High Order ARX Research Articles

I-pIDtune 2.0: An Updated Interactive Tool for Integrated System Identification and PID Control

This paper describes i-pIDtune 2.0, an updated interactive software tool that integrates system identification and PID controller design. The tool supports experimental design and execution under plant-friendly conditions, high-order ARX estimation, and control-relevant model reduction leading to models that comply with IMC-PID tuning rules; these controllers can then be evaluated in simulation. All four stages are depicted simultaneously and interactively in one screen. The updated tool includes two significant new features: 1) support for “bumpless” PI-D and I-PD closed-loop simulations that provide a broader understanding of PID tuning effectiveness in practical settings and 2) extension to plants with integrating dynamics. These new features are described and illustrated with two example problems.

Noise Reduction for Data-driven Control by Correlation Analysis and High-order ARX Identification

Noise Reduction for Data-driven Control by Correlation Analysis and High-order ARX Identification

Optimal order selection for high order ARX models

The objective of System Identification is to estimate models that can reproduce the static and dynamic properties of the process being modeled. Some system identification methods require a high order linear model of the process. These models are adopted as the process most reliable description. In some applications, it is necessary to reduce the order of the high order model. An important question is what is the best order of the high order model? This work presents an optimization strategy that maximizes the fit index in the search for the best high order of SISO and MIMO ARX models. The search for the best time delays related to the high order model is also addressed. The results of the optimization strategy are compared to those obtained by the exhaustive search in two scenarios: comparison of the fit indices of the high order models and comparison of these indices in models with reduced order obtained from the high order models.

ARX Model Estimation of Multivariable Errors-in-Variables Systems

This paper proposes a method for the estimation of ARX (Autoregressive with external input) model of multivariable errors-in-variables (EIV) systems. In parameter estimation, the input noise variances need to be estimated in order to obtain a consistent estimate. Two methods are developed to estimate the input noises variances. One way is to minimize a correlated output error criterion. The other way is to extract the input noises variances from the power spectrum of measured inputs. When the input noise variances are known, a modified least-squares method will give consistent estimate for the EIV system. Numerical simulations are used to illustrate the performance of the method. This work lays down a basis for extending the asymptotic method of Liu and Zhu (2017) to multivariable EIV systems which starts from a high order ARX model estimation.

Open-loop asymptotically efficient model reduction with the Steiglitz–McBride method

In system identification, it is often difficult to use a physical intuition when choosing a noise model structure. The importance of this choice is that, for the prediction error method (PEM) to provide asymptotically efficient estimates, the model orders must be chosen according to the true system. However, if only the plant estimates are of interest and the experiment is performed in open loop, the noise model can be over-parameterized without affecting the asymptotic properties of the plant. The limitation is that, as PEM suffers in general from non-convexity, estimating an unnecessarily large number of parameters will increase the risk of getting trapped in local minima. Here, we consider the following alternative approach. First, estimate a high-order ARX model with least squares, providing non-parametric estimates of the plant and noise model. Second, reduce the high-order model to obtain a parametric model of the plant only. We review existing methods to do this, pointing out limitations and connections between them. Then, we propose a method that connects favorable properties from the previously reviewed approaches. We show that the proposed method provides asymptotically efficient estimates of the plant with open-loop data. Finally, we perform a simulation study suggesting that the proposed method is competitive with state-of-the-art methods.

ARX modeling of unstable linear systems

High-order ARX models can be used to approximate a quite general class of linear systems in a parametric model structure, and well-established methods can then be used to retrieve the true plant and noise models from the ARX polynomials. However, this commonly used approach is only valid when the plant is stable or if the unstable poles are shared with the true noise model. In this contribution, we generalize this approach to allow the unstable poles not to be shared, by introducing modifications to correctly retrieve the noise model and noise variance.

Process Identification with Autoregressive Linear Regression Method using Experimental Data: Review

Objective: To obtain mathematical model and parameters of poorly understood and imprecisely known plant/process. Methods: One solution to this problem is to obtain these using identification techniques. Process identification is a technique where a mathematical model of the process under study is build from process input-output data. Several autoregressive models are used to estimate the present value of the model using its past values of the process. Initially, the data set is generated for the given system and the auto regressive model is fitted to it, for the estimation of the model parameters. Residual error for a system is calculated using auto regression model parameters. Findings: A mathematical model for plant under study can be formulated with different system identification using Linear Regression methods like Auto Regressive eXogenous variable (ARX), Auto Regressive Moving Average with an eXogenous variable (ARMAX), Output Error (OE) and Box-Jenkins (BJ). For high model order ARX model is preferred and takes low computations but only suitable for white noise. The ARMAX model considers disturbance affecting process and provides higher performance index i.e. fitness which reveals percentage variation in output estimated by respective model. The Output-Error (OE) model estimates process model but cannot model disturbance features. The residual analysis of Box- Jenkins model shows that the prediction error is not auto-correlated, correlated and is uncorrelated with the input applied to process, thus showing Box- Jenkins model ability to capture noise dynamics of process. Applications/Improvements: The process model can be identified for the unknown, poorly known or partially known system and formulated model can be used in model Predictive controller design and Adaptive control techniques.

The Box–Jenkins Steiglitz–McBride algorithm

An algorithm for identification of single-input single-output Box–Jenkins models is presented. It consists of four steps: firstly a high order ARX model is estimated; secondly, the input–output data is filtered with the inverse of the estimated disturbance model; thirdly, the filtered data is used in the Steiglitz–McBride method to recover the system dynamics; in the final step, the noise model is recovered by estimating an ARMA model from the residuals of the third step. The relationship to other identification methods, in particular the refined instrumental-variable method, are elaborated upon. A Monte Carlo simulation study with an oscillatory system is presented and these results are complemented with an industrial case study. The algorithm can easily be generalized to multi-input single-output models with common denominator.

Reweighted nuclear norm regularization: A SPARSEVA approach

The aim of this paper is to develop a method to estimate high order FIR and ARX models using least squares with re-weighted nuclear norm regularization. Typically, the choice of the tuning parameter in the reweighting scheme is computationally expensive, hence we propose the use of the SPARSEVA (SPARSe Estimation based on a VAlidation criterion) framework to overcome this problem. Furthermore, we suggest the use of the prediction error criterion (PEC) to select the tuning parameter in the SPARSEVA algorithm. Numerical examples demonstrate the veracity of this method which has close ties with the traditional technique of cross validation, but using much less computations.

Identification of Box-Jenkins models using structured ARX models and nuclear norm relaxation

In this contribution we present a method to estimate structured high order ARX models. By this we mean that the estimated model, despite its high order is close to a low order model. This is achieved by adding two terms to the least-squares cost function. These two terms correspond to nuclear norms of two Hankel matrices. These Hankel matrices are constructed from the impulse response coefficients of the inverse noise model, and the numerator polynomial of the model dynamics, respectively. In a simulation study the method is shown to be competitive as compared to the prediction error method. In particular, in the study the performance degrades more gracefully than for the Prediction Error Method when the signal to noise ratio decreases.

I-pIDtune: An interactive tool for integrated system identification and PID control

This paper describes i-pIDtune, an interactive software tool that integrates system identification and PID controller design. The tool supports experimental design and execution under plant-friendly conditions, high-order ARX estimation, and control-relevant model reduction leading to models that comply with the IMC-PID tuning rules. All these stages are depicted simultaneously and interactively in one screen. Thus, i-pIDtune allows to display both open- and closed-loop responses of the estimated models and important control-relevant validation criteria, what enables the user to readily assess how design variable choices, control performance requirements and model error can impact the achievable closed-loop performance from a restricted complexity model estimated under noisy conditions.

Closed-loop identification using direct approach and high order ARX/GOBF-ARX models

Model accuracy plays a key role in the performance of advanced, model predictive control algorithms. Model fidelity is usually affected by routine operating condition changes, which necessitate reidentification. From several theoretical and practical considerations, it is recommended that such re-identification be performed under closed-loop conditions. The direct approach for closed-loop identification, owing to its simplicity, is better suited for MPC. In order to yield unbiased and consistent parameter estimates, however, this approach requires the noise model to be sufficiently parameterized. Towards this objective, high order ARX models are the most suitable candidates from the viewpoint of ease of parameter estimation. For multivariable systems, however, the identification of high order ARX models would require longer experiments to be performed. This being undesirable from a practical viewpoint, there is a need for a parsimonious parameterization that would retain the benefits of high order ARX models. In this work, we propose to use generalized orthonormal basis filters (GOBFs) to achieve this parsimonous parameterization. Further, we propose an approach to obtain reduced order models by emphasizing important frequencies so as to suitably shape the bias. We also show that the choice of the GOBF parameterization has another important merit, viz. their ability to perform well even with minimal perturbation data or short experiment times. The efficacy of the proposed approach is demonstrated via simulations on the benchmark Shell Control Problem and a laboratory quadruple tank setup.

A Box-Jenkins Method that is Asymptotically Globally Convergent for Open Loop Data

AbstractAn algorithm for the identification of a multi-input single-output (MISO) Box-Jenkins model is developed. The method consists of several simple steps: first a high order ARX model is estimated; then the process model is calculated using the so-called Steiglitz-McBride iteration on the filtered input-output data; and finally the disturbance model is calculated by estimating an ARMA model of the output error residual. It can be proven that the Steiglitz-McBride iteration for the process model part is asymptotically globally convergent. The proposed method outperforms a standard Box-Jenkins method in simulation examples.

Development of ARX Models for Predictive Control Using Fractional Order and Orthonormal Basis Filter Parametrization

Among various industrial vendors of model predictive control (MPC) schemes, ARX appears to be a popular choice of model structure while models for predictive control schemes are being developed. These models are, however, nonparsimonious in the number of model parameters. As a consequence, the length of the data required to keep the variance errors low while using the conventional ARX structure is significantly large. Thus, if it is possible to reparametrize the ARX model such that fewer parameters are required at the identification stage, then it is possible to reduce the length of identification data and thereby reduce the cost involved in model identification exercise. In this work, we explore the possibility of reparametrizing ARX models using the fractional-order differential operators (FO-ARX) and orthonormal basis filters (OBF-ARX). We also propose a novel approach for identification of time delay matrix from multivariate data using ARX and OBF-ARX models. The efficacy of the proposed modeling technique is demonstrated by conducting simulation studies on the benchmark Shell control problem. Analysis of the simulation results reveals that, when compared with the conventional high-order ARX structure, the FO-ARX and the OBF-ARX are better model parametrizations when the data length is less. In particular, OBF-ARX parametrization is able to estimate the time delay matrix from multivariate data quite accurately. The experimental studies establish the feasibility of using the proposed FO-ARX and OBF-ARX models for formulating MPC schemes.

A new robust-control-oriented system identification method

This paper introduces a new robust-control-oriented system identification method, which consists of the following three steps: 1. High-order ARX model identification; 2. Loop shaping weighting functions design based on the high-order ARX model; 3. Control-oriented model reduction by minimizing the weighted L2-gap between the high-order ARX model and the low-order model. This method truly integrates the control objective into the identification step. A robust controller can be readily designed as a result of the identification. Simulation examples are given to show that smaller weighted ν-gap can be achieved by using the proposed method.

Una metodología para la identificación integrada con el diseño de controladores imc-pid

Este artículo describe, en forma de tutorial, una metodología integrada para la identificación y diseño de controladores comenzando con el modelado dinámico basado en datos de planta, y concluyendo con la sintonía de parámetros para controladores de tipo PID de elevadas prestaciones. Al integrar el paso de identificación con el diseño de controladores, el método logra demostrar una funcionalidad que es muy deseada por la comunidad de control industrial. Los pasos principales en esta metodología son: diseño experimental, estimación de modelos tipo ARX de alto orden, y reducción de modelos relevantes al control que están de acuerdo con leyes de sintonía IMC-PID. Cuando se utiliza una entrada de excitación persistente, el estimado del modelo ARX de alto orden es consistente, lo cual hace que el método sea atractivo como modelo intermedio para estimación de modelos de bajo orden relevantes al control. Además, el bajo esfuerzo computacional asociado con la estimación de modelos ARX significa que se pueden usar eficazmente métodos estadísticos sencillos (tales como la validación cruzada) para determinar una estructura adecuada para el modelo ARX sin mucha intervención por parte del usuario. La metodología se ilustra en el caso de una planta con retardo sujeta a una perturbación con elevada deriva.

A comparative study between linear transfer models of low order and ARX model of high order depending on spatial series

A comparative study between linear transfer models of low order and ARX model of high order depends on spatial series. The research includes a comparison between efficiency of the different linear transfer function f stationary time – spatial series with low order models ( ARX11k , ARMAX2221 ,O.E221 , B.J22221 , pss1 ) and high order ARX model according to the percentage rate of coincidence of the model to the data . The comparison is made on a spatial – time series which shows the effect resulted on the biological factors of the soil as a consequence of changes in the chemical factors on one hand and physical factors on the other, and the two factors all are together as an input–output model ( multiple–uni ) in one time and (multiple–multiple) on another time . It appears that ARX models of high order has a high percentage fit compared with the transfer models of low order .

ON THE ROLE OF FUTURE HORIZON IN CLOSED-LOOP SUBSPACE IDENTIFICATION

Closed loop subspace identification has become a focus of interest with several recent developments. Notably are the innovation estimation approach (Qin and Ljung, 2003), the state space approach with ARX pre-estimates (SSARX, (Jansson, 2003)), and the whitening filter approach (Chiuso and Picci, 2004). All these approaches use an extended future horizon to form the projection or regression from which an observable subspace is extracted. Yet there are other methods such as OKID of (Phan and Longman, 1992) and that of (Ljung and McKelvey, 1996) that do not use an extended horizon in the projection or regression step. Instead, a single high order ARX model is used.A natural question is whether the future horizon is necessary and if so what role does it play in these steps. In this paper we investigate the role of the future horizon using the whitening filter approach of (Chiuso and Picci, 2004), which works for both open-loop and closed-loop data. We conclude that the role of future horizon in this algorithm is merely extending the order of a bank of already high order ARX models. The difference from a single ARX model is insignificant if the ARX order or past horizon is sufficiently high. The role of future horizon is mainly in the model reduction step where it serves to elevate the order of the Hankel matrix. We complement the analysis with simulations.

A novel subspace identification approach with enforced causal models

Subspace identification methods (SIMs) for estimating state-space models have been proven to be very useful and numerically efficient. They exist in several variants, but all have one feature in common: as a first step, a collection of high-order ARX models are estimated from vectorized input–output data. In order not to obtain biased estimates, this step must include future outputs. However, all but one of the submodels include non-causal input terms. The coefficients of them will be correctly estimated to zero as more data become available. They still include extra model parameters which give unnecessarily high variance, and also cause bias for closed-loop data. In this paper, a new model formulation is suggested that circumvents the problem. Within the framework, the system matrices ( A , B , C , D ) and Markov parameters can be estimated separately. It is demonstrated through analysis that the new methods generally give smaller variance in the estimate of the observability matrix and it is supported by simulation studies that this gives lower variance also of the system invariants such as the poles.

Subspace Identification and ARX Modeling

In this paper we present a new identification method that points at the close relationship between high order ARX modeling and subspace identification. A high order ARX model is utilized to obtain initial estimates of certain Markov parameters. These parameters are then used to restructure the data model used for subspace identification to facilitate the estimation of the state sequence. Based on the estimated state sequence, the system parameters are estimated by linear regression. The method is shown to be competitive to existing subspace methods by a simulation example. The method can also be used, without modification, on closed loop data in contrast to most previously published subspace identification methods.