# Identification of dynamic errors-in-variables models

Identification of linear dynamic systems from noise-corrupted output measurements is a fundamental research problem which has been investigated in the past decades. On the other hand, the estimation of the parameters for linear dynamic systems when also the input is affected by noise is recognized as a more difficult problem which only recently has received increasing attention.

Representations where errors or measurement noises are present on both the inputs and outputs are usually called *errors-in-variables* models and play an important role when the identification purpose is the determination of the inner laws that describe the process, rather than the prediction of its future behaviour. The class of scientific disciplines which make use of such a kind of representations is very broad, such as time series modeling, array signal processing for direction-of-arrival estimation, blind channel equalization, multivariate calibration in analytical chemistry, image processing, astronomical data reduction, etc.

As a typical model example, consider the following system with noise-corrupted input and output measurements. The noise-free input is denoted by uo(t) and the undisturbed output by yo(t). They are linked through the linear difference equation

A(q^-1) y_o(t) = B(q^-1) u_o(t) , (1)

where A(q-1) and B(q-1) are polynomials of the type

A(q^-1) = 1 + a1 q^-1 + ... + an q^-n

B(q^-1) = b0 + b1 q^-1 + ... + bn q^-n (2)

and q^-1 is the backward shift operator, i.e. q^-1 x(t) = x(t-1). It is not restrictive to assume that the polynomials A(q^-1), B(q^-1) have equal degree n, which represents the order of the system.

In an errors-in-variables environment we assume that the observations are corrupted by additive measurement noises eu(t) and ey(t), at the input and output respectively. Therefore, the available signals are of the form

u(t) = u_o(t) + eu(t)

y(t) = y_o(t) + ey(t). (3)

The problem is to determine the system characteristics, i.e. the transfer function

G(q^-1) = B(q^-1)/A(q^-1) (4)

of the system given the available measurements u(t) and y(t) for t=1,...,N.

The problem has been examined from different point of views. It is possible to device the statistically efficient maximum likelihood estimate [3]. This method is computationally expensive. But the computationally simple methods [4] are often inaccurate. There are subspace fitting approaches [8], [1], which are more accurate in cost of higher computational load. Significant improvement in the accuracy with low computational load is reported in [2]. There are frequency domain approaches using a two dimensional time series model [5]. Some overview and further perspectives can be found in [21], [7], [6], [10]. The case when the unperturbed input is periodic is treated in [11]. The local convergence properties of bias-eliminating least squares (BELS) algorithms for EIV identification were investigated in [12], [13]. For the Frisch and BELS algorithms, the estimation accuracy was analyzed in [14], [22], [15] and [24]. The Frisch scheme was extended to correlated ouput noise cases in [23] and [27], and a simplified form of the BELS method was proposed in [25]. A new method, called the extended compensated least square (ECLS) method, was proposed and further analyzed in [16] and [17]. In [18], some approaches for continuous-time modeling in EIV identification were discussed. A licentiate thesis comprising some of the above results appeared in 2005, [19].

Overviews of many identifiability and estimation aspects was given in [20], [21], [37] and [44].

Recently, in [26], the relations between BELS, Frisch scheme and ECLS methods were considered. More recent studies, [39], [40], [42] show how to imbed all these methods into a joint framework (called the generalized instrumental variable estimator, GIVE), where certain choices of algorithmic parameters can result in the different special cases earlier known as specific methods. Accuracy comparisons for three Frisch methods were made in [28]. A more general study of some bias-compensated least squares problems appeared in [51].

The statistical analysis of a third-order cumulants based algorithm was investigated in [30].

An application of EIV techniques for data driven controller design appears in [42].

The sample maximum likelihood method (SML) is based on periodic excitation, runs in the frequency domain, and allows for an arbitrary, nonparametric noise model. Another ML method is mostly defined in the time-domain, and is based on parametric models both of the measurement noise, and the statistics of the unperturbed input. These two approaches are compared in [29],[32], and it is shown that under some conditions, they can give similar, but not identical, behavior. A comparison of the accuracy of the ML estimates for EIV problems compared to prediction error estimates is investigated in [43].

A covariance matching approach has been proposed and analysed from different viewpoints in [31], [33], [34], [35], [36], [38], [41], [48], [49], [52], [53]. Both discrete-time and continuous-time models are included, and the the case of discrete-time models is so-far analysed in quite some detail. The underlying idea is to first condense the information in the recorded, noisy input-output measurements in a small set of covariances (with some different lag values). Next a parameteric model is fitted to these covariances using a (weighted) nonlinear least squares approach. The method is quite promising and gives much higher accuracy than GIVE, and require a modestly higher computational complexity. It is also shown that the approach is closely related to structural equation modeling (SEM) in multivariate statistics.

The contribution [50] presents a discussion under what conditions errors-in-variables problems can be successivefully handled for systems operating under feedback control.

## Researcher in the project:

## Partners:

Mei Hong Bjerstedt, Volvo Construction Equipment, Eskilstuna, Sweden

Yasir Irshad, Karlstad University, Sweden;

Alireza Karimi, EPFL, Lausanne, Switzerland;

David Kreiberg, Uppsala University, Sweden;

Erik Larsson, Ericsson Research, Sweden;

Kaushik Mahata, University of Newcastle, Australia;

Magnus Mossberg, Karlstad University, Sweden;

Rik Pintelon, Free University of Brussels, Belgium;

Johan Schoukens, Free University of Brussels, Belgium;

Umberto Soverini, University of Bologna, Italy;

Stephane Thil, Nancy-University, France;

Klaske Van Heusden, EPFL, Lausanne, Switzerland;

Fan Wallentin, Uppsala University, Sweden;

Liuping Wang, RMIT University, Melbourne, Australia

Wei Xing Zheng, University of Western Sydney, Australia

Juan Yuz, Universidad Santa Maria, Valparaiso, Chile

Wei Xing Zheng, University of Western Sydney, Penrith South, Australia.

## Previous Researchers:

Mats Cedervall, Anders Eriksson, Joakim Sorelius, Petre Stoica.

Initially, the project was supported by the Swedish Research Council for Engineering Sciences as a part of a multi-project on Statistical Signal Processing. Currently it is supported by a grant from the Swedish Research Council.

## References

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