State Space System Identification Research Articles

Identification of structured nonlinear state–space models for hysteretic systems using neural network hysteresis operators

Hysteretic system behavior is ubiquitous in science and engineering fields including measurement systems and applications. In this paper, we put forth a nonlinear state–space system identification method that combines the state–space equations to capture the system dynamics with a compact and exact artificial neural network (ANN) representation of the classical Prandtl–Ishlinskii (PI) hysteresis. These ANN representations called PI hysteresis operator neurons employ recurrent ANNs with classical activation functions, and thus can be trained with classical neural network learning algorithms. The structured nonlinear state–space model class proposed in this paper, for the first time, offers a flexible interconnection of PI hysteresis operators with a linear state–space model through a linear fractional representation. This results in a comprehensive and flexible model structure. The performance is validated both on numerical simulation and on measurement data.

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.

An EM Algorithm for Lebesgue-sampled State-space Continuous-time System Identification*

This paper concerns the identification of continuous-time systems in state-space form that are subject to Lebesgue sampling. Contrary to equidistant (Riemann) sampling, Lebesgue sampling consists of taking measurements of a continuous-time signal whenever it crosses fixed and regularly partitioned thresholds. The knowledge of the intersample behavior of the output data is exploited in this work to derive an expectation-maximization (EM) algorithm for parameter estimation of the state-space and noise covariance matrices. For this purpose, we use the incremental discrete-time equivalent of the system, which leads to EM iterations of the continuous-time state-space matrices that can be computed by standard filtering and smoothing procedures. The effectiveness of the identification method is tested via Monte Carlo simulations.

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.

Identification of Nonlinear State-Space Systems With Skewed Measurement Noises

In this paper, we consider the identification problem for nonlinear state-space models with skewed measurement noises. The generalized hyperbolic skew Student’s t (GHSkewt) distribution is employed to describe the skewed noises and formulate the hierarchical model of the considered system. A unified framework for estimating unknown states and model parameters is presented based on expectation-maximization (EM) algorithm, in which the forward filtering backward simulation with rejection sampling (RS-FFBSi) is employed to efficiently estimate the smoothing densities of the hidden states, and optimization method is adopted to update model parameters. One numerical study and the electro-mechanical positioning system (EMPS) are employed to verify the effectiveness of the developed approach.

Identification of Nonlinear State-Space Systems via Sparse Bayesian and Stein Approximation Approach

This paper is concerned with the parameter estimation of non-linear discrete-time systems from noisy state measurements in the state-space form. A novel sparse Bayesian convex optimisation algorithm is proposed for the parameter estimation and prediction. The method fully considers the approximation method, parameter prior and posterior, and adds Bayesian sparse learning and optimization for explicit modeling. Different from the previous identification methods, the main identification challenge resides in two aspects: first, a new objective function is obtained by our improved Stein approximation method in the convex optimization problem, so as to capture more information of particle approximation and convergence; second, another objective function is developed with L1-regularization, which is sparse method based on recursive least squares estimation. Compared with the previous study, the new objective function contains more information and can easily mine more important information from the raw data. Three simulation examples are given to demonstrate the proposed algorithm’s effectiveness. Furthermore, the performances of these approaches are analyzed, including parameter estimation of root mean squared error (RMSE), parameter sparsity and prediction of state and output result.

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.

CO2 Removal via an Environmental Green Solvent, K2CO3-Glycine (PCGLY): Investigative Analysis of a Dynamic and Control Study.

Promoted potassium carbonate with glycine has been actively investigated as a chemical solvent for the removal of CO2. Though a vast number of studies have been reported for potassium carbonate, dynamic studies regarding this promoted solvent are not yet extensively reported in the literature. In this work, a steady-state simulation has been performed via an equilibrium stage model in Aspen Plus V10 using the experimental data of an absorber from the bench scale pilot plant (MINI CHAS) located in Universiti Teknologi PETRONAS. In this study, 15 wt % K2CO3 + 3 wt % glycine is utilized as the medium for absorption, and the operating pressure is set at 40 bar to imitate the natural gas treatment process. The removal observed from the pilot plant is about 75% and the steady-state simulation with a tuned vaporization efficiency managed to replicate a similar result. The transient analysis is performed via activating a flow-driven method, and the simulation is transferred into Aspen Dynamic. A simple control strategy using a proportional–integral (PI) controller is installed at the gas outlet to monitor the CO2 composition, and further disturbances are introduced at the inlet gas flow rate using a step test and ramp test. The controller is tuned using a trial-and-error method and a satisfactory response is achieved under varying changes in the inlet gas flow rate. Further investigation is carried out using the model predictive controller (MPC), in which 5000 data points are generated through pseudorandom binary sequence (PRBS) analysis for state-space model system identification. The state-space model identified as the best is then used to design the MPC controller. A disturbance rejection test on the MPC controller is conducted via changing the gas flow rate at 5% and a quick response is observed. In conclusion, both MPC and PI controllers managed to produce a good response once the disturbance was introduced within the CO2–potassium carbonate–glycine (PCGLY) system.

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.

A framework for exergy analysis of process control systems

In this study, a generic framework has been developed for using exergy as a dynamic measure for energy efficiency of process control systems. The proposed framework begins by restating the exergy balance in terms of nonlinear state space; followed by partitioning of input variables. The exergy balance is then linearized, and added to the plant model to construct an extended state space model which has unknown number of non-physical states. The numerical method of sub space state space system identification is applied to determine the extended model matrices along with the system order. Laplace transform is then applied to convert the extended state space into a multi-input-single-output (MISO) formula for estimating the lost work. In the next step, an scalar index is suggested for representing the steady and transient parts of the dynamic lost work. The proposed index which is defined in the time domain is transformed into Laplace domain for consistency with the formulated MISO lost work. The suggested framework is tested for two case studies; (1) selection of the control variables for a CSTR, and (2) selection of the tray number for inferential control of a distillation column. The results confirm the generality of the proposed framework not only for exergetical optimization of control systems but also for disclosing the thermodynamically unrealizable structures.

Recursive identification of state space systems with coloured process noise and measurement noise

This paper concerns the modelling and identification of the state space system, in which both coloured process noise and measurement noise are encountered. By using the state filtering, the state space system with coloured process noise is transformed into a model without correlated noise, and a state filtering-based parameter estimation algorithm is derived on the base of designing a state filter observer using the multi-innovation identification. The validity of the proposed algorithm is verified by given simulation examples.

Recursive identification of state space systems with coloured process noise and measurement noise

Recursive identification of state space systems with coloured process noise and measurement noise

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.