Subspace State Space System Identification Research Articles

System identification based structural damage aggravation detection in a large masonry building

System identification is widely used to estimate the dynamic characteristics of structures using either nonparametric or parametric approaches. Parametric system identification constructs mathematical models to estimate dynamic characteristics such as natural frequency, damping ratio, and mode shapes. Also, the same approach can be used to characterize damage if periodic measurements are available. In this paper, we report damage aggravation in a large neoclassical masonry structure using system identification. We took measurements in 2019 and 2023 in the Institute of Engineering, Dean Office building, which was damaged by the 2015 Gorkha, Nepal earthquake. The building is a neoclassical three storied construction with a central courtyard and an extension along the N-S direction. Without major interventions and strengthening, the building is still in use. Using the numerical algorithms for subspace state space system identification (N4SID), vibration frequencies of the structure are identified for both measurements. Also, mode shapes are identified, and damage aggravation is detected using natural frequency and MAC parameters. The first mode frequency along the N-S direction is reduced by 44% and that along the E-W direction is reduced by 1.15%, which indicates a major damage aggravation along the N-S (Y) direction.

A hybrid rotordynamic modeling method for a rotor system with flexible foundation and nonlinear support force: Numerical and experimental investigation

The dynamic characteristics of a flexible foundation would greatly affect the dynamic behavior of the rotor system. In this paper, a hybrid dynamic model method in time domain combines the finite element (FE) method and the subspace-based identification method to model a rotor system with the consideration of a flexible foundation and nonlinear support forces. The state-space model of flexible foundation is identified by the subspace state-space system identification algorithm with measured frequency response functions (FRFs). To solve the calculation problem of the hybrid dynamic model in time domain, a general algorithm named the hybrid model fusion solution (HMFS) method is proposed, which combines the linear and nonlinear nodes separation method used in the FE model and the trapezoidal rule used in the state-space model, improving the calculation efficiency of solving transient response with nonlinear forces. Simulations for the hybrid dynamic model under constant speed and speed-up cases were carried out, resulting that the dynamic characteristics of the entire system can be represented when the nonlinear factors of rolling bearings and gravity were also taken into consideration. Additionally, owe to the consideration of tested dynamic behaviors of flexible base, experimental results in the laboratory test rig are well agreed with the numerical ones under two different test configurations. The investigation provides a complete modeling and highly efficient solution framework for the rotor system with a flexible foundation and nonlinear rolling bearings, with the abilities of further consideration of more nonlinear components such as squeeze film dampers and rubbing forces.

Experimental modal analysis of a single-link flexible robotic manipulator with curved geometry using applied system identification methods

In this paper, the principal time-domain system identification methods are studied and used to carry out the experimental modal analysis of a single-link flexible manipulator with curved geometry in a comparative fashion. The identification process carried out in this study is aimed at deriving first-order state-space dynamical models and second-order configuration-space dynamical models of the single-link flexible manipulator by using an array of fundamental system identification procedures, which includes the ARX (AutoRegressive eXogenous method), SSEST (State-Space ESTimation method), N4SID (Numerical Algorithm for Subspace State-Space System Identification), ERA/OKID (Eigensystem Realization Algorithm combined with the Observer/Kalman Filter Identification method), and TFEST (Transfer Function ESTimation method) methods. The main goal of this paper is, therefore, to perform a comparative study between the most fundamental system identification procedures applied to the numerical and experimental analysis of the structural vibrations of a flexible manipulator having a curved geometric shape. First, this paper poses the theoretical foundations for the development of the numerical study and the experimental testing to be performed for the case study. Subsequently, the flexible manipulator analyzed in this work is modeled with various levels of complexity, which range from a simple lumped parameter model to a linear finite element model developed by integrating the interconnections between the SOLIDWORKS and ANSYS software. By doing so, the computational procedures applied to the time-domain experimental measurements of the dynamic behavior of the flexible manipulator allow for reconstructing first-order and second-order mechanical models of the structural system of interest in the MATLAB simulation environment. The numerical results and the experimental tests reported in this investigation demonstrate the effectiveness of all the time-domain system identification approaches considered in the paper.

The Accuracy and Computational Efficiency of the Loewner Framework for the System Identification of Mechanical Systems

The Loewner framework has recently been proposed for the system identification of mechanical systems, mitigating the limitations of current frequency domain fitting processes for the extraction of modal parameters. In this work, the Loewner framework computational performance, in terms of the elapsed time till identification, is assessed. This is investigated on a hybrid, numerical and experimental dataset against two well-established system identification methods (least-squares complex exponential, LSCE, and subspace state space system identification, N4SID). Good results are achieved, in terms of better accuracy than LSCE and better computational performance than N4SID.

Constrained temperature and relative humidity predictive control: Agricultural greenhouse case of study

The importance of Model Predictive Control (MPC) has significant applications in the agricultural industry, more specifically for greenhouse’s control tasks. However, the complexity of the greenhouse and its limited prior knowledge prevent an exact mathematical description of the system. Subspace methods provide a promising solution to this issue through their capacity to identify the system’s comportment using the fit between model output and observed data. In this paper, we introduce an application of Constrained Model Predictive Control (CMPC) for a greenhouse temperature and relative humidity. For this purpose, two Multi Input Single Output (MISO) systems, using Numerical Subspace State Space System Identification (N4SID) algorithm, are firstly suggested to identify the temperature and the relative humidity comportment to heating and ventilation actions. In this sense, linear state space models were adopted in order to evaluate the robustness of the control strategy. Once the system is identified, the MPC technique is applied for the temperature and the humidity regulation. Simulation results show that the regulation of the temperature and the relative humidity under constraints was guaranteed, both parameters respect the ranges 15 °C ≤ Tint ≤ 30 °C and 50 % ≤ Hint ≤ 70 % respectively. On the other hand, the control signals uf and uh applied to the fan and the heater, respect the hard constraints notion, the control signals for the fan and the heater did not exceed 0 ≤ uf ≤ 4.3 Volts and 0 ≤ uh ≤ 5 Volts, respectively, which proves the effectiveness of the MPC and the tracking tasks. Moreover, we show that with the proposed technique, using a new optimization toolbox, the computational complexity has been significantly reduced. The greenhouse in question is devoted to Schefflera Arboricola cultivation.

Subspace State-Space Identification of Nonlinear Dynamical System Using Deep Neural Network with a Bottleneck

This paper proposes a new type of subspace state-space system identification method for nonlinear dynamical systems, which generates a model consisting of a state estimator and a predictor that can be directly used for model predictive control (MPC). The main feature of the proposed method is that it uses a neural network with a bottleneck layer between the state estimator and predictor to represent the input-output dynamics, and it is proven that the state of the dynamical system can be extracted from the bottleneck layer based on the observability of the target system. The training of the network is shown to be a natural nonlinear extension of the subspace state-space system identification method established for linear dynamical systems. This correspondence provides interpretability and optimality to the resulting model based on linear control theory. The usefulness of the proposed method and the interpretability of the model are demonstrated through an illustrative example of MPC.

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.

Advanced Constrained Model Predictive Control of Vapor Pressure Deficit in Agricultural Greenhouses

The agricultural greenhouse is an open system to the external environment. The challenge is to track the reference trajectory of relevant climatic parameters and regulate the internal climate, despite the strong meteorological disturbances. This study focuses on the modelling and control of Vapor Pressure Deficit (VPD) coupled with temperature and hygrometry in greenhouses. The importance of this parameter is related to plant growth and its sensitivity to external climate disturbances. The VPD is a primordial parameter for preventing plant diseases caused by an inappropriate interior climate. This work provides a methodology for efficient control of VPD with a Multi-setting parameterization of the controller according to the needs of the plant and the actuators’ power available in the greenhouse. For this purpose, the VPD is indirectly evaluated through the ambient air temperature and humidity using the psychometric chart embedded in a fuzzy inference algorithm. For dynamic modelling of VPD, a discrete state-space model is obtained by considering an agricultural greenhouse as a black-box system. Numerical algorithms for Subspace State Space System Identification (N4SID) of Matlab is applied to parametric identification based on experimental measurements. A Constrained Model Predictive Control (CMPC) method is applied for VPD control design, considering physical and operational constraints on Input/output variables. Here, the CMPC strategy is improved by information on the greenhouse’s outside climatic conditions. Promising numerical simulation results show the feasibility of the proposed modelling and control strategy. They show good performances, despite the climatic exchange phenomenon between inside and outside the greenhouse and strong external weather disturbances. The inside VPD tracks nearly reference trajectories with a small overshooting as desired.

Digital twin approach for on-load tap changers using data-driven dynamic model updating and optimization-based operating condition estimation

On-load tap changers (OLTCs), which are found in power transformers, are mechanically operating components. The vibration signal of an OLTC can provide effective observed data for estimation of the mechanical state transition and a faulty operating condition. Data-driven methods (e.g., deep learning, machine learning) and physics-based methods (e.g., the finite element method, the lumped parameter model) for health-state estimation require sufficient prior knowledge – such as various observed data about fault states, modeling information (including geometry, material properties), and operating conditions – to build a valid digital twin approach. However, prior knowledge for various OLTCs and transformer models is hard to obtain. To begin to address the shortcomings of existing methods, this study proposes a digital twin approach for OLTCs using 1) pre-processing of the vibration signal, 2) data-driven dynamic model updating, and 3) optimization-based operating condition estimation. First, the time–frequency domain features are extracted from the reference signal using a minimum entropy deconvolution (MED) filter, to extract the impulsive vibration signal from OLTC operation. The initial operating conditions that arise from tap changing and diverter switching are assumed as impulsive force using extracted features. The dynamic model is driven by the numerical algorithm for subspace state-space system identification (N4SID), the reference signal, and the excitation impulse force. Next, the phase and magnitude modulation are updated to estimate uncertain operating condition and refine the dynamic model using optimization-based parameter tuning. Analysis results from the proposed digital twin approach are demonstrated for both a numerical and experimental example to verify the effectiveness of the proposed approach. In the numerical case study, i) the simplified physics-based modelling and ii) the joint-input state estimation method were compared with the proposed method. In the experimental case study, the proposed method was applied to an OLTC vibration signal of both inactive and active power transformers. The inflection points in the Dynamic Resistance Measurement (DRM) graph show synchronization with the experimental results of the estimated excitation forces derived using the proposed method.

A high-fidelity building performance simulation test bed for the development and evaluation of advanced controls

We present an open-source building performance simulation test bed, the Advanced Controls Test Bed (ACTB), that interfaces high-fidelity Spawn of EnergyPlus building models, with advanced controllers implemented in Python. The ACTB leverages the Building Optimization Testing and Alfalfa platforms for managing simulations, providing an external clock, a representational state transfer (REST) application programming interface (API), and key performance indicators for evaluating the effectiveness of control strategies. The REST API allows the development of external controllers programmed in languages such as Python, which provides flexibility and a rich choice of scientific libraries for designing control sequences. We present three test cases based on the U.S. Department of Energy's Reference Small Office Building to demonstrate the ACTB's capabilities: (a) rule-based controls compliant with ASHRAE Guideline 36 control sequences; (b) an economic model predictive control implemented using do-mpc; and (c) a deep Q-network reinforcement learning agent implemented using OpenAI Gym. Abbreviations: ACTB: Advanced Controller Test Bed; AHU: Air Handling Unit; AI:Artificial Intelligence; API: Application Programming Interface; BEM: Building EnergyModeling; BSS: Best Subset Selection; DOE: Department of Energy; DQN: Deep-QNetwork; EKF: Extended Kalman Filter; FMI: Functional Mock-up Interface; FMU:Functional Mock-up Unit; FSS: Forward Stepwise Selection; HVAC: Heating; Ventilationand Air Conditioning; KPI: Key Performance Indicator; LTI: Linear Time-Invariant; MBL: Modelica Buildings Library; MHE: Moving Horizon Estimator; MPC: ModelPredictive Control; N4SID: Numerical Subspace State-Space System Identification; REST: Representational State Transfer; RL: Reinforcement Learning; ROM: Reducedorder model

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.

Modal identification of high-rise buildings under earthquake excitations via an improved subspace methodology

The subspace state-space system identification method has drawn extensive attention in structural modal identification, which is generally involved with the stabilization diagram for estimating structural modal parameters. However, the conventional stabilization diagram has an inherent problem, i.e., some spurious modes may be identified as stable results, leading to the adverse effect on structural modal identification. To address this issue, this paper proposes an improved subspace algorithm, in which a Monte Carlo-based stabilization diagram is involved. The performance of the Monte Carlo-based stabilization diagram for discriminating the poles denoting the physical modes from those representing spurious modes is demonstrated through a numerical study. The simulation results further prove that the proposed method can accurately estimate the time-varying structural modal parameters. Moreover, the proposed method is applied to field measurements on a 218-m-tall building during the 1994 Northridge earthquake event, and the identified results verify the applicability and effectiveness of the proposed method in field measurements. This paper aims to provide an effective tool for modal identification of high-rise buildings under earthquake excitations.

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.

Stochastic subspace identification based data‐driven approach for monitoring electromechanical dynamics from phasor measurement units

This study proposes to extend the numerical algorithm for subspace state space system identification (N4SID) for power grid electromechanical dynamics monitoring using the multi-channel noisy synchrophasor measurements. The oscillation modes, mode shapes, participation factors and coherent groups are estimated in a comprehensive manner. It consists of five key steps: (i) estimation of power system state subspace model via N4SID; (ii) electromechanical modes discrimination from the trivial modes by developing the generalised inverse and mode assurance criterion; (iii) calculations of the mode shapes using the estimated states and output subspace matrices; (iv) estimation of participation factors estimated by the calculated mode shapes and (v) the generator coherency identification by developing the direction cosine denoted coherency inductor. Test results on the IEEE 16-machine 68-bus test system and the practical China Southern power Grid using field synchrophasor measurements are performed. Comparison results with other existing methods show that the proposed method achieves better accuracy and efficiency for power system electromechanical dynamics monitoring in the presence of measurement noise.

Prediction of Power Output of Wind Turbines Using System Identification Techniques

In this paper, a generalized State Space Model for prediction of wind turbine output power has been developed via System Identification techniques. The techniques involved in this paper are the subspace methods, mainly, Numerical algorithms for Subspace State Space System Identification based on advanced linear algebra, and Prediction Error Method that based on iterative optimization. Predicting the power output of wind turbines in general is very useful in many applications, such as online monitoring of wind turbines, increasing the share of wind power from the wind farms to grids, operator training simulations, control system upgrades and so on. In order to attain sufficient accuracy for the model, many influencing inputs are taken into account, which are pitch angle, rotor speed with several other meteorological inputs as humidity, temperature, air pressure and wind speed have been considered in the system identification prediction; the result has shown that each input is sufficient and affects the model accuracy. Furthermore, the effect of model order on simulation accuracy has been investigated. In addition, a comparative study has been conducted between Prediction Error Method and Numerical algorithms for Subspace State Space System Identification methods in terms of complexity, accuracy, and speed of simulation. Simulations have demonstrated each methods’ ability to predict wind power with some slight differences in accuracy and computational requirements.

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.

Robust stabilization for discrete-time Takagi-Sugeno fuzzy system based on N4SID models

PurposeNonlinear systems identification from experimental data without any prior knowledge of the system parameters is a challenge in control and process diagnostic. It determines mathematical model parameters that are able to reproduce the dynamic behavior of a system. This paper aims to combine two fundamental research areas: MIMO state space system identification and nonlinear control system. This combination produces a technique that leads to robust stabilization of a nonlinear Takagi–Sugeno fuzzy system (T-S).Design/methodology/approachThe first part of this paper describes the identification based on the Numerical algorithm for Subspace State Space System IDentification (N4SID). The second part, from the identified models of first part, explains how we use the interpolation of linear time invariants models to build a nonlinear multiple model system, T-S model. For demonstration purposes, conditions on stability and stabilization of discrete time, T-S model were discussed.FindingsStability analysis based on the quadratic Lyapunov function to simplify implementation was explained in this paper. The linear matrix inequalities technique obtained from the linearization of the bilinear matrix inequalities was computed. The suggested N4SID2 algorithm had the smallest error value compared to other algorithms for all estimated system matrices.Originality/valueThe stabilization of the closed-loop discrete time T-S system, using the improved parallel distributed compensation control law, was discussed to reconstruct the state from nonlinear Luenberger observers.

Subspace-Aided Closed-Loop System Identification With Application to DC Motor System

A novel closed-loop numerical algorithm for subspace state space system identification (CN4SID) is proposed in this paper. Different from standard schemes, CN4SID algorithm can extend the standard LQ decomposition to the closed-loop cases by incorporating the controller information. Moreover, the proposed method can deliver unbiased pole estimation in the closed-loop framework. To this end, CN4SID algorithm shows superior pole estimation performance compared with a class of subspace identification method via principal component analysis algorithms. The effectiveness of the proposed CN4SID algorithm is finally verified through a practical dc motor system.

Model Predictive Control Method of Simulated Moving Bed Chromatographic Separation Process Based on Subspace System Identification

Simulated moving bed (SMB) chromatographic separation is a new type of separation technology based on traditional fixed bed adsorption operation and true moving bed (TMB) chromatographic separation technology, which includes inlet‐outlet liquid, liquid circulation, and feed liquid separation. The input‐output data matrices were constructed based on SMB chromatographic separation process data. The SMB chromatographic separation process was modeled by utilizing two subspace system identification algorithms: multivariable output‐error state‐space (MOESP) identification algorithm and numerical algorithms for subspace state‐space system identification (N4SID), so as to obtain the 3rd‐order and 4th‐order state‐space yield models of the SMB chromatographic separation process, respectively. The model predictive control method based on the established state‐space models is used in the SMB chromatographic separation process. The influence of different control indicators on the predictive control system response performance is discussed. The output response curves of the yield models were obtained by changing the related parameters so that the yield model parameters are optimized set meanwhile. Finally, the simulation results showed that the yield models are successfully controlled based on the each control period and given yield range.