Subspace System Identification Method Research Articles

Identification of the dynamic model of second-life lithium-ion cells through Subspace System Identification and similarity transformation

When lithium-ion batteries (LiBs) reach the end of their first useful life in electric vehicles (EVs), they can still be used in applications with lower power demands, a process known as second-life. However, to ensure that LiBs – or cells – removed from EVs operate safely, efficiently, and reliably in a second application, various tests and procedures must be applied to study their internal conditions. Some information about the conditions of the lithium-ion cells can be obtained from the parameters of its equivalent circuit model(ECM). However, the tests to obtain blackthe ECM parameters usually require taking the LiBs out of service and testing their cells in laboratory benches with special equipment, which is costly and time-consuming. Therefore, this work proposes a methodology to identify the ECM of lithium-ion cells based on Subspace System Identification (SSI) methods. This procedure can be executed online, while the LiBs are being used in the specified application. SSI methods have been widely adopted to identify the dynamic model of lithium-ion cells, but they return models without a physical meaning, called non-parametric models. This type of model cannot be used to address the internal conditions of a lithium cell, because its parameters do not have any physical meaning. To solve this problem, a novel method for estimating the lithium-ion cell physical parameters, in the ECM format, and from its non-parametric model, is proposed in this work. The procedure is based on similarity transformation techniques and special characteristics of the ECM. To validate the proposed method, nine samples of cells from the same manufacturer were studied. These cells had distinct degradation conditions and were removed from heavy-duty EVs at the end of their first life. The results showed that: (a) a good approximation between the identified model voltage output and the actual cells’ experimental voltage output was achieved, with root mean square error values as small as 2.26 mV; (b) it was possible to identify the ECM of the tested cells and their parameters using the proposed similarity transformation; and (c) the identified parameters can be used to detect the internal conditions of the cells.

Online state of health estimation of lithium-ion batteries through subspace system identification methods

When Lithium-ion Batteries (LiBs) reach the end of their first life in electric vehicles (EVs), they can still be used in applications with lower power demands, a process known as second-life. However, to ensure that LiBs – or cells – removed from EVs operate safely, efficiently and reliably in a second application, several tests and procedures must be applied to study their internal conditions. Naturally, one of the most important parameters to be determined is the state of health (SoH). However, the available processes for determining the SoH of lithium-ion cells are limited by high costs, relatively long test times and the need for specific equipment, limiting the second-life market. Hence, this work proposes a methodology to estimate the SoH of lithium-ion cells, based on subspace system identification (SSI) methods, where the parameters estimated for the equivalent circuit model (ECM) of a given cell are associated with its SoH. To validate the proposed methodology, nine cell samples from the same manufacturer were considered, which were removed from heavy-duty EVs at the end of their first life. The obtained results showed that: (a) good approximations between the identified models and the actual cells were achieved, with root mean square error (RMSE) values as small as 1.32 mV; (b) SSI methods can be applied online, while the LiBs are still operating in the EV during their first life, eliminating the need of additional tests; and (c) there is a clear association between ECM parameters and the SoH, so it was possible to estimate the SoH of the samples with RMSE values varying from 2.11% to 3.34%. Therefore, the proposed methodology offers significant improvements when compared to the conventional capacity tests, including the possibility of estimating the SoH relatively fast, online and without the need for specific equipment.

Data Driven Positive Subspace System Identification

This paper considers the data-driven subspace system Identification for positive systems. The application of subspace system Identification (SSID) method to collected data of positive systems does not guarantee the positivity of the identified system. Thus, two approaches are proposed to obtain the state space parameters of the positive system using the data collected from its input and output. The first approach offers a procedure based on nonnegative least square algorithm applied to data-driven state space representation. The second approach applies a nonnegative matrix factorization to the data collected Hankel matrix followed by generalized realization algorithm. The effectiveness of two approaches in positive SSID are demonstrated by numerical examples.

High Dimensional Data Reduction in Modal Analysis with Stochastic Subspace Identification

Subspace system identification methods are widely used in output-only vibration analysis of civil structures, known as operational modal analysis. With the advent of new sensor technologies, such as video camera-based full field displacement or velocity measurements, the number of measured outputs is quickly increasing. In this paper, we propose principal component analysis-based data size reduction methods for efficient application of subspace methods, while preserving the high spatial resolution of the identified mode shapes for detailed modal analysis.

Identification of modal parameters of long-span bridges under various wind velocities

The modal parameters identification of bridges under non-stationary environmental excitation has caught the attention of researchers. This paper studies the non-stationarity of wind velocity, and extracts the time-varying mean wind velocity based on a discrete wavelet transform and recursive quantitative analysis. The calculated turbulence intensity and turbulence integral scale under the non-stationary model are smaller than those under the stationary model, especially the turbulence integral scale. The empirical wavelet transform is used to identify the modal parameters of long-span bridges, and the power spectral density spectrum is proposed as a replacement for the Fourier spectrum as the basis of the frequency band selection. The bridge modal parameters are then compared using the covariance-driven stochastic subspace system identification method (SSI-COV) and the Hilbert transform method based on an improved empirical wavelet transform (EWT-HT). Both methods can accurately identify the modal frequency, and the absolute difference between these two methods is equal to 0.003 Hz. The wind velocity results in a change of less than 1% in the modal frequency. The absolute difference between the modal damping ratios identified using SSI-COV and EWT-HT is significant and can reach 0.587%. The modal damping ratios are positively correlated with the mean wind velocities, which aligns with the quasi-steady assumption. In addition, the applicability of SSI-COV and EWT-HT is also evaluated using the standard deviation, coefficient of variation, and range dispersion indicators. The results show that the EWT-HT is more applicable to the identification of the modal parameters of long-span bridges under non-stationary wind velocities.

Molecular Weight Distribution Control for Polymerization Processes Based on the Moment-Generating Function.

The molecular weight distribution is an important factor that affects the properties of polymers. A control algorithm based on the moment-generating function was proposed to regulate the molecular weight distribution for polymerization processes in this work. The B-spline model was used to approximate the molecular weight distribution, and the weight state space equation of the system was identified by the subspace state space system identification method based on the paired date of B-spline weights and control inputs. Then, a new performance criterion mainly consisting of the moment-generating function was constructed to obtain the optimal control input. The effectiveness of the proposed control method was tested in a styrene polymerization process. The molecular weight distribution of the styrene polymers can be approximated by the B-spline model effectively, and it can also be regulated towards the desired one under the proposed control method.

Kalman predictor subspace residual for mechanical system damage detection

For mechanical system structural health monitoring, a new residual generation method is proposed in this paper, inspired by a recent result on subspace system identification. It improves statistical properties of the existing subspace residual, which has been naturally derived from the standard subspace system identification method. Replacing the monitored system state-space model by the Kalman filter one-step ahead predictor is the key element of the improvement in statistical properties, as originally proposed by Verhaegen and Hansson in the design of a new subspace system identification method.

On subspace system identification methods

On subspace system identification methods

Efficient Forced Response Computations of Acoustical Systems with a State-Space Approach

State-space models have been successfully employed for model order reduction and control purposes in acoustics in the past. However, due to the cubic complexity of the singular value decomposition, which makes up the core of many subspace system identification (SSID) methods, the construction of large scale state-space models from high-dimensional measurement data has been problematic in the past. Recent advances of numerical linear algebra have brought forth computationally efficient randomized rank-revealing matrix factorizations and it has been shown that these factorizations can be used to enhance SSID methods such as the Eigensystem Realization Algorithm (ERA). In this paper, we demonstrate the applicability of the so-called generalized ERA to acoustical systems and high-dimensional input data by means of an example. Furthermore, we introduce a new efficient method of forced response computation that relies on a state-space model in quasi-diagonal form. Numerical experiments reveal that our proposed method is more efficient than previous state-space methods and can even outperform frequency domain convolutions in certain scenarios.

Modeling and Validation of a Four‐Tank System for Level Control Process Using Black Box and White Box Model Approaches

Mathematical modeling of a system plays a vital role in controller design, tuning to meet the desired specifications and process parameter optimization. In this paper, different identification algorithms are implemented in the four‐tank system to demonstrate the superiority of the recommended subspace system identification method (Numeric Algorithms for Subspace State Space System Identification [N4SID]) for the different input design. A system identification method was utilized to identify the model of a dynamic system. It is modeled using the First‐Principle Method (FPM), Prediction Error Method (PEM) and Numeric Algorithms for N4SID method. The identification is performed through MATLAB. The plant input and output is acquired through LabVIEW. The developed models are validated with a new dataset based on the model performance specifications: Mean Absolute Error (MAE), Root Mean Squared Error (RMSE) and Best Fit Rate (BFR). The results show that the proposed N4SID method with the amplitude‐modulated pseudorandom binary signal (APRBS) signal is better than the other methods with different inputs. © 2020 Institute of Electrical Engineers of Japan. Published by Wiley Periodicals LLC.

Dynamic Characterization of Structures from Limited Measurements Using a Subspace System Identification Method

In this study, we investigate a subspace-system identification method that estimates the modal characteristics and damping ratio of structures using a small number of sensor measurement responses. With this method, a Kalman state vector is first constructed from the input and measurement data, and then the state-space system matrix is optimized from it. For this purpose, a block Hankel matrix is constructed using the input data of the load applied to the structure and the measurement data of the response, and then the matrix is decomposed through QR factorization. Then, singular value decomposition is used to obtain the extended observability matrix and state-space system order $$n$$ , and the state-space system matrix constituting the Kalman state vector is obtained via least-squares optimization. The natural frequency and damping ratio of the structural system are calculated from the eigenvalue of the obtained system matrix. To verify the effectiveness of this subspace state-space system identification method, the natural frequency and mode shape of a simply supported beam and cantilever beam were calculated using a limited number of measurement data, and the results were compared with an exact solution to evaluate the accuracy of the estimation results of the dynamic characteristics. Furthermore, the modal characteristics and damping ratio of a real bridge were estimated from acceleration data. The stability chart calculated from the example shows that the subspace state-space system identification method discussed in this study can accurately identify the modal characteristics of structures using only a small number of measurement data.

Application of the Subspace-Based Methods in Health Monitoring of Civil Structures: A Systematic Review and Meta-Analysis

A large number of research studies in structural health monitoring (SHM) have presented, extended, and used subspace system identification. However, there is a lack of research on systematic literature reviews and surveys of studies in this field. Therefore, the current study is undertaken to systematically review the literature published on the development and application of subspace system identification methods. In this regard, major databases in SHM, including Scopus, Google Scholar, and Web of Science, have been selected and preferred reporting items for systematic reviews and meta-analyses (PRISMA) has been applied to ensure complete and transparent reporting of systematic reviews. Along this line, the presented review addresses the available studies that employed subspace-based techniques in the vibration-based damage detection (VDD) of civil structures. The selected papers in this review were categorized into authors, publication year, name of journal, applied techniques, research objectives, research gap, proposed solutions and models, and findings. This study can assist practitioners and academicians for better condition assessment of structures and to gain insight into the literature.

Health Monitoring of Civil Infrastructures by Subspace System Identification Method: An Overview

Structural health monitoring (SHM) is the main contributor of the future’s smart city to deal with the need for safety, lower maintenance costs, and reliable condition assessment of structures. Among the algorithms used for SHM to identify the system parameters of structures, subspace system identification (SSI) is a reliable method in the time-domain that takes advantages of using extended observability matrices. Considerable numbers of studies have specifically concentrated on practical applications of SSI in recent years. To the best of author’s knowledge, no study has been undertaken to review and investigate the application of SSI in the monitoring of civil engineering structures. This paper aims to review studies that have used the SSI algorithm for the damage identification and modal analysis of structures. The fundamental focus is on data-driven and covariance-driven SSI algorithms. In this review, we consider the subspace algorithm to resolve the problem of a real-world application for SHM. With regard to performance, a comparison between SSI and other methods is provided in order to investigate its advantages and disadvantages. The applied methods of SHM in civil engineering structures are categorized into three classes, from simple one-dimensional (1D) to very complex structures, and the detectability of the SSI for different damage scenarios are reported. Finally, the available software incorporating SSI as their system identification technique are investigated.

Pseudo-State-Space Fractional System Identification

When implementing continuous-time subspace methods for system identification in a stochastic context, it is required to compute filtered derivatives of continuous-time signals. These filters are introduced to avoid amplification of noise in high frequencies. Different types of filters are studied in this paper on the basis of MOESP and PO-MOESP subspace algorithms.

Sequential Monte Carlo Filter for State-of-Charge Estimation of Lithium-Ion Batteries Based on Auto Regressive Exogenous Model

The state of charge (SOC) of lithium-ion batteries (LIBs) is an important evaluation index in battery management system of energy storage systems. However, it is always a challenging task to accurately estimate SOC of LIBs, because of the existence of nonlinear characteristics and significant temperature effects. To improve the SOC estimation accuracy and robustness, a model-based estimation approach for SOC and impedance of LIBs is proposed against uncertain loading profiles and ambient temperatures. First an auto regressive exogenous model for online model order determining and parameters identification is established to monitor parameters variations based on numerical subspace state space system identification method. Second, a sequential Monte Carlo filter is employed to overcome the nonlinear and non-Gaussian error distribution state estimation problems caused by complicated open-circuit voltage characteristics. Finally, evaluation of the adaptability and generality of the proposed method are verified by different LIBs under different operating conditions. Experimental results indicate that the proposed method shows great performance, whose estimation value converges to real SOC within an error of $\pm \text{3}\%$ , and the battery model can simulate battery dynamics robustly with high accuracy.

Construction of a dynamic adaptive optics system based on the subspace identification

Adaptive optics (AO) systems have been used in many applications, such as ground-based astronomical telescopes for improving the resolution by counteracting the effects of atmospheric turbulence. However, the traditional AO system model is not good at dealing with noise interference in the closed-loop correction. Therefore, our paper proposes a subspace system identification method based on errors-in-variables to build an accurate dynamic model of the AO system from measurement data with closed-loop system identification. Experimental simulation results show that the root mean square of the residual wavefront is smaller than that of the traditional method, whether photon noise and camera readout noise exist or not. We conclude that the identified model has good accuracy and noise disturbance compensation ability to deal with dynamic wavefront correction compared with the traditional method.

Large-scale dynamic modeling of task-fMRI signals via subspace system identification

Objective. We analyze task-based fMRI time series to produce large-scale dynamical models that are capable of approximating the observed signal with good accuracy. Approach. We extend subspace system identification methods for deterministic and stochastic state-space models with external inputs. The dynamic behavior of the generated models is characterized using control-theoretic analysis tools. To validate their effectiveness, we perform a probabilistic inversion of the identified input–output relationships via joint state-input maximum likelihood estimation. Our experimental setup explores a large dataset generated using state-of-the-art acquisition and pre-processing methods from the Human Connectome Project. Main results. We analyze both anatomically parcellated and spatially dense time series, and propose an efficient algorithm to address the high-dimensional optimization problem resulting from the latter. Our results enable the quantification of input–output transfer functions between each task condition and each region of the cortex, as exemplified by a motor task. Further, the identified models produce impulse response functions between task conditions and cortical regions that are compatible with typical hemodynamic response functions. We then extend subspace methods to account for multi-subject experimental configurations, identifying models that capture common dynamical characteristics across subjects. Finally, we show that system inversion via maximum-likelihood allows the time-of-occurrence of the task stimuli to be estimated from the observed outputs. Significance. The ability to produce dynamical input–output models might have an impact in the expanding field of neurofeedback. In particular, the models we produce allow the partial quantification of the effect of external task-related inputs on the metabolic response of the brain, conditioned on its current state. Such a notion provides a basis for leveraging control-theoretic approaches to neuromodulation and self-regulation in therapeutic applications.

Observer-based repetitive model predictive control in active vibration suppression

In this paper, an observer-based feedback/feedforward model predictive control (MPC) algorithm is developed for addressing the active vibration control (AVC) of lightly damped structures. For this purpose, the feedback control design process is formulated in the framework of disturbance rejection control (DRC) and a repetitive MPC is adapted to guarantee the robust asymptotic stability of the closed-loop system. To this end, a recursive least squares (RLS) algorithm is engaged for online estimation of the disturbance signal, and the estimated disturbance is feed-forwarded through the control channels. The mismatched disturbance is considered as a broadband energy bounded unknown signal independent of the control input, and the internal model principle is adjusted to account for its governing dynamics. For the sake of relieving the computational burden of online optimization in MPC scheme, within the broad prediction horizons, a set of orthonormal Laguerre functions is utilized. The controller design is carried out on a reduced-order model of the experimental system in the nominal frequency range of operation. Accordingly, the system model is constructed following the frequency-domain subspace system identification method. Comprehensive experimental analyses in both time-/frequency-domain are performed to investigate the robustness of the AVC system regarding the unmodeled dynamics, parametric uncertainties, and external noises. Additionally, the spillover effect of the actuation authorities and saturation of the active elements as two common issues of AVC systems are addressed.

Covariance Based Estimation for Reduced Order Models of Microgrid Power Flow Dynamics

Identification using system data becomes a great tool for modeling a system when model derivation using physical laws is complex or impossible. Covariance Based Realization Algorithm (CoBRA), one kind of subspace methods for system identification, allows the use of finite size covariance data matrices to reduce storage requirement. In addition, the covariance pre-processing reduces noise effect and allows CoBRA to concentrate on the identification of possibly low order deterministic dynamics. This paper aims at providing practical analysis of CoBRA in the area of power systems. This includes persistent excitation, identification from arbitrary data segments, the effect of sample approximation and imperfect data. Finally, an application of CoBRA to the identification of microgrid power flow dynamics is demonstrated using available synchrophasor measurements.

위상동기신호를 이용한 한전계통의 저주파진동 검출과 고유치해석

This paper describes the results of a low‐frequency oscillation analysis using data measured in PMU installed in the KEPCO system, and the comparison with eigenvalues computed from the linear model. The dominant oscillation modes are estimated by applying various algorithms. The algorithms are: the extended Prony method; multiple time interval parameter estimation method; subspace system identification method; and spectral analysis. From the measurement data, modes of frequency 0.68[㎐] and 0.92[㎐] were estimated, and modes of frequency 0.63[㎐] and 0.80[㎐] were computed from the eigenvalue calculation. There was a difference between the mode estimated from measurement data and that from the linear model. This is possibly because of an error in the dynamic data of the KEPCO system used in eigenvalue calculation. Because wide area modes exist in the KEPCO system, these modes should be monitored continuously for the reliable operation of the system. In order to prevent total blackouts caused by wide area oscillation, moreover, contingency analysis should be performed in relation to this mode and appropriate measures should be established.