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.

Multimodal subspace identification for modeling discrete-continuous spiking and field potential population activity

Objective. Learning dynamical latent state models for multimodal spiking and field potential activity can reveal their collective low-dimensional dynamics and enable better decoding of behavior through multimodal fusion. Toward this goal, developing unsupervised learning methods that are computationally efficient is important, especially for real-time learning applications such as brain–machine interfaces (BMIs). However, efficient learning remains elusive for multimodal spike-field data due to their heterogeneous discrete-continuous distributions and different timescales. Approach. Here, we develop a multiscale subspace identification (multiscale SID) algorithm that enables computationally efficient learning for modeling and dimensionality reduction for multimodal discrete-continuous spike-field data. We describe the spike-field activity as combined Poisson and Gaussian observations, for which we derive a new analytical SID method. Importantly, we also introduce a novel constrained optimization approach to learn valid noise statistics, which is critical for multimodal statistical inference of the latent state, neural activity, and behavior. We validate the method using numerical simulations and with spiking and local field potential population activity recorded during a naturalistic reach and grasp behavior. Main results. We find that multiscale SID accurately learned dynamical models of spike-field signals and extracted low-dimensional dynamics from these multimodal signals. Further, it fused multimodal information, thus better identifying the dynamical modes and predicting behavior compared to using a single modality. Finally, compared to existing multiscale expectation-maximization learning for Poisson–Gaussian observations, multiscale SID had a much lower training time while being better in identifying the dynamical modes and having a better or similar accuracy in predicting neural activity and behavior. Significance. Overall, multiscale SID is an accurate learning method that is particularly beneficial when efficient learning is of interest, such as for online adaptive BMIs to track non-stationary dynamics or for reducing offline training time in neuroscience investigations.

Structural flexibility identification from impact test data through a subband estimation method

SummaryFlexibility is an important parameter reflecting bridge load‐carrying capacity. Dynamic testing is a fast and effective method to obtain the structural modal flexibility of small‐ and medium‐span bridges. The Deterministic‐stochastic subspace identification (DSI) algorithm is a well‐established structure identification method in the time domain. However, the estimation of damping, especially the modal scaling factor, is not always reliable due to inevitable measurement noise, which directly affects the identification accuracy of flexibility. This paper proposes a maximum likelihood estimation method in subbands (SMLE), which can be regarded as an add‐on method of the DSI algorithm because the initial parameters are obtained from the DSI algorithm. The processing of frequency band division is implemented first, and the frequency response function curve in each subband of the whole frequency range is fit separately. Then, subband cyclic iteration is proposed to improve the identification accuracy in a closely spaced mode system. The proposed SMLE method maintains the advantages of the DSI algorithm while improving the accuracy of parameter estimation and flexibility identification. Two lumped mass models are used to verify that the proposed method can effectively improve estimates to obtain a precise flexibility matrix and predict the displacement of the structure under static loading. Experimental example of a continuous girder bridge is considered to verify the availability and effectiveness of the proposed method in practice.

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.

Continuous-Time Subspace Identification with Prior Information Using Generalized Orthonormal Basis Functions

This paper presents a continuous-time subspace identification method utilizing prior information and generalized orthonormal basis functions. A generalized orthonormal basis is constructed by a rational inner function, and the transformed noises have ergodic properties. The lifting approach and the Hambo system transform are used to establish the equivalent nature of continuous and transformed discrete-time stochastic systems. The constrained least squares method is adopted to investigate the incorporation of prior knowledge in order to further increase the subspace identification algorithm’s accuracy. The input–output algebraic equation derives an optimal multistep forward predictor, and prior knowledge is expressed as equality constraints. In order to solve an optimization problem with equality constraints characterizing the prior knowledge, the proposed method reduces the computational burden. The effectiveness of the proposed method is provided by numerical simulations.

Battery leakage fault diagnosis based on multi-modality multi-classifier fusion decision algorithm

With the rapid development of the new energy vehicle industry and the overall number of electric vehicles, the thermal runaway problem of lithium-ion batteries has become a major obstacle to the promotion of electric vehicles. During actual usage, the battery leakage problem leads to the degradation of the system performance, which may cause arcing, external short circuit or even thermal runaway. Therefore, it is essential to analyze the internal mechanism of electrolyte leakage phenomenon and design the corresponding fault diagnosis algorithm. This work tests the disassembled leaking battery module of the practical vehicle. The incremental capacity analysis of the charging process indicated that the battery had capacity loss, and the voltage signal trend analysis of the discharging process found that the leaking battery had higher voltage difference slope. The detection results based on the electrochemical impedance spectrum (EIS) test corroborate the anomaly internal impedance of the battery. Moreover, an incremental capacity bar graph analysis method based on cloud data is designed and the ohm resistance is calculated by a simplified subspace identification algorithm. On this basis, the threshold alarm information is incorporated to form a feature matrix, and a machine learning fault diagnosis algorithm based on multi-modality multi-classifier fusion decision framework is proposed, which is capable of scoring and quantifying the hazard level of fault types such as electrolyte leakage, and achieve up to 26 days advance warning on cloud-based real-vehicle data.

Stability by assigning structures by applying the multivariable subspace identification algorithm for a wind system with a DFIG

In this paper, a controller for a doubly fed induction generator (DFIG) connected to a wind system is proposed. This control assigning its own structures as an optimal control method, the electric model in the DFIG state space is also shown, for which it is expected to estimate a linear model through the subspace technique and thus to design the controller. It will be possible to show that a structure assignment controller is undoubtedly a good option for the control of multivariable systems. The results of the output signals will be analyzed when applying the controller, assigning their own structures, which will allow us to observe that the response and disturbance times are below two tenths of a second.

A Hybrid Hubspace-RNN based approach for Modelling of Non-Linear Batch Processes

The manuscript addresses the problem of developing a modelling strategy that can accurately capture the dynamics of a non-linear batch process, demonstrated on a uni-axial rotational molding process. To this end, this work presents a strategy that utilizes the nonlinear modeling capability of a recurrent neural network (RNN) model, while leveraging the rigor of subspace-based approaches. The proposed modeling strategy is as follows: a subspace identification algorithm is first utilized on all the training batches to obtain the state sequence for these batches. Then the state sequences from all the training batches and the corresponding input sequences are given as the output and the input to a neural network, respectively. This step essentially builds a non-linear state space model, albeit using the state trajectory identified by the subspace model. The output equation obtained from the above-mentioned subspace identification step along with the newly obtained non-linear state equation is now ready to be used for predicting the output trajectory of the system. The results from the experiments illustrate the improved prediction performance of a model obtained by the proposed hybrid Subspace-RNN approach in comparison to both a subspace-based approach and an RNN-based approach.

Indirect measurement of propeller fluctuating forces using an inverse analysis: Numerical and experimental studies

The propeller fluctuating force is being recognized as the dominant source of underwater radiated noise, its accurate acquisition is thus necessitated for a reliable acoustic-structure analysis and optimization of marine ships. However, such dynamic operational forces are frequently impossible or expensive to directly measure due to various practical challenges. To alleviate this limitation, this paper presents an inverse method for the indirect estimation of propeller forces using measured shafting vibration responses. The force reconstruction procedure is constructed based on the state-space model of the propeller–shafting system, which can be derived by a data-driven subspace identification algorithm in conjunction with either the numerical or experimental models of the system. To ensure the robustness of the inverse analysis in the presence of inevitable measurement noise, a modified regularization strategy based on homotopy continuation is introduced. The effectiveness of the proposed algorithm is demonstrated through numerical simulations and experimental investigations. The obtained results demonstrate the potential feasibility of indirect measurement for accurately identifying propeller fluctuating forces in the time domain.

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.

Modelling of an interacting series process for rector cooling system using system identification method

This research presents the modelling of cooling system plant in the TRIGA PUSPATI reactor (RTP). The RTP cooling system involves the interacting series process which are reactor model and heat exchanger model. The behaviour process is complex due to the nonlinearity, uncertainty, and time variation of the system. Due to the nonlinearity and interacting series process of system, the numerical algorithm for subspace state space system identification (N4SID) was proposed in the modelling of RTP cooling system. However, the model was converted to the transfer function form. The output response from the simulation model of RTP cooling subsystems were analysed by comparing the model with the real operational data. The responses of the model were within the respective range of a good model data. The residual analysis for each output was also calculated. The comparison and residual analysis were done to validate the model accuracy to determine the behaviour of the model. Overall, the model accuracy assessment shows that all models are still fit in the good model region (<10%) and has a good representation of a real plant.

Practical Implementation of Recursive Subspace Identification on Seismically Excited Structures with Fixed Window

As one of the most catastrophic natural disasters worldwide, earthquakes and their effect on structures are very important to structural health monitoring (SHM), particularly for the ones living around the Pacific ring of fire. In this regard, SHM techniques with real-time or online processing can be used to identify states of structures, track modal parameters, provide a warning message about damage, and help post-earthquake reconnaissance and rehabilitation. For instance, a recursive formulation based on subspace identification (SI) has been demonstrated that it is capable to track system changes. In this study, a recursive subspace identification (RSI) algorithm with a fixed window is proposed to investigate the time-varying dynamic characteristics under seismic excitations. Subsequently, some suggestion is described and discussed for practical implementation. For verifying the proposed algorithm, different datasets from full-scale experiments are applied to examine its applicability. In other words, the practicability of implementing RSI in real-time or online has been developed and examined in this paper.

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.

Time-varying frequency parameter identification of space solar power satellites based on an improved recursive subspace algorithm and optimal sensor placement

This study focuses on the time-varying modal frequency identification of space solar power satellites. Accordingly, an improved recursive subspace identification algorithm based on the variable forgetting factor is proposed to enhance the tracking adaptability of the original method in time-varying systems. Considering the changes in the structural configuration induced by the rotation of the solar concentrator mirror, a time-varying attitude-vibration coupling dynamic model of a space solar power satellite with an integrated symmetrical concentrator configuration is developed based on the modal synthesis technology and the substructure method. Moreover, based on the projection subspace theory, the time-varying forgetting factor is determined by minimizing the mean square deviation of the system estimator. Subsequently, the variable forgetting factor subspace algorithm is adopted to recursively identify the time-varying pseudo modal frequency parameters. A finite element model of the space solar power satellite is established via numerical simulations, and several optimal sensor placement methods are applied to determine the positions of the vibration sensors. Based on the output signals obtained from the optimal sensor placement results, the time-varying frequency parameters of the system are obtained using the proposed algorithm. The computational results reveal that a decrease in the average relative error by 4.2% compared with that obtained via the conventional recursive subspace method is observed when the signal-to-noise-ratio of the measured output is selected as 15 dB. In addition, the results also reveal that an average relative error reduction of approximately 0.4%–1.4%, over two other methods with a fixed forgetting factor, can be achieved using the proposed algorithm when the rotation speed of the solar concentrator mirror is increased. The findings of this study confirm that the proposed algorithm demonstrates a high noise-immune ability and tracking performance for fast time-varying systems.

Nonlinear system identification based on restoring force transmissibility of vibrating structures

In vibrating elastic structures, the motion at locations with no external load is induced by internal forces and due to transmissibility. This capability is utilized in this paper for parameter estimation of nonlinear structures in the absence of input measurements. It is demonstrated that the nonlinearities are detectable at degrees of freedom away from the excitation with no need for input measurements and mass matrix information. To estimate the linear and nonlinear parameters within the physical equations of motion, the proposed approach is utilized in conjunction with a new data-driven subspace identification algorithm. The idea is to project the output measurements onto the basis function that is used for predicting the restoring force, and then separate the nonlinear dynamics from the underlying linear system. The proposed approach is implemented and examined for different structural nonlinearity types and compared to other methods. Moreover, the vibro-impact parameters of three parallel cantilever beams with non-equidistant gaps are experimentally investigated. The results address the feasibility range of the transmissibility-based subspace algorithm for the identification of a broad class of structural nonlinearities when the excitation loads are not measurable.

Uncertainty quantification of modal parameter estimates obtained from subspace identification: An experimental validation on a laboratory test of a large-scale wind turbine blade

The uncertainty afflicting modal parameter estimates stems from e.g., the finite data length, unknown, or partly measured inputs and the choice of the identification algorithm. Quantification of the related errors with the statistical Delta method is a recent tool, useful in many modern modal analysis applications e.g., damage diagnosis, reliability analysis, model calibration. In this paper, the Delta method-based uncertainty quantification methodology is validated for obtaining the uncertainty of the modal parameter and the modal indicator estimates in the context of several well-known subspace identification algorithms. The focus of this study is to validate the quality of each Delta method-based approximation with respect to the experimental Monte Carlo distributions of parameter estimates using a statistical distance measure. On top of that, the accuracy in obtaining the related confidence intervals is empirically assessed. The case study is based on data obtained from an extensive experimental campaign of a large scale wind turbine blade tested in a laboratory environment. The results confirm that the Delta method is, on average, adequate to characterize the distribution of the considered estimates solely based on the quantities obtained from one data set, validating the use of this statistical framework for uncertainty quantification in practice.

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.