Stochastic Subspace Identification Algorithm Research Articles

Automated operational modal analysis for supertall buildings based on a three-stage strategy and modified hierarchical clustering

Several automated operational modal analysis (AOMA) methods have been developed to evaluate the dynamic characteristics of large-scale structures in real-time under environmental and operational variations. Typically, these methods need to manually set some thresholds to achieve automatic identification. However, there is a lack of uniform standards for setting these thresholds, which may result in the adoption of inappropriate settings in AOMA, thereby introducing errors in modal identification. To address this issue, this paper proposes a novel AOMA framework to automatically identify modal parameters, which mainly involves three steps. Firstly, the covariance-driven stochastic subspace identification (Cov-SSI) algorithm is employed to perform modal estimation based on measured ambient vibration records. Then, a three-stage strategy, including hard validation criteria, Gaussian mixture model clustering-driven soft validation criteria, and the local outlier factor algorithm, is presented to remove spurious modes and determine real physical modes. Finally, a modified hierarchical clustering based on the k-nearest neighbor information is applied to group the determined physical modes to achieve automatic modal identification. The effectiveness of the proposed AOMA framework in automatically identifying modal parameters is validated through a numerical simulation study for a five-floor frame structure. Furthermore, the proposed framework is applied to evaluate the modal properties of a 600 m high supertall building based on long-term monitoring records. The results show that the AOMA framework can effectively remove spurious modes and group physical modes, thus identifying the modal parameters with high accuracy. In short, the proposed AOMA framework is an effective tool for modal identification applied in field measurements or structural health monitoring.

A robust probabilistic approach to stochastic subspace identification

Modal parameter estimation of operational structures is often a challenging task when confronted with unwanted distortions (outliers) in field measurements. Atypical observations present a problem to operational modal analysis (OMA) algorithms, such as stochastic subspace identification (SSI), severely biasing parameter estimates and resulting in misidentification of the system. Despite this predicament, no simple mechanism currently exists capable of dealing with such anomalies in SSI. Addressing this problem, this paper first introduces a novel probabilistic formulation of stochastic subspace identification (Prob-SSI), realised using probabilistic projections. Mathematically, the equivalence between this model and the classic algorithm is demonstrated. This fresh perspective, viewing SSI as a problem in probabilistic inference, lays the necessary mathematical foundation to enable a collection of new OMA approaches. To this end, a statistically robust SSI algorithm (robust Prob-SSI) is developed, capable of providing a principled and automatic way of handling outlying or anomalous data in the measured timeseries, such as may occur in field recordings, e.g. intermittent sensor dropout. Robust Prob-SSI is shown to outperform conventional SSI when confronted with ‘corrupted’ data, exhibiting improved identification performance and higher levels of confidence in the found poles when viewing stabilisation diagrams. Similar benefits are also demonstrated on the Z24 Bridge benchmark dataset, highlighting enhanced performance on measured systems.

A hybrid output-only scheme for precise modal estimation and uncertainty quantification of large-scale structure through vibration-based measurements

In structural health monitoring or operational modal analysis, owing to unknown or unmeasured ambient excitation information, only structural responses are typically available for modal identification. In this context, bias and variance (uncertainty) errors inevitably exist in modal estimates (especially damping estimates), resulting in inaccurate determinations of structural dynamic properties. To this end, a hybrid output-only scheme based on the second-order blind identification, block bootstrap, and enhanced covariance-driven stochastic subspace identification algorithm with first-order perturbation is presented in this paper to perform precise modal estimation and uncertainty quantification. The accuracy and effectiveness of the hybrid scheme in performing modal identification are verified by numerical simulation study of a framework structure. Furthermore, the hybrid method is applied to analyze recorded acceleration responses of a supertall building with 600 m height during Typhoon Kompasu by considering different sensor setups to verify its applicability and robustness in practical applications, and further investigates the dynamic properties of the skyscraper and their variations during the typhoon. The numerical simulation and field measurement studies demonstrate that the hybrid scheme can effectively reduce the bias errors and quantify the variance errors in modal estimates through a single ambient vibration test, thereby improving the confidence and accuracy of modal identification. Notably, the hybrid method is an effective tool to perform high-accuracy modal estimation and uncertainty quantification for large-scale structures under ambient excitations.

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.

Removal of AM-FM harmonics using VMD technology for operational modal analysis of milling robot

The dynamics of a milling robot have a significant impact on machining efficiency and quality. Traditional experimental modal analysis (EMA) is used to study the dynamics of milling robots in the static state. However, the dynamic properties of milling robots change when they are in the operational state. To investigate the dynamics of milling robots during cutting, operational modal analysis (OMA) is used. During normal cutting, milling robots are primarily excited by cutting forces that are dominated by amplitude and frequency-modulated (AM-FM) harmonic components that are caused by the fluctuation of cutting loads and spindle speeds. Such cutting forces make the dynamic parameters identified by OMA inaccurate. To address this issue, the variational mode decomposition-OMA (VMD-OMA) method is proposed herein, which involves removing AM-FM harmonic components from the response signal using the VMD technology and subsequently using the stochastic subspace identification (SSI) algorithm to identify dynamic parameters. To verify the effectiveness of this method, simulations and experimental research have been conducted. The simulation results demonstrate that the dynamic parameters identified by VMD-OMA are generally closer to the theoretical values. Experimental research conducted on a milling robot named TriMule shows that VMD-OMA can effectively eliminate AM-FM harmonics in response signals and identify the dynamic parameters of TriMule during cutting.

Static and Dynamic Response Characteristics of a Full-Scale Long-Cantilever Tower under Various Loading Conditions

Long-cantilever transmission towers are drawing more and more attention in ultrahigh-voltage (UHV) transmission projects. To meet the requirements of high-voltage safety clearance, the structural characteristics of long-cantilever transmission towers are significantly different from traditional ones. Aiming to ensure the safety of long-cantilever transmission towers during service, this paper investigates their static and dynamic characteristics under various extreme loading conditions based on full-scale tests and numerical simulation. With the strain data tested in seven typical cases, the bending moments and axial forces of the main members were back-calculated and the initial imperfections were calculated based on these results. Using a stochastic subspace identification algorithm, the evolution of the modal parameters during the full-scale test was revealed. The damage position of the main members was identified according to the mode shape changes, and the dynamic collapse process was analyzed via the measured dynamic responses and recorded videos. Numerical simulation was conducted in ANSYS using the calculated initial imperfections, and the results were in good agreement with the full-scale test. The results show that torsion action was not dominant for this tower and the failure mode was still the bending failure at the tower body. The proposed imperfection quantification method makes up for the deficiency of the empirical values, and using experimental results to guide numerical simulation proves practical. The modal parameters vary greatly under different loading conditions, which can be attributed to the damage after each loading case.

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.

Investigation of long-term modal properties of a supertall building under environmental and operational variations

The information on long-term modal properties of civil structures under environmental and operational variations is generally required in structural health monitoring and vibration control. In operational modal analysis (OMA), damping estimates usually present larger fluctuations and uncertainties than natural frequency and mode shape estimates, which affects the precise determination of the long-term damping characteristics of civil structures. To this end, a combined OMA scheme is adopted in this paper based on the variational mode extraction technique and covariance-driven stochastic subspace identification algorithm to reduce errors in modal identification, especially for damping estimates. Through numerical simulation study of a three-story frame structure, the effectiveness and accuracy of the combined scheme for modal estimations are validated. Moreover, the combined approach is further utilized to evaluate the long-term modal properties of a 600 m high skyscraper under environmental and operational variations from the measured response records of 183 days. Based on the reliable modal estimates by the combined method, the effects of environmental factors (air temperature and relative humidity), wind conditions, and building response amplitudes on the long-term modal properties and their statistical characteristics are investigated in detail. This paper aims to provide an effective tool for analyzing the long-term modal properties of civil structures and further reveal the time evolution of the dynamical characteristics of supertall buildings under environmental and operation variations. • Present a combined scheme based on variational mode extraction technique and covariance-driven stochastic subspace identification algorithm to reduce errors in structural modal estimates. • Suggest an empirical formula for determining minimum length of responses required for accurate modal estimates. • Validate effectiveness and accuracy of the combined scheme via numerical simulation study. • Investigate long-term evolution of dynamical characteristics of a 600 m high skyscraper under environmental and operational variations.

Finite element model of a cable-stayed bridge updated with vibration measurements and its application to investigate the variation of modal frequencies in monitoring

Ambient vibration measurements of a cable-stayed bridge in Taiwan were first conducted to identify its dominant modal parameters with a reliable stochastic subspace identification algorithm recently developed. A finite element model is then constructed based on the identified frequencies and shape vectors of four vertically bending, one torsional, and one horizontally bending modes. This model is also updated by adjusting the girder-pier connection with the addition of a rotational spring to reflect the actual conditions and its accuracy is demonstrated with less than 1% of error for the first two vertically bending modes. As a solid application for probing the environmental effects in structural health monitoring, the vibration measurements of 24 hours were further conducted and combined with the well calibrated model to investigate the controlling factor for its variation in modal frequencies. It is found that the frequencies of the first two vertically bending modes are closely correlated to the root-mean-square velocity representing the traffic excitation intensity. Moreover, it is shown with the updated model that the daily variations in the frequencies of the first two vertically bending modes can be reasonably simulated by modifying the coefficient of the rotational spring merely in a limited range.

Synthetical Modal Parameters Identification Method of Damped Oscillation Signals in Power System

It is vital to improve the stability of the power system by accurately identifying the modal parameters of damped low-frequency oscillations (DLFO) and controlling the oscillation in time. A new method based on empirical mode decomposition (EMD), stochastic subspace identification (SSI), and Prony algorithms, called synthetical modal parameters identification (SMPI) method, is developed by efficiently matching the modal parameters of DLFO which are acquired from the SSI and Prony algorithm. In this approach, EMD is used for denoising the raw oscillation signals thereby enhancing the noise resistance, and then using the SSI and Prony algorithms to identify the precise modal parameters assisted by parameter matching. It is demonstrated that the proposed SMPI method holds great accuracy in identifying full modal parameters including natural frequencies, damping ratios, amplitudes, and phase angles with simulated signals with known modal parameters and real-time signals from some power system case studies. The strategy of SMPI has effectively overcome the weakness of a single approach, and the identification results are promising to heighten the stabilization of power systems. Besides, SMPI shows the potential to troubleshoot in different fields, such as construction, aeronautics, and marine, for its satisfactory robustness and generalization ability.

A three-stage automated modal identification framework for bridge parameters based on frequency uncertainty and density clustering

As the automated modal analysis is crucial for a continuous monitoring system, this study proposes a framework for automated modal identification of bridge parameters based on the uncertainty of estimated frequencies and density-based clustering algorithm, which consists of the following three stages: First, the modal parameters and standard deviations of the estimated frequencies are calculated in a wide range of model orders to construct the stabilization diagram using the reference-based covariance-driven stochastic subspace identification algorithm. Second, the criteria of frequency uncertainty and stabilization are adopted to eliminate the spurious modes. Third, for present purpose, the modified version of an unsupervised density-based clustering algorithm is introduced to group physical modes and detect outliers to reach automated identification of bridge modal parameters. From the analysis, it has shown that the proposed framework is powerful in eliminating the spurious modes and robust in the presence of interference caused by spurious modes while a simple procedure for clustering physical modes with desired statistical reliability is employed.

Deflection estimation of beam structures based on the measured strain mode shape

Deflection is of great significance for evaluating the safety performance of long-span bridges, while it is difficult or expensive to be directly measured in actual projects despite numerous displacement sensors have been developed. This paper proposes a displacement reconstruction framework to obtain the dynamic deflection of beam structures using strain data. The framework defines the judgment criteria of curvature symbols and standard side first, and then stochastic subspace identification algorithm is used to calculate the strain mode shapes involved in the vibration, which solves the unclear calculation process and the need for considerable measurement points of the traditional mode superposition method. Furthermore, the displacement mode shapes and corresponding modal coordinates are obtained, and then the dynamic deflections are derived. Numerical simulations of cantilever and simply supported beams illustrate the effectiveness of the proposed framework, and the results show that the error is only 0.69% with only four strain measurement points. The two corresponding model experiments have also been carried out, and the results demonstrate that the deflection estimation error is only 0.14% with five measurement points, which further proves that the developed displacement reconstruction framework can accurately reconstruct the dynamic deflection of the beam structures.

A quality-based automated procedure for operational modal analysis

Many efforts have been made in the last decade to define Automated Operational Modal Analysis (AOMA) procedures able to process large datasets from long-term monitoring systems. However, some issues are still open and further studies on this topic are needed; in particular following points should be better investigated: i) controlling the tuning parameters to minimize the modelling errors, ii) defining suitable automatization method for processing large datasets and iii) validating the extracted modal parameters. These points are investigated in this paper with the aim of enhancing the current AOMA procedures based on the Stochastic Subspace Identification (SSI) techniques, and the following novelties are introduced: i) a minimization approach for the tuning of the initial parameters in the SSI algorithm, ii) a statistical method to automatically define the cut-off threshold in the hierarchical clustering phase, and iii) a Modal Quality Index (MQI) – ranging from 0 to 1 – to validate the identified modes. The above novelties represent key aspects that allow a real-time check of the modal parameters provided by a monitoring system within a Continuous Structural Health monitoring (CSHM) framework.

Modal parameters identification of bridge by improved stochastic subspace identification method with Grubbs criterion

In the wind tunnel test of a long-span bridge model, to ensure that the dynamic characteristics of the model can satisfy the test design requirements, it is particularly important to accurately identify the modal parameters of the model. First, the stochastic subspace identification algorithm was used to analyze the modal parameters of the model in the wind tunnel test; then, Grubbs criterion was introduced to effectively eliminate outliers in the damping ratio matrix. Stochastic subspace identification algorithm with Grubbs criterion improved the accuracy of the modal parameter identification and the ability to determine system matrix order and prevented the modal omissions caused by determining the stable condition of the damping ratio in the stability diagram. Finally, Oujiang Bridge was used as an example to verify the stochastic subspace identification algorithm with Grubbs criterion and compare with the results of the finite element method. The example shows that the improved method can be effectively applied to the modal parameter identification of bridges.

Output-only modal analysis for non-synchronous data using stochastic sub-space identification

The stochastic subspace identification (SSI) method has been recognized as the most dominant and popular system identification technique in the time domain. Nevertheless, it cannot cope with the non-synchronicity of dynamic response measurements that happens sometimes in structural health monitoring practice. To overcome this, this study proposes a strategy for the SSI algorithm to realize system identification from non-synchronous dynamic response measurements. The modal identification is carried out in a pairwise manner by pairing a time-shifted signal with a common reference signal. The state space model is employed to fit the pair of reference and time-shifted signals and the SSI algorithm is exploited to extract the modal parameters. The core of the strategy is the tactical usage of the mean phase deviation (MPD) of the mode for seeking out the actual time lag as well as the actual mode shape components simultaneously. By so doing, the strategy fulfills integrated time lag estimation and modal extraction for non-synchronous dynamic response measurements. Furthermore, the strategy also overcomes the difficulty to determine the model order with the use of the stabilization diagram and reduces the computational cost substantially with the help of the periodicity of the MPD. To examine the performance of the strategy, intensive validations are conducted by making use of the non-synchronous acceleration measurements of the Jiangyin Bridge subjected to a ship collision and the artificially misaligned acceleration measurements of the Canton Tower struck by an earthquake. Both the time lags and the mode shapes identified by the strategy can be well validated, indicating that the proposed strategy is competent for modal identification of non-synchronous dynamic response measurements.

A Hybrid Coordinated Design Method for Power System Stabilizer and FACTS Device Based on Synchrosqueezed Wavelet Transform and Stochastic Subspace Identification

The occurrence of low-frequency electromechanical oscillations is a major problem in the effective operation of power systems. The scrutiny of these oscillations provides substantial information about power system stability and security. In this paper, a new method is introduced based on a combination of synchrosqueezed wavelet transform and the stochastic subspace identification (SSI) algorithm to investigate the low-frequency electromechanical oscillations of large-scale power systems. Then, the estimated modes of the power system are used for the design of the power system stabilizer and the flexible alternating current transmission system (FACTS) device. In this optimization problem, the control parameters are set using a hybrid approach composed of the Prony and residual methods and the modified fruit fly optimization algorithm. The proposed mode estimation method and the controller design are simulated in MATLAB using two test case systems, namely IEEE 2-ar-ea 4-generator and New England-New York 68-bus 16-genera-tor systems. The simulation results demonstrate the high performance of the proposed method in estimation of local and interarea modes, and indicate the improvements in oscillation damping and power system stability.

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 Response Characterization of Floating Structures Based on Numerical Simulations

Output-only methods are widely used to characterize the dynamic behavior of very diverse structures. However, their application to floating structures may be limited due to their strong nonlinear behavior. Therefore, since there is very little experience on the application of these experimental tools to these very peculiar structures, it is very important to develop studies, either based on numerical simulations or on real experimental data, to better understand their potential and limitations. In an initial phase, the use of numerical simulations permits a better control of all the involved variables. In this work, the Covariance-driven Stochastic Subspace Identification (SSI-COV) algorithm is applied to numerically simulated data of two different solutions to Floating Offshore Wind Turbines (FOWT) and for its capability of tracking the rigid body motion modal properties and susceptibility to different modeling restrictions and environmental conditions tested. The feasibility of applying the methods in an automated fashion in the processing of a large number of datasets is also evaluated. While the structure natural frequencies were consistently obtained from all the simulations, some difficulties were observed in the estimation of the mode shape components in the most changeling scenarios. The estimated modal damping coefficients were in good agreement with the expected results. From all the results, it can be concluded that output-only methods are capable of characterizing the dynamic behavior of a floating structure, even in the context of continuous dynamic monitoring using automated tracking of the modal properties, and should now be tested under uncontrolled environmental loads.

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.

A Genetic Algorithm Procedure for the Automatic Updating of FEM Based on Ambient Vibration Tests.

The dynamic identification of the modal parameters of a structure, in order to gain control of its functionality under operating conditions, is currently under discussion from a scientific and technical point of views. The experimental observations obtained through structural health monitoring (SHM) are a useful calibration reference of numerical models (NMs). In this paper, the procedures for the identification of modal parameters in historical bell towers using a stochastic subspace identification (SSI) algorithm are presented. Then, NMs are manually calibrated on the identification’s results. Finally, the applicability of a genetic algorithm for the automatic calibration of the elastic parameters is considered with the aim of searching for the properties of the autochthonous material, in order to reduce modelling error following the model assurance criterion (MAC). In this regard, several material values on the same model are examined to see how to approach the evolution and the distribution of these features, comparing the characterization proposed by the genetic algorithm with the results considered by the manual iterative procedure.