge has very low vibration frequencies, below 1 Hz for all significant modes. The outcomes indicate that the estimation of dynamic characteristics and the corresponding accuracy is influenced by several factors, including the length of the accelerometric acquisitions and the followed specific procedures. The impact of environmental factors, with particular reference to the temperature, is examined. Compared to the damping ratio, the natural frequency shows higher estimation accuracy and marked sensitivity to the environmental factor. Consequently, in the selection of benchmark for damage detection, it should be taken into account that the two modal parameters have specific criticalities. Once that the modal parameter has been chosen, values outside the estimated uncertainty range can be considered as alarm triggers.

Operational Modal Analysis of the Curved JUST Footbridge Induced by Human

This paper investigated the vibration levels of curved library footbridge that is located in a composite structure building in the center of Jordan University of Science and Technology (JUST) induced by human. Operational modal analysis (OMA) through ambient vibration test was performed to extract the structural dynamic properties (frequencies, damping ratios and mode shapes) using data driven stochastic subspace identification (DD-SSI) algorithm in time domain. The unbiased modal parameters were identified by using proper signal pre-processing. Finally, this study considers conducting the OMA of JUST library footbridge to identify with confidence the full modal properties (damping ratios, frequencies, and mode shapes) in the frequency range up to 50 Hz and for seven modes.

Analysis of vibration monitoring data of an onshore wind turbine under different operational conditions

A wind turbine structure works under the complicated environment and its operational state is complex. In this study, the operational state of a 1.5 MW wind turbine and the vibration response of the tower have been monitored for a long time. The vibration characteristic of the wind turbine under different operational conditions is discussed in detail, and it was observed that the rated rotation speed condition has the highest vibration level. The Campbell diagrams of wind turbine both in the FA direction and SS direction are drawn based on the natural frequencies results, which are identified by data-driven stochastic subspace identification method. It was observed from the Campbell diagrams that when the rotation speed is close to the grid-connected rotation speed, the blade-passing frequency 3f (0.43 Hz) is equal to the fundamental frequency of wind turbine which will easily lead to the resonance. This phenomenon is confirmed by the time history diagrams of the operational condition and vibration speed of a certain startup process. Lastly, the relationship between first-order, second-order modal parameters and operational conditions of the wind turbine are discussed in detail respectively. The mean value of first-order natural frequency is affected slightly by the different operational conditions while the value of second-order natural frequency increases significantly with the increase of rotation speed. The work in this paper can help us to better understand the response of the wind turbine under different operational conditions, which can also lay the foundation for the structure design, modal validation and damage diagnosis of wind turbine.

Damping estimation of a double layer grid with ball joint system by output-only modal identification

Modal parameters of large civil engineering structures such as modal damping ratios (MDRs) are determined mainly by output-only modal identification. In this paper, MDRs of a double layer grid were obtained using output-only modal identification. For this purpose, a double layer grid constructed from ball-joint system was tested. Doing some random tapping on the structure, the acceleration response in multiple locations was measured. The acquired data was processed using output-only modal identification to arrive at MDRs. The MDRs corresponding to the first eight modes of the grid were extracted by five output-only modal identification techniques; namely enhanced frequency domain decomposition (EFDD), curve-fit frequency domain decomposition (CFDD) and three different methods of data-driven stochastic subspace identification. To determine the appropriate model order used in SSI methods, a sensitivity analysis was carried out and the resulting number of order was 200. The proper frequency resolution of 1600 was also determined to estimate the MDRs of the grid by EFDD and CFDD. The results showed that the MDRs of the grid, obtained from different methods, are in good agreement with each other. The grid has very low MDRs, as the MDRs of the modes measured from different methods varied from 0.06% to 0.11%.

Improved automated operational modal identification of structures based on clustering

A robust automated method for operational modal analysis (OMA) is in a great demand for processing a large amount of structural health monitoring data from engineering structures. This paper proposes an improved automated OMA approach based on data-driven Stochastic Subspace Identification (SSI) and clustering techniques with novel criteria. The framework of the proposed approach includes two main components, namely, “modal identification by SSI” and “automated interpretation of SSI output.” Three procedures including hard validation criteria removal, an improved statistics-based clustering procedure, and a developed cluster merging procedure are combined in the second component for automatically interpreting the stabilization diagram from the SSI output, without a priori knowledge on the modal parameters and no manual tuning during interpreting. Numerical validation results on a frame structure model demonstrate that the proposed approach is capable of identifying the vibration modes accurately, under a significant noise effect. No spurious modes are observed, and the physical modes can be accurately identified. Experimental studies on a steel frame structure in the laboratory and a real footbridge are conducted to demonstrate the robustness and applicability of using the proposed approach for automated OMA and modal tracking. Identification results are compared with baselines and those from an existing reference method to demonstrate the improvement and contribution made in the proposed approach on the automated OMA.

New signal subspace approach to estimate the inter‐area oscillatory modes in power system using TLS‐ESPRIT algorithm

The authors propose here a new low-frequency oscillatory (LFO) modes estimation scheme by applying Karhunen–Loeve transform (KLT) with total least square estimation of signal parameter employing rotational invariance technique (TLS-ESPRIT). The proposed KLT provides the perfect decorrelation of the signal from the noise at low signal-to-noise ratio and for both white and coloured noise. KLT also maximally compress the energy of the estimated signal. KLT helps to estimate the signal subspace perfectly for any non-stationary ambient power signal with the fastest computational time. First, the authors applied the KLT denoise to the filtered signal buried with highly correlated coloured Gaussian noise through the first eigenvectors of the correlation matrix and estimated the signal subspace. Finally, TLS-ESPRIT is applied to estimate the modes of the power system. A comparative analysis of the KLT-TLS-ESPRIT with the modified TLS-ESPRIT, multi-channel wavelet transform, data driven-stochastic subspace identification, TLS-ESPRIT, Root-MUSIC, and improved Prony method is presented. To explore the real-time application of the KLT-TLS-ESPRIT, the authors considered Kundur's two-area test system and 16-machine 68-bus system using 50,000 Monte-Carlo simulations. The obtained results show that the KLT-TLS-ESPRIT accurately estimates the LFO with least mean and standard deviation as compared to the remaining six methods.

A Method of Modal Parameter Identification for Wind Turbine Blade Based on Binocular Dynamic Photogrammetry

The identification of operational modal parameters of a wind turbine blade is fundamental for online damage detection. In this paper, we use binocular photogrammetry technology instead of traditional contact sensors to measure the vibration of blade and apply the advanced stochastic system identification technique to identify the blade modal frequencies automatically when only output data are available. Image feature extraction and target point tracking (PT) are carried out to acquire the displacement of labeled targets on the wind turbine blade. The vibration responses of the target points are obtained. The data-driven stochastic subspace identification (SSI-Data) method based on the Kalman filter prediction sequence is explored to extract modal parameters from vibration response under unknown excitation. Hankel matrixes are reconstructed with different dimensions, so different modal parameters are produced. Similarity of these modal parameters is compared and used to cluster modes into groups. Under appropriate tolerance thresholds, spurious modes can be eliminated. Experiment results show that good effects and stable accuracy can also be achieved with the presented photogrammetry vibration measurement and automatic modal identification algorithm.