ARX Model Research Articles

RETRACTED ARTICLE: ARX and ARMAX modelling of SBCNC-60 machine for surface roughness and MRR with optimization of system response using PSO

RETRACTED ARTICLE: ARX and ARMAX modelling of SBCNC-60 machine for surface roughness and MRR with optimization of system response using PSO

Control of Precalciner Temperature in the Cement Industry: A Novel Method of Hammerstein Model Predictive Control with ISSA

As the most critical equipment in the pre-calcination process of dry cement production, the temperature of the precalciner is an essential factor affecting the quality of cement. However, the cement calcination system is time-delayed, nonlinear, and multi-disturbance, which makes it difficult to predict and control the precalciner temperature. In this study, a deep learning-based Hammerstein model is proposed, and a model predictive control system is built to predict and control the precalciner temperature. In the prediction model, the CNN-GRU network architecture is used to extract the operating states of the precalciner, and an attention mechanism is employed to find and emphasize the important historical information in the extracted states. Then, an ARX model is built to predict the temperature of the precalciner using the extracted operating state information. The complex nonlinear model solution in the control system is formed into a linear control problem and an inverse solution problem. The generalized predictive control (GPC) is used for linear control, and the improved sparrow search algorithm (ISSA) is used for the problem of an inverse solution. Tested with data from a cement plant in Hebei, China, the prediction accuracy of the model proposed in this paper is 99%, and the established control algorithm has less overshoot compared to PID and better stability in anti-disturbance tests. It is demonstrated that the prediction model developed in this study has better accuracy and the control strategy based on this model has good robustness.

A construction-free coordinate-descent augmented-Lagrangian method for embedded linear MPC based on ARX models

This paper proposes a construction-free algorithm for solving linear MPC problems based on autoregressive with exogenous terms (ARX) input-output models. The solution algorithm relies on a coordinate-descent augmented Lagrangian (CDAL) method previously proposed by the authors, which we adapt here to exploit the special structure of ARX-based MPC. The CDAL-ARX algorithm enjoys the construction-free feature, in that it avoids explicitly constructing the quadratic programming (QP) problem associated with MPC, which would eliminate construction cost when the ARX model changes/adapts online. For example, the ARX model parameters are dependent on linear parameter-varying (LPV) scheduling signals, or recursively adapted from streaming input-output data with cheap computation cost, which make the ARX model widely used in adaptive control. Moreover, the implementation of the resulting CDAL-ARX algorithm is matrix-free and library-free, and hence amenable for deployment in industrial embedded platforms. We show the efficiency of CDAL-ARX in two numerical examples, also in comparison with MPC implementations based on other general-purpose quadratic programming solvers.

Superstabilizing Control of Discrete-Time ARX Models under Error in Variables

This paper applies a polynomial optimization-based framework towards the superstabilizing control of an Autoregressive with Exogenous Input (ARX) model given noisy data observations. The recorded input and output values are corrupted with L-infinity-bounded noise where the bounds are known. This is an instance of Error in Variables (EIV) in which true internal state of the ARX system remains unknown. The consistency set of ARX models compatible with noisy data has a bilinearity between unknown plant parameters and unknown noise terms. The requirement for a dynamic compensator to superstabilize all consistent plants is expressed using polynomial nonnegativity constraints, and solved using sum-of-squares (SOS) methods in a converging hierarchy of semidefinite programs in increasing size. The computational complexity of this method may be reduced by applying a Theorem of Alternatives to eliminate the noise terms. The effectiveness of this method is demonstrated on control of example ARX models.

Nonlinear Dynamic System Identification of ARX Model for Speech Signal Identification

System Identification becomes very crucial in the field of nonlinear and dynamic systems or practical systems. As most practical systems don’t have prior information about the system behaviour thus, mathematical modelling is required. The authors have proposed a stacked Bidirectional Long-Short Term Memory (Bi-LSTM) model to handle the problem of nonlinear dynamic system identification in this paper. The proposed model has the ability of faster learning and accurate modelling as it can be trained in both forward and backward directions. The main advantage of Bi-LSTM over other algorithms is that it processes inputs in two ways: one from the past to the future, and the other from the future to the past. In this proposed model a backward-running Long-Short Term Memory (LSTM) can store information from the future along with application of two hidden states together allows for storing information from the past and future at any moment in time. The proposed model is tested with a recorded speech signal to prove its superiority with the performance being evaluated through Mean Square Error (MSE) and Root Means Square Error (RMSE). The RMSE and MSE performances obtained by the proposed model are found to be 0.0218 and 0.0162 respectively for 500 Epochs. The comparison of results and further analysis illustrates that the proposed model achieves better performance over other models and can obtain higher prediction accuracy along with faster convergence speed.

Density Estimation in the Regressor Space for Nonlinear System Identification

In this paper, the functioning and performance of a recently proposed weighting approach for dynamic datasets is demonstrated for nonlinear system identification. The overall goal of the new method is to improve data-driven model performance. This is achieved by increasing the weight of training data points in areas of the regressor space that are underrepresented during training, while decreasing the weight for overrepresented data points. The method is based on estimating the parameters of a model using weighted error norms, where the weighting is carried out with the inverse values of the kernel density estimated in the regressor space. The performance of the new method is investigated using nonlinear ARX models for the identification of (i) an artificial Hammerstein process and (ii) a nonlinear real-world process. Overall, the new weighting method shows significantly improved performance on the simulation error. It is demonstrated that performance is improved most notably in areas that are underrepresented in the training data.

Input Spectrum Design for Identification of a Thermostat System

This paper considers the shaping of the amplitude spectra of perturbation signals for the identification of a thermostat system. The current approach in control engineering practice utilizes flat spectrum signals, which may not result in the highest possible accuracy. This research aims to investigate the effectiveness of optimal signals with amplitude spectra designed using two state-of-the-art software approaches, namely the model-based optimal signal excitation 2 (MOOSE2) design and the optimal excitation (optexcit) design, in improving estimation accuracy. Such a comparison on a real system is currently lacking. In particular, there exists a research gap on how the combined choice of signal and model structure affects performance measures. In this research, two model structures are used, which are the autoregressive with exogenous input (ARX) and the output error (OE) model structures. Four performance measures are compared, namely the determinant of the covariance matrix of the parameter estimates and the minimum error, mean error and maximum error in the frequency response. Results show that the optimal signals are effective in reducing the determinant of the covariance matrix and the maximum error in the frequency response for the thermostat system, when applied in combination with the ARX model structure. The flat spectrum signal remains very useful as a general broadband perturbation signal as it provides a good overall fit of the frequency response. The findings from this work highlight the benefits of applying optimal signals especially if the identification results are to be used for control, since these signals improve key performance measures which have direct implications on controller design.

Machine Learning Models and Intra-Daily Market Information for the Prediction of Italian Electricity Prices

In this paper we assess how intra-day electricity prices can improve the prediction of zonal day-ahead wholesale electricity prices in Italy. We consider linear autoregressive models with exogenous variables (ARX) with and without interactions among predictors, and non-parametric models taken from the machine learning literature. In particular, we implement Random Forests and support vector machines, which should automatically capture the relevant interactions among predictors. Given the large number of predictors, ARX models are also estimated using LASSO regularization, which improves predictions when regressors are many and selects the important variables. In addition to zonal intra-day prices, among the predictors we include also the official demand forecasts and wind generation expectations. Our results show that the prediction performance of the simple ARX model is mostly superior to those of machine learning models. The analysis of the relevance of exogenous variables, using variable importance measures, reveals that intra-day market information successfully contributes to the forecasting performance, although the impact differs among the estimated models.

Equalization Control of Nonlinear Systems by Discrete Models of the Volterra Operators

The Associated linear equations (ALEs) are parametric models of the Volterra operators. Whit them, a Volterra inverse is constructed with the objective of been used as open loop control of continuous nonlinear systems. However, most of the actual control systems are of discrete nature, this work introduces the novel discrete version of the ALEs. This discrete version is a series of ARX models of the Volterra operators for both the direct and the inverse series. These discrete models of the ALEs are used for an equalization strategy in order to create an open loop control on a reported simulated Duffing Oscillator and a physical Duffing system constructed by analog circuits.

Development and assessment of a model predictive controller enabling anticipatory control strategies for a heat-pipe system

To support the reliable and resilient operation of modular reactors and microreactors, anticipatory control strategies have been proposed for achieving faster-than-real-time predictions and decision-making capabilities in anticipation of potential anomalies, including setpoint changes and cyber incidents. This work presents how anticipatory control strategies can be implemented via model predictive control (MPC) for a single heat pipe's temperature. Considering the uncertainty in developing and applying MPC, this work evaluates MPC performance given three different model forms: a linear response surface model, an artificial neural network (ANN), and an autoregressive model with exogenous input (ARX). This work also evaluates the impacts of different input biases and variance on MPC performance in order to account for potential sensor reading variations due to cyber incidents. We observe that the ANN and ARX result in more fluctuated control actions compared to the MPC with linear response surface model. However, when the cyber incidents are of large magnitudes, the linear response surface model produces smaller feasible regions than the ANN and ARX models under identical constraints.

Hourly identification and simulation of the TGE S.A. Day-Ahead Market system

Abstract The paper presents selected research results concerning the identification and simulation of the TGE S.A. Day-Ahead Market (DAM) system of the day for electricity delivered and sold, listed for the following hours: 5:01-6:00, 11:01-12:00, 17:01-18:00 and 23:01-24:00 in 2019, which were obtained in the MATLAB and Simulink environment using the System Identification Toolbox. As a result of identification, four respective discrete parametric arx models were obtained, which were then subject to quality assessment. Then, a simulation model was built in the Simulink environment, which was used for simulation tests and for assessing the sensitivity of the model created using the data from 2019 as the basis and the data from 2020 for verification. The obtained results confirm the correctness of both the performed discrete parametric identification and the possibility of testing the quality of the model and its sensitivity with the use of the DAM system model in the MATLAB and Simulink environment.

Time‐delay and parameter estimation for an ARX model based on copula theory and Armijo criterion and their applications in the modeling of the dynamics of the UAV

AbstractTime‐delay dynamic systems are widely existed in industrial applications owing to the measure, control or other processes. To carry out system analysis, control and fault diagnosis, the identification of time‐delay systems becomes more and more important. This article considers the identification of time‐delay ARX models based on a novel two‐stage algorithm. Firstly, a 2‐copula criterion based time‐delay estimation method is presented by using a measure of dependence between the model input and output. This method can obtain the time delay without the estimates of the parameters. Secondly, a multi‐gradient algorithm with adaptive stacking length is studied. This algorithm accelerates traditional stochastic gradient algorithm by taking several recent gradients in each iteration. The stacking length, that is, the number of the gradient used in a step, is determined by the Armijo criterion. The proposed algorithm is validated by numerical experiments and the modeling of unmanned aerial vehicle data.

Second‐order optimization methods for time‐delay Autoregressive eXogenous models: Nature gradient descent method and its two modified methods

SummaryThis article proposes several second‐order optimization methods for time‐delay ARX model. Since the time‐delay in the information vector makes the traditional identification algorithms be inefficient, a redundant rule based method is utilized to transformed the model into a redundant model. Then, the nature gradient descent (NGD) algorithm is developed for such a model. To reduce the computational efforts of the NGD algorithm and to adaptively update each element in the parameter vector, two modified NGD algorithms are also presented. The simulation examples verify the effectiveness of the proposed algorithms.

System Identification for the design of behavioral controllers in crowd evacuations

Behavioral modification using active instructions is a promising interventional method to optimize crowd evacuations. However, existing research efforts have been more focused on eliciting general principles of optimal behavior than providing explicit mechanisms to dynamically induce the desired behaviors, which could be claimed as a significant knowledge gap in crowd evacuation optimization. In particular, we propose using dynamic distance-keeping instructions to regulate pedestrian flows and improve safety and evacuation time. We investigate the viability of using Model Predictive Control (MPC) techniques to develop a behavioral controller that obtains the optimal distance-keeping instructions to modulate the pedestrian density at bottlenecks. System Identification is proposed as a general methodology to model crowd dynamics and build prediction models. Thus, for a testbed evacuation scenario and input–output data generated from designed microscopic simulations, we estimate a linear AutoRegressive eXogenous model (ARX), which is used as the prediction model in the MPC controller. A microscopic simulation framework is used to validate the proposal that embeds the designed MPC controller, tuned and refined in closed-loop using the ARX model as the Plant model. As a significant contribution, the proposed combination of MPC control and System Identification to model crowd dynamics appears ideally suited to develop realistic and practical control systems for controlling crowd motion. The flexibility of MPC control technology to impose constraints on control variables and include different disturbance models in the prediction model has confirmed its suitability in the design of behavioral controllers in crowd evacuations. We found that an adequate selection of output disturbance models in the predictor is critical in the type of responses given by the controller. Interestingly, it is expected that this proposal can be extended to different evacuation scenarios, control variables, control systems, and multiple-input multiple-output control structures.

The design of nonlinear damped building isolation systems by using mobility analysis

Additional power law nonlinear damping has shown advantages in seismic isolation of buildings. The design of nonlinear damped building isolation systems can be conducted in the frequency domain based on the output frequency response function (OFRF) of the building’s frequency output responses. But this often requires many runs of simulations of nonlinear building systems, which will spend a long time when the finite element (FE) model or differential equation model of the building system is complex. In this study, the issue will be resolved by an integrated mobility analysis and equivalent linearization approach. In this approach, the complex linear building system without additional nonlinear damping is represented by several data-driven autoregressive with exogenous input (ARX) models. Mobilities of the building system can be evaluated from these ARX models, so that a mobility-based frequency domain representation of the linear building system can be achieved. After that, an equivalent linearization approach is applied to simulate the building system with additional nonlinear damping. Finally, the OFRF-based design can be conducted based on the proposed mobility analysis and equivalent linearization approach. A 4-degree of freedom (DoF) building system is discussed to demonstrate the advantages of the proposed method. The results indicate that the new approach can significantly increase the efficiency of the design and be effectively applied to the design of complex nonlinear building isolation systems.

Robust multimodel control for uncertain nonlinear MIMO systems based on ARX-Laguerre multimodel and LSDP approach

In this paper, we propose a new alternative for the robust control of nonlinear MIMO (multiple-input multiple-output) systems based on the multimodel approach and LSDP (Loop-Shaping Design Procedure)approach in the discrete case. First, the multimodel approach is exploited by representing each linear sub-model in the form of the recent linear MIMO ARX (Auto-Regressive with eXogenous inputs) -Laguerre model which guarantees a significant parameters number reduction compared with the MIMO ARX model. Thus, the resulting multivariable multimodel, entitled MIMO ARX-Laguerre multimodel, ensures a reduction of the parametric complexity with a simple recursive vector representation. However, this reduction is conditioned by an optimal choice of the pole characterising each Laguerre base. To deal with this, we propose the identification, using the genetic algorithm, of the Laguerre poles as well as the weighting function parameters of the MIMO ARX-Laguerre multimodel. This latter allows to generate a set of linear MIMO ARX-Laguerre sub-models which are exploited to propose a robust multimodel control for uncertain nonlinear MIMO systems. In fact, for each identified sub-model, we combine the LSDP approach and the RGA (Relative Gain Array) theory to develop a local robust controller. Then, a multivariable roust multimodel control algorithm is developed where the overall control strategy by multimodel approach amounts to incorporating the same weighting functions used in the considered multivariable multimodel in order to interpolate the control laws calculated from each local robust controller. The optimisation of Laguerre poles and weighting functions parameters as well as the proposed multivariable robust multimodel control algorithm are validated on the experimental prototype Quanser Aero system.

Online structural damage state identification of concrete arch dams under dynamic loads using a recursive TVARX approach

Concrete arch dams have been widely constructed worldwide, and many of these dams are located in areas with high seismic activity. However, strong seismic loading poses a severe threat to concrete arch dams, and dams have the risk of suffering damage and the possibility of an ensuing dam break. In past decades, seismic and structural health monitoring has undergone rapid growth in the context of arch dams to track the operation state of arch dams under strong seismic loading. In fact, arch dams with damage caused by seismic loading normally exhibit nonlinear dynamic behaviors and time-variable dynamic properties. To effectively identify the structural damage state development of arch dams under seismic loading, an online damage identification method using a novel recursive time-variable autoregressive with exogenous (TVARX) input model was presented in this paper. To improve the tracking capability for the time-variable coefficients of the TVARX model, an adaptive adjustment algorithm of variable weighting factors was incorporated into the recursive least-squares method. In the proposed method, the predicted model residual, a damage index named the fit ratio (FR), the difference between fit ratios (DF), and a damage index based on the change in ARX model parameters (CH) were proposed to identify the presence of damage, damage area, and relative damage extent for arch dams. Eventually, the proposed approach was used for the online damage state identification of arch dams in a numerical simulation example and a shaking table test, and the identification results demonstrated the effectiveness of the proposed approach.

Robust Standard Gradient Descent Algorithm for ARX Models Using Aitken Acceleration Technique.

A robust standard gradient descent (SGD) algorithm for ARX models using the Aitken acceleration method is developed. Considering that the SGD algorithm has slow convergence rates and is sensitive to the step size, a robust and accelerative SGD (RA-SGD) algorithm is derived. This algorithm is based on the Aitken acceleration method, and its convergence rate is improved from linear convergence to at least quadratic convergence in general. Furthermore, the RA-SGD algorithm is always convergent with no limitation of the step size. Both the convergence analysis and the simulation examples demonstrate that the presented algorithm is effective.

The ARX and ARMAX Models for Thermoelectric Cooling on Glass Windows: A Comparative Study

Thermoelectric cooling (TEC), in particular, can be combined with a heat sink for local cooling, but they can also be integrated into electronic chips for point-to-point cooling. The study aims to develop a dynamic model of a cooling system integrated with TEC for glass window. The main target of this study is to develop a dynamic model of a cooling system integrated with TEC. The black box modelling approach in producing a mathematical model was selected based on the ARMAX and ARX model that corresponds to the actual dynamic state of the cooling system. The best model was finalized based on the best match on curve patterns when comparing the real and estimated models using the system identification tools in MATLAB, and also having the least error. The accuracy of the models was compared and analysed. The results showed that the 4th order of the ARMAX model produced a higher best fitting and standard deviation values of 80.23% and 0.027592 compared to the 4th order of the ARX model of 78.14% and 0.030769 respectively. This system accuracy is almost within the acceptable range for most error calculations in the validation method. Yet, this cooling system integrated with TEC is found more suitable for the 4th order of the ARMAX model when compared to the ARX model due to the noise parameter in the ARMAX model. Nevertheless, the noise order in this system is not dominant, therefore, whenever the noise order of the system in the ARMAX model is high than the second structure (nb), the number of errors is also high. In addition, the ARMAX model is found incapable of achieving the highest fitting due to the losses from the dynamic environment and losses from the TEC itself. Still, the use of this black box model used in this study is a significant variation where system parameters can be identified even offline.

Testing for Serial Correlation in Autoregressive Exogenous Models with Possible GARCH Errors.

Autoregressive exogenous, hereafter ARX, models are widely adopted in time series-related domains as they can be regarded as the combination of an autoregressive process and a predictive regression. Within a more complex structure, extant diagnostic checking methods face difficulties in remaining validity in many conditions existing in real applications, such as heteroscedasticity and error correlations exhibited between the ARX model itself and its exogenous processes. For these reasons, we propose a new serial correlation test method based on the profile empirical likelihood. Simulation results, as well as two real data examples, show that our method has a good performance in all mentioned conditions.