Identification of Errors-in-Variables Systems via Extended Compensated Least Squares for the Case of Coloured Output Noise (original) (raw)

Recursive Extended Compensated Least Squares based Algorithm for Errors-in-Variables Identification

An algorithm for the recursive identification of single-input single-output linear discrete-time time-invariant errors-in-variables system models in the case of white input and coloured output noise is presented. The approach is based on a bilinear parametrisation technique which allows the model parameters to be estimated together with the auto-correlation elements of the input/output noise sequences. In order to compensate for the bias in the recursively obtained least squares estimates, the extended bias compensated least squares method is used. An alternative for the online update of the associated pseudo-inverse of the extended observation covariance matrix is investigated, namely an approach based on the matrix pseudo-inverse lemma and an approach based on the recursive extended instrumental variables technique. A Monte-Carlo simulation study demonstrates the appropriateness and the robustness against noise of the proposed scheme.

Errors-in-variables identification using maximum likelihood estimation in the frequency domain

Automatica

This report deals with the identification of errors-in-variables (EIV) models corrupted by additive and uncorrelated white Gaussian noises when the noise-free input is an arbitrary signal, not required to be periodic. In particular, a frequency domain maximum likelihood (ML) estimator is proposed and analyzed in some detail. As some other EIV estimators, this method assumes that the ratio of the noise variances is known. The estimation problem is formulated in the frequency domain. It is shown that the parameter estimates are consistent. An explicit algorithm for computing the asymptotic covariance matrix of the parameter estimates is derived. The possibility to effectively use lowpass filtered data by using only part of the frequency domain is discussed, analyzed and illustrated.

Identification of dynamic errors-in-variables models: Approaches based on two-dimensional ARMA modeling of the data

Automatica, 2003

This paper studies the linear dynamic errors-in-variables (EIV) problem in a fairly general condition where the input-output disturbing noises are colored and the input is quasi-stationary. A novel formulation of the extended frequency domain maximum likelihood (ML) estimator is developed which reduces the number of nonlinear normal equations to be solved. Sufficient conditions are provided to achieve local identifiability of the EIV model for specified noise cases of interest. The parameter estimates are calculated via a numerically stable Gauss-Newton minimization scheme started by an initial value generation strategy. Also, both the consistency and accuracy of the extended ML estimate are analyzed in detail. The performance of the proposed method is finally demonstrated on simulated dynamic systems.

On instrumental variable-based methods for errors-in-variables model identification

IFAC Proceedings Volumes, 2008

In this paper, the problem of identifying stochastic linear discrete-time systems from noisy input/output data is addressed. The input noise is supposed to be white, while the output noise is assumed to be coloured. Some methods based on instrumental variable techniques are studied and compared to a least squares bias compensation scheme with the help of Monte Carlo simulations.

Identification of errors-in-variables models as a quadratic eigenvalue problem

The paper proposes a new approach for identifying linear dynamic errors-in-variables (EIV) models, whose input and output are affected by additive white noise. The method is based on a nonlinear system of equations consisting of part of the compensated normal equations and of a set of high order Yule-Walker equations. This system allows mapping the EIV identification problem into a quadratic eigenvalue problem that, in turn, can be mapped into a linear generalized eigenvalue problem. The system parameters are thus estimated without requiring the use of iterative identification algorithms. The effectiveness of the method has been tested by means of Monte Carlo simulations and compared with those of other EIV identification methods.

Error assessment of the estimated coefficients obtained in system identification technique

Signal, Image and Video Processing, 2014

System identification technique plays an important role in many electrical devices. In this technique, an adaptive filter models the unknown system with a Finite Impulse Response (FIR) or an Infinite Inverse Response (IIR) filter. This paper concentrates on the system identification technique based on the least squares criterion and evaluates the relationship between each estimated coefficient (obtained by the adaptive filter) and its corresponding coefficient in the unknown system. By logically classifying the variables, the amount of error between these two corresponding coefficients is evaluated and precisely expressed based on the auto-correlation lags of the input signal of the system and the coefficients of the unknown system. Also, the computed error is simplified for two particular cases in which the input signal of the system is an ideal zero-mean white Gaussian noise or a windowed (short-time) zero-mean white Gaussian noise. Experimental results provided in the simulation part verify the arithmetic expressions presented in the paper.

Improvement Instrumental Variables Method for Output Error Model

Procedia Computer Science, 2019

The identification of parameters system can be done in multiple ways by alternating methods (Least Squares, Instrumental Variables) and models (ARX, OE). The Ordinary Least Squares method which was presented in several research papers gives biased estimates for Output Error model. For example, the interpretation results in the first order OE model, the autocorrelation function shows that the residuals are not white noise, and the LS estimator is well biased, and it is not a suitable estimator for identification of output error model. In this paper, we will present a detailed study of Recursive Instrumental Variables method and its instruments choice to identify the parameters of Output Error model with unbiased estimates. The idea is simultaneously to identify the parameters system similar to an ARX model with Least Squares method. It is a successfully applied to identify with unbiased estimates parameters of a second order system based on whitening error prediction, and we showed that, if the instruments are weak, then the estimator may be biased and confidence intervals and hypothesis tests unreliable. Also, for every data observation set, the estimates parameters can be compared with those originally generated by MATLAB functions. Finally, a numerical example illustrates the efficiency and performances of the proposed method.

Unifying some higher-order statistic-based methods for errors-in-variables model identification

Automatica, 2009

In this paper, the problem of identifying linear discrete-time systems from noisy input and output data is addressed. Several existing methods based on higher-order statistics are presented. It is shown that they stem from the same set of equations and can thus be united from the viewpoint of extended instrumental variable methods. A numerical example is presented which confirms the theoretical results. Some possible extensions of the methods are then given.

Identification of Errors-In-Variables Models Using the EM Algorithm

Proceedings of the 17th IFAC World Congress, 2008, 2008

This paper advocates a new subspace system identification algorithm for the errorsin-variables (EIV) state space model via the EM algorithm. To initialize the EM algorithm an initial estimate is obtained by the errors-in-variables subspace system identification method: EIV-MOESP (Chou et al. [1997]) and EIV-N4SID (Gustafsson [2001]). The EM algorithm is an algorithm to compute the maximum value for the likelihood function that is consists of two steps; namely the E-and M-steps. The E-and M-steps in the EM algorithm are calculated by computing the conditional expectation under the assumption that the input-output data is completely observed. Numerical example shows that the EM algorithm can monotonically improve the initial estimates obtained by subspace identification methods.

Accuracy analysis of bias-eliminating least squares estimates for errors-in-variables systems

Automatica, 2007

The bias-eliminating least squares (BELS) method is one of the consistent estimators for identifying dynamic errors-in-variables systems. The attraction of the BELS method lies in its good accuracy and its modest computational cost. In this report, we investigate the accuracy properties of the BELS estimates. It is shown that the estimated system parameters and the estimated noise variances are asymptotically Gaussian distributed. An explicit expression for the normalized covariance matrix of the estimated parameters is derived and supported by some numerical examples.