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Several estimation methods have been proposed for identifying errors-in-variables systems, where both input and output measurements are corrupted by noise. One of the more interesting approaches is the Frisch scheme. The method can be applied using either time or frequency domain representations. This paper investigates the general mathematical and geometrical aspects of the Frisch scheme, illustrating the analogies and the differences between the time and frequency domain formulations.

Integrity of signals is an important issue for aerospace navigation systems and, in particular, for satellite navigation positioning. In this paper integrity monitoring techniques are processed with a new FDI technique implemented by a snapshot RAIM algorithm, based on linearized models, and position domain tests. The approach consists of the joint exploitation, in an Errors In Variables (EIV) framework, of all the possible Least Squares (LS) solutions under the hypothesis of a single fault on a pseudorange measurement. The characteristics of the behavior of the different LS position and bias estimates, by varying the fault size and the faulty satellite, are investigated. The non linear dependence of the locus of the solutions from the fault size is considered. The analysis of the loci properties results in a new criterion and algorithm for the detection and isolation of a faulty satellite signal. The effectiveness of the proposed method has been compared with respect to a classic FDI method by means of Montecarlo simulations based on a Galileo constellation simulator.

This paper concerns the identification of multivariable errors-in-variables (EIV) models, i.e. models where all inputs and outputs are assumed as affected by additive errors. The identification of MIMO EIV models introduces challenges not present in SISO and MISO cases. The approach proposed in the paper is based on the extension of the dynamic Frisch scheme to the MIMO case. In particular, the described identification procedure relies on the association of EIV models with directions in the noise space and on the properties of a set of high order Yule–Walker equations. A method for estimating the system structure is also described.

This paper describes a new approach for identifying ARX models from a finite number of measurements, in presence of additive and uncorrelated white noise. The proposed algorithm is based on some theoretical results concerning the so–called dynamic Frisch Scheme. As a major novelty, the proposed approach deals with frequency domain data. In some aspects, the method resembles the characteristics of other identification algorithms, originally developed in the time domain. The proposed method is compared with other techniques by means of Monte Carlo simulations. The benefits of filtering the data and using only part of the frequency domain is highlighted by means of a numerical example.

This paper describes a new approach for identifying autoregressive models from a finite number of measurements, in presence of additive and uncorrelated white noise. As a major novelty, the proposed approach deals with frequency domain data. In particular, two different frequency domain algorithms are proposed. The first algorithm is based on some theoretical results concerning the so--called dynamic Frisch Scheme. The second algorithm maps the AR identification problem into a quadratic eigenvalue problem. Both methods resemble in many aspects some other identification algorithms, originally developed in the time domain. The features of the proposed methods are compared each other and with those of other time domain algorithms by means of Monte Carlo simulations.

This paper deals with the identification of errors–in–variables models where the additive input and output noises are mutually correlated white processes. The proposed solution is based on the extension of the dynamic Frisch scheme introduced in (Beghelli et al., 1990). First, a geometric characterization of the whole set of admissible solutions in the noise space is described. Then, a criterion that allows to select the solution of the identification problem inside the locus is proposed. This criterion relies on the properties of a set of high–order Yule–Walker equations. The effectiveness of this identification approach is tested by means of Monte Carlo simulations.

This paper deals with the identification of multi–input multi–output errors–in–variables (EIV) models in the Frisch scheme context. The extension of the Frisch scheme to MIMO models introduces some problems not present in the SISO case. The approach proposed in this paper relies on the association of EIV models to directions in the noise space and on the statistical properties of the residual of the EIV model. In particular, a selection criterion based on the comparison of the theoretical statistical properties of the residual with those computed from the data is introduced. The performance of the proposed identification algorithm is evaluated by means of numerical simulations.

The extension of the Frisch scheme from the original algebraic case to the dynamic one leads to the use of errors–in–variables models where the measurements of the input and output are affected by additive white and independent noises. This problem admits a single solution when the assumptions of the scheme are exactly fulfilled but its application to real processes requires the introduction of specific model selection criteria. This paper analyzes the additional problems encountered in the extension of Frisch identification to the multivariable case and introduces a geometric approach for its solution.

This paper describes an identification procedure for minimally parametrized multivariable models in the Errors–in–Variables (EIV) context of the Frisch scheme that considers additive white observation noise on the process inputs and outputs. This procedure relies on the geometric approach described in (Guidorzi and Diversi, 2009) that associates EIV models to directions in the noise space. The proposed procedure has been tested by means of a Monte Carlo simulation that confirms its effectiveness.

An important problem in sensor array processing is the estimation of the number of transmitted signals. Most of the proposed solutions rely on the assumption of uniform additive white noise on the measured signals. In this paper, an approach for estimating the number of sources in the presence of nonuniform white noise is proposed. The method is based on the computation of the maximal corank of the covariance matrix of the noisy data in the Frisch scheme context. The effectiveness of the method is tested by means of Monte Carlo simulations.

This paper proposes a two-step identification procedure for the direction-of-arrival estimation problem in the presence of nonuniform white noise. The first step consists in estimating the unknown sensor noise variances by exploiting the properties of the Frisch scheme. Once that the noise covariance matrix has been identified, the angles of arrival are computed by using the classical ESPRIT algorithm. The effectiveness of the whole procedure is tested by means of Monte Carlo simulations.

In this paper the problem of identifying a fuzzy model from noisy data is addressed. The piecewise-affine fuzzy model structure is used as non-linear prototype for a multi–input, single–output unknown system. The consequents of the fuzzy model are identified from noisy data which are collected from experiments on the real system. The identification procedure is formulated within the Frisch scheme, well established for linear systems, which is extended so that it applies to piecewise-affine, constrained models.

Errors-in-Variables (EIV) models consider the presence of additive errors on the measures of all measurable attributes of a process. Traditional identification procedures for these processes rely on the properties of the covariance matrix of the observations and on its decomposition into the sum of a matrix associated with the (unknown) noiseless sequences and of the observation noise covariance matrix. This paper makes reference, in a behavioural framework, to the observed noisy trajectories and performs a decomposition of these trajectories into a regular part, defininig the associated behaviour of the process, and a noise part. The Monte Carlo simulations that have been performed show that the proposed approach leads to accurate estimates of both the system parameters and the noise variances.

Multivariable output-error identification does not constitute, in any way, a straightforward extension of the scalar case. The aim of this paper is twofold: 1) Introduction of a new minimal parametrization for multivariable output error models leading to an easily implementable prediction error identification procedure; 2) Comparison of PEM and errors-in-variables approaches based on the dynamic Frisch scheme in the identification of MIMO output error processes.

This paper considers the problem of determining linear relations from data affected by additive noise in the context of the Frisch scheme. The loci of solutions of the Frisch scheme and their properties are first described in the algebraic case. In this context two main problems are analyzed: the evaluation of the maximal number of linear relations compatible with data affected by errors and the determination of the linear relation actually linking the noiseless data. Subsequently the extension of the Frisch scheme to the identification of dynamical systems is considered for both SISO and MIMO cases and the problem of its application to real processes is investigated.

Deriving linear relations from data affected by additive noise is a problem of high practical relevance in many fields. All available procedures are based on sets of assumptions concerning the noise process; the results obtained from their use depend, consequently, from the distance between the actual process and the specific assumptions. The Frisch scheme introduces less a priori assumptions than other commonly used methods like, for instance, Least Squares, but leads to a whole family of models compatible with a set of noisy data so that its use is impractical in most cases. This paper introduces, on the basis of definitions and properties previously described, a robust and consistent procedure to extract a single model from two independent sets of noisy data in the context of the Frisch scheme. This allows the use of this approach also in all practical cases requiring a single model to describe the process behind the data.

The fuzzy model identification problem from noisy data is addressed. The piecewise linear fuzzy model structure is used as a nonlinear prototype for a multi-input, single-output unknown system. The consequent of the fuzzy model is identified using noisy data, e.g. collected from experiments on a real system. The identification procedure is formulated within the Frisch scheme, well established for linear systems, which has been modified and improved to be applied in fuzzy systems field.

This paper considers the problem of determining linear relations from data affected by additive noise in the context of the Frisch scheme. The loci of solutions of the Frisch scheme and their properties are first described in the algebraic case. In this context two main problems are analyzed: the evaluation of the maximal number of linear relations compatible with data affected by errors and the determination of the linear relation actually linking the noiseless data. Subsequently the extension of the Frisch scheme to the identification of dynamical systems is considered for both SISO and MIMO cases and the problem of its application to real processes is investigated. For this purpose suitable identification criteria and model parametrizations are described. Finally two classical identification problems are mapped into the Frisch scheme, the blind identification of FIR channels and the identification of AR + noise models. This allows some theoretical and practical extensions.