Prediction Error Method Research Articles

Covariance analysis in SISO linear systems identification

In this paper, we analyze the asymptotic covariance of models of causal single-input single-output linear time invariant systems. Expressions for the asymptotic (co)variance of system properties estimated using the prediction error method are derived. These expressions delineate the impacts of model structure, model order, true system dynamics, and experimental conditions. A connection to results on frequency function estimation is established. Also, simple model structure independent upper bounds are derived. Explicit variance expressions and bounds are provided for common system properties such as impulse response coefficients and non-minimum phase zeros. As an illustration of the insights the expressions provide, they are used to derive conditions on the input spectrum which make the asymptotic variance of non-minimum phase zero estimates independent of the model order and model structure.

Variance analysis of linear SIMO models with spatially correlated noise

In this paper we address the identification of linear time-invariant single-input multi-output (SIMO) systems. In particular, we assess the performance of the prediction error method by quantifying the variance of the parameter estimates. Using an orthonormal representation for the modules composing the SIMO structure, we show that the parameter estimate of a module depends on the model structure of the other modules, and on the correlation structure of the output disturbances. We provide novel results which quantify the variance-error of the parameter estimates for finite model orders, where the effects of noise correlation structure, model structure and input spectrum are visible. In particular, we show that a sensor does not increase the accuracy of a module if common dynamics have to be estimated. When a module is identified using less parameters than the other modules, we derive the noise correlation structure that gives the minimum total variance. The implications of our results are illustrated through numerical examples and simulations.

Speech dereverberation using weighted prediction error with correlated inter-frame speech components

In this paper, we propose a new dereverberation approach based on the weighted prediction error (WPE) method implemented in the short-time Fourier transform (STFT) domain. Our main contribution is to model the temporal correlation of the STFT coefficients across analysis frames, referred to as inter-frame correlation (IFC), and exploit it in the dereverberation process. Since accurate modeling of the IFC is not tractable, we consider an approximate model wherein only a finite number of consecutive speech frames are considered correlated. It is shown that, given an estimate of the IFC matrix, the proposed approach results in a convex quadratic optimization problem with respect to the reverberation prediction weights, and a closed-form solution can be accordingly derived. Furthermore, an efficient method for the estimation of the underlying IFC matrix is developed based on the extension of a recently proposed speech variance estimator. We evaluate the performance of our approach incorporating the estimated IFC matrix and compare it to the original and several variants of the WPE method. The results reveal lower residual reverberation and higher overall quality of the enhanced speech when the proposed method is employed.

Adaptive Input Design for LTI Systems

Optimal input design for parameter estimation has obtained extensive coverage in the past. A key problem here is that the optimal input depends on some unknown system parameters that are to be identified. Adaptive design is one of the fundamental routes to handle this problem. Although there exist a rich collection of results on this problem, there are few results that address dynamical systems. This paper presents sufficient conditions for convergence/consistency and asymptotic optimality for a class of adaptive systems consisting of a recursive prediction error estimator and an input generator depending on the time-varying parameter estimates. The results apply to a general family of single input single output linear time-invariant systems. An important application is adaptive input design for which the results imply that, asymptotically in the sample size, the adaptive scheme recovers the same accuracy as the off-line prediction error method that uses data from an experiment where perfect knowledge of the system has been used to design an optimal input spectrum.

Refined instrumental variable parameter estimation of continuous‐time Box–Jenkins models from irregularly sampled data

This study investigates the estimation of continuous-time Box–Jenkins model parameters from irregularly sampled data. The Box–Jenkins structure has been successful in describing systems subject to coloured noise, since it contains two sub-models that feature the characteristics of both plant and noise systems. Based on plant-noise model decomposition, a two-step iterative procedure is proposed to solve the estimation problem, which consists of an instrumental variable method for the plant model and a prediction error method for the noise model. The proposed method is of low complexity and shows good estimation robustness and accuracy. Implementation issues are discussed to improve the computational efficiency. Numerical examples are presented to demonstrate the effectiveness of the proposed method.

An error prediction model of NC machining process considering multiple error sources

In flank milling process, machined surface is formed by the edge of the cutting tool while the position of tool is influenced by the movement of computer numerical control (CNC) machining axis and the status of tool is influenced by the cutting force; thus, the quality of the workpiece is affected by multiple error sources. In this paper, a new error prediction method is proposed to integrate errors of the process system, including tool rotation error, machine geometric error, and tool deformation error. All these errors are synthesized to generate the movement of tool edge in the workpiece coordinate system. Then, the machining error is obtained by projecting the point cloud of tool edge to the workpiece normal vector. Case studies are performed at the end, and the influence of tool rotation error on workpiece error is discussed. The model has been validated with an experiment.

Cost function shaping of the output error criterion

Identification of an output error model using the prediction error method leads to an optimization problem built on input/output data collected from the system to be identified. It is often hard to find the global solution of this optimization problem because in most cases both the corresponding objective function and the search space are nonconvex. The difficulty in solving the optimization problem depends mainly on the experimental conditions, more specifically on the spectra of the input/output data collected from the system. It is therefore possible to improve the convergence of the algorithms by properly choosing the data prefilters; in this paper we show how to perform this choice. We present the application of the proposed approach to case studies where the standard algorithms tend to fail to converge to the global minimum.

Low-Complexity Simultaneous Estimation of Head-Related Transfer Functions by Prediction Error Method

Low-Complexity Simultaneous Estimation of Head-Related Transfer Functions by Prediction Error Method

The error prediction of inventory reconciliation during storage and transportation process based on PLS and MFOA-LSSVM

Process optimization and fault diagnosis technology, which is represented by production process monitoring, design and production condition adjustment, play an important role in the modern petroleum industry. Accurate inventory reconciliation model is the basis of process optimization and fault diagnosis. To eliminate the impact of the inventory reconciliation error caused by different metering systems, the error prediction method of inventory reconciliation during storage and transportation process based on partial least squares (PLS) and least squares support vector machine optimized by modified fruit fly optimization algorithm (MFOA-LSSVM) is proposed. The general error prediction method flow of inventory reconciliation is provided. The principles of PLS and MFOA-LSSVM are elaborated in detail. Firstly, the algorithm of PLS is used to exclude the interference of unrelated factors and extract the most relevant factors that influence the error of inventory reconciliation. Then the modified three-dimensional fruit fly optimization algorithm with diminishing steps as well as a good global search capability is adopted to select the LSSVM model parameters and build the error prediction model. Finally, the sample data were revised by using the predictive value to verify the validity of the proposed method. The experimental modeling was carried out by PLS and MFOA-LSSVM. Compared with other forecasting methods, this method not only has the advantage of faster calculations, but also can well predict the error of reconciliation.

Identification and characterization of the hydraulic unit in an anti-lock brake system

The present paper identifies single-input single-output linear minimum-order models for the hydraulic unit of an anti-lock brake system in the increase mode and the decrease mode. The input variable and the output variable in the models for the increase mode and the decrease mode are the pressure of master cylinder and the pressure of the brake caliper respectively. In the process of model identification, excitation signals are designed by introducing a novel method to change the states of the solenoid supply valve and the solenoid discharge valve and to generate different operating modes. It is shown that the excitation signals are persistently exciting of a certain order and cover the bandwidth of the system. In this study, the structure models (the output error model, the autoregressive with exogenous input model, the autoregressive moving-average with exogenous input model and the box Jenkins model) are identified for the increase mode and the decrease mode by employing the experimental data of a test automobile and using the ordinary least-squares method and the prediction error method. After validation of the model outcome by comparison with the experimental results, the best models are presented for the increase mode and the decrease mode of the hydraulic unit in a test automobile.

Global identification of stochastic dynamical systems under different pseudo-static operating conditions: The functionally pooled ARMAX case

The identification of a single global model for a stochastic dynamical system operating under various conditions is considered. Each operating condition is assumed to have a pseudo-static effect on the dynamics and be characterized by a single measurable scheduling variable. Identification is accomplished within a recently introduced Functionally Pooled (FP) framework, which offers a number of advantages over Linear Parameter Varying (LPV) identification techniques. The focus of the work is on the extension of the framework to include the important FP-ARMAX model case. Compared to their simpler FP–ARX counterparts, FP–ARMAX models are much more general and offer improved flexibility in describing various types of stochastic noise, but at the same time lead to a more complicated, non-quadratic, estimation problem. Prediction Error (PE), Maximum Likelihood (ML), and multi-stage estimation methods are postulated, and the PE estimator optimality, in terms of consistency and asymptotic efficiency, is analytically established. The postulated estimators are numerically assessed via Monte Carlo experiments, while the effectiveness of the approach and its superiority over its FP–ARX counterpart are demonstrated via an application case study pertaining to simulated railway vehicle suspension dynamics under various mass loading conditions.

System identification of the upper part of Murray River

Data based modelling is an important tool for obtaining models of rivers. In this paper we consider several identification methods, namely Prediction Error Method, Maximum Likelihood, continuous time system identification, Refined Instrumental Variable method (used within the context of Data-Based Mechanistic modelling) and Subspace Identification Method, and apply them to real data from the Murray River in Australia. Both Multiple-Input Single-Output models where the output is the water level in a lake and Multiple-Input, Multiple-Output models where in addition a flow is also modelled, are considered. The models are compared in terms of their accuracy on validation data and on how easily the methods can incorporate prior knowledge.

Closed-loop identification of unstable systems using noncausal FIR models

ABSTRACTNoncausal finite impulse response (FIR) models are used for closed-loop identification of unstable multi-input, multi-output plants. These models are shown to approximate the Laurent series inside the annulus between the asymptotically stable pole of the largest modulus and the unstable pole of the smallest modulus. By delaying the measured output relative to the measured input, the identified FIR model is a noncausal approximation of the unstable plant. We present examples to compare the accuracy of the identified model obtained using least squares, instrumental variables methods, and prediction error methods for both infinite impulse response (IIR) and noncausal FIR models under arbitrary noise that is fed back into the loop. Finally, we reconstruct an IIR model of the system from its stable and unstable parts using the eigensystem realisation algorithm.

Parametric estimation of interconnected nonlinear systems described by input-output mathematical models

In this paper, two types of mathematical models are developed to describe the dynamics of large-scale nonlinear systems, which are composed of several interconnected nonlinear subsystems. Each subsystem can be described by an input-output nonlinear discrete-time mathematical model, with unknown, but constant or slowly time-varying parameters. Then, two recursive estimation methods are used to solve the parametric estimation problem for the considered class of the interconnected nonlinear systems. These methods are based on the recursive least squares techniques and the prediction error method. Convergence analysis is provided using the hyper-stability and positivity method and the differential equation approach. A numerical simulation example of the parametric estimation of a stochastic interconnected nonlinear hydraulic system is treated.

Regularized linear system identification using atomic, nuclear and kernel-based norms: The role of the stability constraint

Inspired by ideas taken from the machine learning literature, new regularization techniques have been recently introduced in linear system identification. In particular, all the adopted estimators solve a regularized least squares problem, differing in the nature of the penalty term assigned to the impulse response. Popular choices include atomic and nuclear norms (applied to Hankel matrices) as well as norms induced by the so called stable spline kernels. In this paper, a comparative study of estimators based on these different types of regularizers is reported. Our findings reveal that stable spline kernels outperform approaches based on atomic and nuclear norms since they suitably embed information on impulse response stability and smoothness. This point is illustrated using the Bayesian interpretation of regularization. We also design a new class of regularizers defined by “integral” versions of stable spline/TC kernels. Under quite realistic experimental conditions, the new estimators outperform classical prediction error methods also when the latter are equipped with an oracle for model order selection.

Development and validation of a new efficient SRTM DEM-based horizon model combined with optimization and error prediction methods

Several post-processing methods have been developed over the last years in order to take into consideration topography within satellite-based solar radiation maps using digital elevation models (DEM). If the main part of these procedures is to estimate the obstructed horizon around each DEM point of a given region so as to consider terrain-based shading effects, the size of the area can also limit this implementation. That is why we have developed a new efficient horizon model based on the DEM retrieved during the Shuttle Radar Topography Mission (SRTM). In order to be usable at any world location with the same expected accuracy, this model is only derived from mathematical statements without any kind of empirical approximation. Validation against in situ horizons and comparison with some other models have finally shown this one presents both better accuracy (RMSE of 1.555° against 1.712° or more) and lower computation time (at least 4 times faster). Furthermore, in the case of very large areas, we propose an optimization procedure allowing the user to knowingly alter the modeling error in order to reduce processing time. Finally, using in situ data, we have also developed a method for predicting the repercussion of the original SRTM DEM error on the final horizon precision.

Stiffness modeling of machine tools based on machining space analysis

Machining stiffness, which serves the overall stability of a machine tool, plays a significant role in determining the resulting machining errors in a machine tool. Therefore, the machine tools can thus be improved to get high levels of precisions directly from what we behold as the most important determinant, the overall stiffness. In this paper, a novel approach modeling the static stiffness field of the machining space is being discussed. Within this method, a parametric model, considering six-directional static stiffness, is established to design and evaluate the static stiffness model of a high-performance computerized numerical control (CNC) machine tool. The result is then used for further discussions on the estimating and reducing of the machining errors. The machining errors are predicted by using the same CNC model under various load bearings considering the practical machining positions and stress conditions. Moreover, the effect of the six-directional static stiffness on the machining errors according to each position in the machining space is researched, where we investigate the possible solutions to reduce the machining errors with the processing technology put into consideration. In the end, by experimenting on boring-milling machining center, we verify the accuracy and efficiency of the proposed error prediction method by comparing the results from experimental data and those from finite element analysis. The proposed model is then proved to have great efficiency in optimizing the machine tool’s layout and structural design based. This improved modeling strongly suggests a favorable application for future manufacturing processes of machine tools.

Comparison of subspace and prediction error methods of system identification for cement grinding process

Maintaining product quality in cement grinding process in the presence of clinker heterogeneity is a challenging task. Model predictive controllers (MPC) are argued to be one possible solution to handle the variability, and the lack of models that relates clinker heterogeneity with product quality makes the MPC design challenging. This investigation addresses the suitability of two data-driven modelling approaches for cement grinding process-prediction error and subspace identification methods. Data collected from cement grinding process is used to build the model of the same. The collected data is used to build different candidate state-space models using the prediction error and subspace identification methods. The candidate models were validated using Akaike's information criterion and mean square error to study the suitability of these modelling techniques. The validation tests are used to identify the most suitable candidate models for the prediction error and subspace methods. The models developed in this investigation are inputs to design predictive controllers for cement industries and assure product quality in the presence of clinker grindability variations.

A Separable Prediction Error Method for Robot Identification

The Prediction Error Method, developed in the field of system identification, handles the identification of discrete time noise model for systems linear with respect to the states and the parameters. However, robots are represented by continuous time models, which are not linear with respect to the states. In this article, we consider the issue of robot identification, taking into account the physical parameters as well as the noise model in order to improve the accuracy of the estimates. Thus, we developed a new technique to tackle this problem. The experimental results tend to show a real improvement in the estimation accuracy.

Adaptive State-space Model for Ultra-precision Feed Axis

One method to produce surfaces with optical properties is the ultra precision cutting. This method allows to create surfaces not only with a roughness of few nanometers but also with low form deviations. Beside the use of advanced cutting technologies with diamond tools the path of the cutting edge is controlled very precisely. The main measures used to attain this precision are the increase of the base accuracy of the machine tool, the protection against environmental influences especially the temperature, and the prevention of dynamic loads on the machine tool. However, the last point in particular is realized by slow and smooth motions of the machine axes, which result in very long processing times of multiple hours to days per work piece. One method to increase the velocities and the accelerations is to predict and compensate the resulting tool path error. This requires a precise model of the machine tool. The parameters of this model need to be identified accurately. Furthermore, the precision of the model can be increased if the parameters are not only identified once, but repeatedly. This enables to adapt the model to parameter changes, which occur due to external and internal influences like temperature shifts, mass change and wear. For this purpose a model is built, which consists of two state-space submodels that represent the motion band and residual band features separately. The parameters of this model are adjusted by the prediction error method. The reaction time between the change of a parameter in the physical world and the adjustment of the related parameter value in the model must be short enough, so that the dynamic tool path error is kept inside the tolerance. This time period limits the bandwidth of available measurements. With this limitation of the dataset, the parameter identification becomes even more difficult. Still, to achieve accurate estimations of the parameters the search space for the identification is reduced by limiting the single parameters of the model. In this work this method is applied to an experimental setup. The precision of this approach is analyzed under varied conditions.