Comparison of three Frisch methods for errors-in-variables identification

Abstract

The errors-in-variables framework concerns static or dynamic systems whose input and output variables are affected by additive noise. Several estimation methods have been proposed for identifying dynamic errors-invariables models. One of the more promising approaches is the so-called Frisch scheme. This report decribes three different estimation criteria within the Frisch context and compares their estimation accuracy on the basis of the asymptotic covariance matrices of the estimates. Some final numerical examples support the theoretical results and analyze the behaviour of the methods in case of finite number of data.

Frisch scheme identification for dynamic diagonal bilinear models

“…The corrupting additive input-output noise sequences, denoted u˜k andỹ k , respectively, are zero mean, ergodic, white signals with unknown variances, defined by ũ ¼ E ½ũ 2 k and ỹ ¼ E ½ỹ 2 k , respectively; furthermore, they are mutually uncorrelated and uncorrelated with the noise-free signals, denoted u 0 k and y 0 k . With reference to the existing linear case, the assumptions postulated here are in common with the welldocumented EIV framework (So¨derstro¨m 2007). It is to be noted that the whiteness property of the input stated in Assumption A3 is also frequently imposed, especially when considering bilinear systems (Carravetta et al 1997;Favoreel, De Moor, and Overschee 1999;Tsoulkas, Koukoulas, and Kalouptsidis 1999;Verdult and Verhaegen 2000).…”

“…Bias compensated least squares and Frisch scheme for linear systems This section provides a brief review of the bias compensated least squares technique (So¨derstro¨m 2007), as well as of the Frisch scheme (Beghelli et al 1990), which have been developed for linear systems. Consider a linear system, i.e.…”

“…The EIV approach has gained increased attention during the last few decades and various solutions with regard to specific assumptions have been proposed. A detailed survey of existing approaches, mainly regarding linear model structures, can be found in Van Huffel and Lemmerling (2002), Markovsky and Van Huffel (2007) and So¨derstro¨m (2007). In general, models and hence the EIV problems can be classified into two categories, namely the static class and the dynamic class; see Markovsky, Willems, Van Huffel, and De Moor (2006).…”

“…This method was initially developed for static problems (Frisch 1934) and extended subsequently to handle dynamic systems in Beghelli et al (1990). A further analysis concerning the estimation accuracy of the scheme was carried out in So¨derstro¨m (2007) and Hong, So¨derstro¨m, Soverini, and Diversi (2007).…”

“…EIV systems are of great practical interest when the description of the underlying internal system behaviour is the prime goal, rather than that of the prediction of external signals (So¨derstro¨m 2007). The EIV approach has gained increased attention during the last few decades and various solutions with regard to specific assumptions have been proposed.…”