A user perspective on errors-in-variables methods in system identification

The paper starts with a brief survey of errors-in-variables methods in system identification. Background and motivation are given, and it is illustrated why the identification problem can be difficult. Under general weak assumptions, the system is not identifiable, but can be parameterized using one degree of freedom. Examples where identifiability is achieved under additional assumptions are also provided. A number of approaches for parameter estimation of errors-in-variables models are reviewed. The underlying assumptions and principles for each approach are highlighted. The paper then continues by discussing from a user’s perspective on how to proceed when practical problems are handled.     

A generalized instrumental variable estimation method for errors-in-variables identification problems

The errors-in-variables identification problem concerns dynamic systems whose input and output variables are affected by additive noise. Several estimation methods have been proposed for identifying dynamic errors-in-variables models. In this paper it is shown how a number of common methods for errors-in-variables methods can be put into a general framework, resulting into a Generalized Instrumental Variable Estimator (GIVE). Various computational aspects of GIVE are presented, and the asymptotic distribution of the parameter estimates is derived.     

Subspace identification using instrumental variable techniques

Subspace-based algorithms for system identification have lately been suggested as alternatives to more traditional techniques. Variants of the MOESP type of subspace algorithms are in addition to open-loop identification applicable to closed-loop and errors-in-variables identification. In this paper, a new instrumental variable approach to subspace identification is presented. It is shown how existing MOESP-algorithms can be derived within the proposed framework, simply by changing instruments and weighting matrices. A noteworthy outcome of the analysis is that an improvement of an existing MOESP method for errors-in-variables identification can be proposed.     

Accuracy analysis of time domain maximum likelihood method and sample maximum likelihood method for errors-in-variables and output error identification

For identifying errors-in-variables models, the time domain maximum likelihood (TML) method and the sample maximum likelihood (SML) method are two approaches. Both methods give optimal estimation accuracy but under different assumptions. In the TML method, an important assumption is that the noise-free input signal is modelled as a stationary process with rational spectrum. For SML, the noise-free input needs to be periodic. It is interesting to know which of these assumptions contain more information to boost the estimation performance. In this paper, the estimation accuracy of the two methods is analyzed statistically for both errors-in-variables (EIV) and output error models (OEM). Numerical comparisons between these two estimates are also done under different signal-to-noise ratios (SNRs). The results suggest that TML and SML have similar estimation accuracy at moderate or high SNR for EIV. For OEM identification, these two methods have the same accuracy at any SNR.     

A covariance matching approach for identifying errors-in-variables systems

The errors-in-variables identification problem concerns dynamic systems whose input and output variables are affected by additive noise. Several estimation methods have been proposed for identifying dynamic errors-in-variables models. In this paper a covariance matching approach is proposed to solve the identification problem. It applies for general types of input signals. The method utilizes a small set of covariances of the measured input–output data. This property applies also for some other methods, such as the Frisch scheme and the bias-eliminating least squares method. Algorithmic details for the proposed method are provided. User choices, for example specification of which input–output covariances to utilize, are discussed in some detail. The method is evaluated by using numerical examples, and is shown to have competitive properties as compared to alternative methods.     

Correlation Analysis-based Stochastic Gradient and Least Squares Identification Methods for Errors-in-variables Systems Using the Multiinnovation

This paper deals with the identification problem of discrete-time linear time-invariant errors-in-variables systems for the case of the colored output noise. Based on the correlation analysis, the multi-innovation theory is introduced to the errors-in-variables systems where both input and output data are noisy. A correlation analysis-based multi-innovation stochastic gradient algorithm and a correlation analysis-based multi-innovation least squares algorithm are proposed by means of the multi-innovation theory in order to improve the parameter accuracy. The simulation results confirm that these two algorithms are effective.

     

Application of structured total least squares for system identification and model reduction

The following identification problem is considered: Minimize the /spl lscr//sub 2/ norm of the difference between a given time series and an approximating one under the constraint that the approximating time series is a trajectory of a linear time invariant system of a fixed complexity. The complexity is measured by the input dimension and the maximum lag. The question leads to a problem that is known as the global total least squares problem and alternatively can be viewed as maximum likelihood identification in the errors-in-variables setup. Multiple time series and latent variables can be considered in the same setting. Special cases of the problem are autonomous system identification, approximate realization, and finite time optimal /spl lscr//sub 2/ model reduction. The identification problem is related to the structured total least squares problem. This paper presents an efficient software package that implements the theory. The proposed method and software are tested on data sets from the database for the identification of systems DAISY.     

Frisch scheme identification for dynamic diagonal bilinear models

The article addresses the problem of dynamic system identification in the errors-in-variables framework for a class of discrete-time time-invariant input–output bilinear models when subjected to a white input signal. The proposed algorithm is based on an extension of the bias-compensated least squares method and utilises the Frisch scheme equations to determine the parameter vector together with the variances of the input and output noise sequences. The appropriateness of the approach is analysed and its performance evaluated when compared to other errors-in-variables identification techniques by means of a Monte Carlo simulation. The results obtained demonstrate the accuracy of the proposed method and the performance in terms of noise robustness is also observed.     

Identification of EIV models by compensated PEM

This paper deals with identification of discrete-time errors-in-variables models where the input and output data are both perturbed by different additive noises. The goal is to study the effects of input noise on the model which is estimated based on the prediction error method. The obtained model is then improved by modifying the results and implementing the instrumental variable method. It is proved that the identification of the errors-in-variables models based on the proposed approach could result in an unbiased estimation in the presence of independent colour noises on the input and output data with adequate accuracy and mediocre complexity.     

L-two-optimal identification of errors-in-variables models: a frequency-domain approach

This paper proposes an L-two-optimal identification approach to cope with errors-in-variables model (EIVM) identification. With normalized coprime factor model (NCFM) representations, L-two-optimal approximate models are derived from the framework of an EIVM according to the kernel and image representations of related signals. Based on the optimal approximate models, the v-gap metric is employed as a minimization criterion to optimize the parameters of a system model, and thus the resulting optimization problem can be solved by linear matrix inequalities (LMIs). In terms of the optimized system model, the noise model (NM) can be readily obtained by right multiplication of an inner. Compared with other EIVM identification methods, the proposed one has a wider scope of applications because the statistical properties of disturbing noises are not demanded. It is also capable of giving identifiability. Finally, a numerical simulation is used to verify the effectiveness of the proposed method.     

Calibration procedures when both measurements are subject to error: A comparative simulation study of the unreplicated case

The problem of statistical calibration, when both standard and non-standard measurements are subject to error, is formulated as a predictive errors-in-variables model. Under the assumptions of unreplicated observations, a simulation study was conducted to compare relative performances of the ordinary least squares and the maximum likelihood estimation method, each combined with classical and inverse prediction. The Fuller method is also included for comparison. The mean squared error of prediction and the probability of concentration are adopted as criteria, based upon which guidelines are provided for selecting appropriate calibration procedures for the given situation.     

Identification of continuous-time errors-in-variables models

A novel direct approach for identifying continuous-time linear dynamic errors-in-variables models is presented in this paper. The effects of the noise on the state-variable filter outputs are analyzed. Subsequently, a few algorithms to obtain consistent continuous-time parameter estimates in the errors-in-variables framework are derived. It is also possible to design search-free algorithms within our framework. The algorithms can be used for non-uniformly sampled data. The asymptotic distributions of the estimates are derived. The performances of the proposed algorithms are illustrated with some numerical simulation examples.     

Robust control oriented identification of errors-in-variables models based on normalised coprime factors

A robust control oriented identification approach is proposed to deal with the identification of errors-in-variables models (EIVMs), which are corrupted with input and output noises. Based on normalised coprime factor model (NCFM) representations, a frequency-domain perturbed NCFM for an EIVM is derived according to a geometrical explanation for the v-gap metric. As a result, identification of the EIVM is converted into that of the NCFM. Besides an identified nominal NCFM, its worst case error has to be quantified. Unlike other traditional control-oriented identification methods, the v-gap metric is employed to measure the uncertainties including a priori information on the disturbing noises and the worst case error for the resulting nominal NCFM. Since this metric is also used as an optimisation criterion, the associate parameter estimation problem can be effectively solved by linear matrix inequalities. Finally, a numerical simulation shows the effectiveness of the proposed method.     

Identification of nonlinear errors-in-variables models

The paper is about a generalization of a classical eigenvalue-decomposition method originally developed for errors-in-variables linear system identification to handle an important class of nonlinear problems. A number of examples are presented to call the attention to the most critical part of the procedure turning the identification problem to a generalized eigenvalue–eigenvector calculation problem with symmetrical matrices. The elaborated method generates consistent parameter estimation. Simulation results demonstrate the effectiveness of the proposed algorithm.     

L 2-optimal identification of MIMO errors-in-variables models from the v-gap geometrical interpretation

An L ₂-optimal identification method is extended to cope with MIMO errors-in-variables (EIV) model estimation based on a geometrical interpretation for the v-gap metric. The L ₂-optimal approximate models are composed of system and noise models and characterised by a normalised right graph symbol (NRGS) and its complementary inner factor (CIF), respectively. This metric can be evaluated as the supreme of sine values of the maximal principal angles between NRGS frequency responses of two concerned models. In order to make full use of the angular cosine formula for complex vectors to reduce computational loads, a CIF of the NRGS of the perturbed model is introduced and thus, the system parameter optimisation can be efficiently solved by sequential quadratic programming methods. With the estimated system model, the associated noise model can be built by right multiplication of an inner matrix. Finally, a simulation example demonstrates the effectiveness of the proposed identification method.     

Error quantification of the normalised right graph symbol for an errors-in-variables system

This paper proposes a novel method to quantify the error of a nominal normalized right graph symbol (NRGS) for an errors-in-variables (EIV) system corrupted with bounded noise. Following an identification framework for estimation of a perturbation model set, a worst-case v-gap error bound for the estimated nominal NRGS can be first determined from \textit{a priori} and \textit{a posteriori} information on the underlying EIV system. Then, an NRGS perturbation model set can be derived from a close relation between the v-gap metric of two models and ${\rm H}_\infty$-norm of their NRGSs' difference. The obtained NRGS perturbation model set paves the way for robust controller design using an ${\rm H}_\infty$ loop-shaping method because it is a standard form of the well-known NCF (normalized coprime factor) perturbation model set. Finally, a numerical simulation is used to demonstrate the effectiveness of the proposed identification method.     

Accuracy analysis of a covariance matching approach for identifying errors-in-variables systems

A system identification method for errors-in-variables problems based on covariance matching was recently proposed. In the first step, a small amount of covariances of noisy input–output data are computed, and then a parametric model is fitted to these covariances. In this paper, the method is further analyzed and the asymptotic accuracy of the parameter estimates is derived. An explicit algorithm for computing the asymptotic covariance matrix of the parameter estimates is given, and the identification method is shown to be asymptotically statistically efficient assuming that the given information is the computed covariances. As an important byproduct, an efficient algorithm is presented for computing the covariance matrix of the computed input–output covariances.     

On the accuracy of a covariance matching method for continuous-time errors-in-variables identification

An analysis of a covariance matching method for continuous-time errors-in-variables system identification from discrete-time data is made. In the covariance matching method, the noise-free input signal is not explicitly modeled and only assumed to be a stationary process. The asymptotic normalized covariance matrix, valid for a large number of data and a small sampling interval, is derived. This involves the evaluation of a covariance matrix of estimated covariance elements and estimated derivatives of such elements, and large parts of the paper are devoted to this task. The latter covariance matrix consists of two parts, where the first part contains integrals that are approximations of Riemann sums, and the second part depends on the measurement noise variances.