Automatica 43 (2007) 1418–1425
Marco Dalaia, Erik Weyerb, Marco C. Campia,∗
aDepartment of Electrical Engineering and Automation, University of Brescia, Via Branze 38, 25123 Brescia, Italy
bDepartment of Electrical and Electronic Engineering, The University of Melbourne, Parkville VIC 3010, Australia
Received 6 June 2006; received in revised form 27 October 2006; accepted 19 January 2007
Available online 19 June 2007
In this paper we consider the problem of constructing confidence regions for the parameters of nonlinear dynamical systems. The proposed
method uses higher order statistics and extends the LSCR (leave-out sign-dominant correlation regions) algorithm for linear systems introduced
in Campi and Weyer [2005, Guaranteed non-asymptotic confidence regions in system identification. Automatica 41(10), 1751–1764. Extended
version available at ?http://www.ing.unibs.it/∼campi?]. The confidence regions contain the true parameter value with a guaranteed probability
for any finite number of data points. Moreover, the confidence regions shrink around the true parameter value as the number of data points
increases. The usefulness of the proposed approach is illustrated on some simple examples.
? 2007 Elsevier Ltd. All rights reserved.
Keywords: Confidence sets; Finite sample results; Nonlinear system identification
It is well known that a model of a dynamical system is of
limited use if no quality tag which describes the accuracy of
the model is attached. Confidence regions for the system pa-
rameters are commonly used as quality tags, and asymptotic
ever, in practice one always has a finite number of samples,
and—even though the asymptotic theory delivers sensible re-
sults in many cases—there are also examples (Garatti, Campi,
& Bittanti, 2004) where it fails when applied to a finite num-
ber of data points. Thus, there is a need for techniques which
deliver confidence regions with guaranteed probabilities when
only a finite number of data points are available.
?This paper was not presented at any IFAC meeting. This paper was
recommended for publication in revised form by Associate Editor Antonio
Vicino under the direction of Editor Torsten Söderström.
∗Corresponding author. Tel.: +390303715458; fax: +39030380014.
E-mail addresses: email@example.com (M. Dalai),
firstname.lastname@example.org (E. Weyer), email@example.com
0005-1098/$-see front matter ? 2007 Elsevier Ltd. All rights reserved.
In Campi and Weyer (2005) a method called LSCR (leave-
confidence regions to which the parameters of a linear system
belong with guaranteed probability. See also Campi and Weyer
(2006) for a comprehensive presentation of LSCR. LSCR ex-
tends earlier work by Hartigan (1969, 1970) to a dynamical
system setting, and it has two important features: first, the prob-
ability that the confidence region contains the true parameters
is guaranteed for any finite amount of data samples; second, the
confidence region concentrates around the true parameter value
when the number of samples increases. In Campi and Weyer
(2005), second order statistics were explored for the construc-
tion of the confidence regions. In the present paper, we consider
nonlinear systems. It is well known (see for example, Ljung,
2001 for a general discussion, or Subba Rao, 1981 for the par-
ticular case of bilinear systems) that second order statistics are
insufficient for the identification of nonlinear systems. Here we
show that it is possible to extend the framework of LSCR to
higher order statistics, and hence to consider the problem of
nonlinear system identification within this setting.
The focus of this paper is on time series, that is the system
to be identified has no exogenous inputs which are measured.
The outline of the paper is as follows. In the next section, we
M. Dalai et al. / Automatica 43 (2007) 1418–1425
the incidence matrix of GN, i.e. the matrix with generic
element QN(i,j) = 1 if j ∈ IN
and zero otherwise.
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regions in system identification. Automatica, 41(10), 1751–1764 Extended
version available at ?http://www.ing.unibs.it/∼campi?.
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Marco Dalai was born in 1979 in Manerbio,
Italy. He obtained his Dr. Eng. degree in Elec-
tronic Engineering in 2003 from the University
of Brescia, Italy, and, since 2004, he has been
a Ph.D. student in Information Engineering at
the Department of Electronics for Automation
of this same university.
Erik Weyer received the Siv. Ing. degree in
1988 and the Ph.D. in 1993, both from the
Norwegian Institute of Technology, Trondheim,
Norway. From 1994 to 1996 he was a Research
Fellow at the University of Queensland, and
since 1997 he has been with the Department
of Electrical and Electronic Engineering, the
University of Melbourne, where he is currently
a Senior Lecturer. His research interests are in
the area of system identification and control.
Marco Claudio Campi is Professor of Auto-
matic Control at the University of Brescia, Italy.
He was born in Tradate, Italy, on December 7,
1963. In 1988, he received the Doctor degree
in electronic engineering from the Politecnico
di Milano, Milano, Italy. From 1988 to 1989,
he was a Research Assistant at the Department
of Electrical Engineering of the Politecnico di
Milano. From 1989 to 1992, he worked as a
Researcher at the Centro di Teoria dei Sistemi
of the National Research Council (CNR) in Mi-
lano. Since 1992, he has been with the Univer-
sity of Brescia, Italy.
Marco Campi is an Associate Editor of Systems and Control Letters, and a
past Associate Editor of Automatica and the European Journal of Control.
Serves as Chair of the Technical Committee IFAC on Stochastic Systems (SS)
and is a member of the Technical Committee IFAC on Modeling, Identifica-
tion and Signal Processing (MISP) and of the Technical Committee IFAC on
Cost Oriented Automation. Moreover, he is a Distinguished Lecturer under
the IEEE Control Systems Society (CSS) Program. His doctoral thesis was
awarded the “Giorgio Quazza” prize as the best original thesis for year 1988.
He has held visiting and teaching positions at many universities and institu-
tions including the Australian National University, Canberra, Australia; the
University of Illinois at Urbana-Champaign, USA; the Centre for Artificial
Intelligence and Robotics, Bangalore, India; the University of Melbourne,
Australia; the Kyoto University, Japan.
The research interests of Marco Campi include: system identification, stochas-
tic systems, adaptive and data-based control, robust convex optimization,
robust control and estimation, and learning theory.