Automatica 47 (2011) 1656–1666
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A generalized instrumental variable estimation method for errors-in-variables
Division of Systems and Control, Department of Information Technology, Uppsala University, P O Box 337, SE-75105 Uppsala, Sweden
a r t i c l e i n f o
Received 11 February 2010
Received in revised form
3 February 2011
Accepted 22 February 2011
Available online 15 June 2011
a b s t r a c t
The errors-in-variables identification problem concerns dynamic systems whose input and output vari-
ables 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 Instrumen-
tal Variable Estimator (GIVE). Various computational aspects of GIVE are presented, and the asymptotic
distribution of the parameter estimates is derived.
© 2011 Elsevier Ltd. All rights reserved.
A system where both the input and output signals are affected
by additive noise is called an errors-in-variables (EIV) system. The
problem of estimating the parameters of such a system from the
noisy input and output data must be solved in many practical
applications of signal processing and control (Van Huffel, 1997;
Van Huffel & Lemmerling, 2002). Many different solutions to the
problem are found in the literature and an overview is given
in Söderström (2007b). Some of these methods use covariance
elements of the noisy data, for example, the bias-eliminating least
squares (BELS) method (Zheng, 1998, 2002), the Frisch scheme
(Beghelli, Castaldi, Guidorzi, & Soverini, 1993; Beghelli, Guidorzi, &
Soverini, 1990; Diversi, Guidorzi, & Soverini, 2003, 2004; Diversi,
Soverini, & Guidorzi, 2006; Söderström, 2008), and the extended
compensated least squares (ECLS) method (Ekman, 2005; Ekman,
Hong, & Söderström, 2006). In Hong and Söderström (2009),
it is shown that these methods are similar and, under weak
assumptions, in several cases equivalent.
The Frisch method exists in a number of different variants,
(Hong, Söderström, Soverini, & Diversi, 2008). In addition to
This research was partially supported by The Swedish Research Council,
contract 621-2007-6364. The material in this paper was partially presented at
the 49th IEEE CDC, December 15–17, 2010, Atlanta, Georgia, USA. This paper was
recommended for publication in revised form by Associate Editor George Yin under
the direction of Editor Ian R. Petersen.
Tel.: +46184713075; fax: +4618511925.
the so-called Frisch equations (or modified normal equations)
some additional equations are needed to define the parameter
estimates. One may use Yule–Walker equations (denoted F–YW
in the following), (Diversi et al., 2006), or the Frisch equations for
an extended model structure (will be labeled F-EM here), (Beghelli
et al., 1993; Diversi et al., 2004). These two alternatives are shown
in Hong and Söderström (2009) to give a set of equations that is
equivalent to the equations used for BELS and for ECLS. Another
option for the Frisch method is to use properties for the residual
correlation, (Diversi et al., 2003). It turns out that this Frisch
variant, which we here label as F-CR, cannot be reformulated into
the ECLS framework.
We will here introduce a wider class of estimators that will
include all the above methods, including F-CR, and label this
class Generalized Instrumental Variable Estimators (GIVE). We will
show how different user choices in GIVE will lead to the different
known methods. We will also provide a general accuracy analysis
leading to the asymptotic covariance matrix of the parameter
estimates. This will extend the previous known results applicable
for F-CR in Söderström (2007a) and for BELS in Hong, Söderström,
and Zheng (2007).
We remark in passing that there are also other alternative
methods for errors-in-variables problems that are based on
obtaining a small set of input–output covariances as a first step.
The covariance matching approach, (Söderström, Mossberg, &
Hong, 2009), makes in fact optimal use of the information in the
covariances, but the corresponding algorithm is considerably more
complex computationally than GIVE.
The paper is organized as follows. In the next section we
describe the problem to be treated, and introduce assumptions
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