Automatica 44 (2008) 3003–3013
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Uncertainty propagation in dynamical systems
, Thordur Runolfsson
Department of Mechanical Engineering, University of California, Santa Barbara, CA 93105-5070, United States
School of Electrical and Computer Engineering, The University of Oklahoma, Norman, OK 73019-1023, United States
a r t i c l e i n f o
Received 27 April 2007
Received in revised form
11 January 2008
Accepted 16 April 2008
Available online 1 November 2008
a b s t r a c t
Dynamical propagation of parametric and initial condition uncertainty is studied. The notion of input
measure of an observable is defined and its propagation to output measure of the observable is studied
by means of transfer operators. Uncertainty of these measures is defined in terms of their cumulative
probability distributions. Comparison with alternative uncertainty metrics such as variance and entropy
is pursued. The developed formalism is illustrated through an analysis of the effect of pitchfork bifurcation
on uncertainty. Finally, the implications of these concepts in the design of nonlinear systems are
© 2008 Elsevier Ltd. All rights reserved.
Uncertainty analysis is a topic of research that has received
much attention in recent years. Indeed, the increased use of
physics based models in the study of the dynamical behavior of
systems in a wide range of applications calls for the analysis and
quantification of model predictions in terms of uncertainties in
model descriptions and model operating environments (Helton,
1994). In this paper we consider systems that can be modeled
by discrete maps and study, abstractly, uncertainty propagation
in such systems, following an approach to the study of measure
propagation developed in Lasota and Mackey (1994), Dellnitz and
Junge (2002) and Mezić and Banaszuk (2004).
The analysis of uncertainty sources and classification of
uncertainty types in mathematical models have received much
attention (Oberkampf, DeLand, Rutherford, Diegert, & Alvin, 2002).
Frequently uncertainty is classified into two types: reducible, or
epistemic, and irreducible, or aleatory. An example of epistemic
uncertainty is uncertainty in initial conditions that can possibly
This paper was not presented at any IFAC meeting. This paper was
recommended for publication in revised form by Associate Editor Yoshito Ohta
under the direction of Editor Roberto Tempo. The work was partially sponsored by a
DARPA seed contract and AFOSR grants FA9550-06-1-0088 and FA9550-05-1-0441.
Both authors were at the United Technologies Research Center, East Hartford, CT
during a part of this work.
Corresponding author. Tel.: +1 405 325 5735; fax: +1 405 325 7066.
E-mail addresses: firstname.lastname@example.org (I. Mezić), email@example.com
be reduced by improved measurements. Aleatory uncertainty is
an uncertainty in the system parameters that is the result of the
intrinsic stochasticity of the system. We make a further distinction
that both of these can be a priori and a posteriori. In particular,
a priori uncertainty is any uncertainty (epistemic or aleatory)
that can be captured in an input description of the system and
a posteriori uncertainty is an uncertainty that is inherent to the
process dynamics and observations.
The most common approach for propagating uncertainty in
mathematical and computational models is to use Monte Carlo
type methods (Halton, 1970; Hanson, 1999). Due to the fact
that Monte Carlo methods are basically ‘‘wrapper’’ methods
they have the advantage that they apply to a large class
of problems but suffer from slow convergence rate and in
many problems the computational burden may be prohibitive.
Approaches for improving the computational speed of Monte
Carlo methods through improved sampling techniques have been
developed (Helton, 1994). An alternative approach for uncertainty
propagation is Polynomial Chaos methods (also called Stochastic
Finite Elements) (Ghanem & Red-Horse, 1999; Ghanem & Spanos,
1991). Polynomial Chaos is an analytical approach based on
expansions of the uncertain quantities of terms of prescribed
random basis functions. It has been demonstrated that for certain
classes of problems Polynomial Chaos can be considerably (up to
several orders of magnitude) faster than Monte Carlo methods.
Furthermore, the analytical representation in Polynomial Chaos
can be of great benefit in analysis.
Monte Carlo and Polynomial Chaos are methods for uncertainty
propagation and have to be combined with other analysis methods
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