21-11-2012, 06:19 PM
Afuzzy dynamic model based state estimator
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Abstract
Systems containing uncertainty are traditionally analyzed with probabilistic methods. However, for non-linear, non-Gaussian
systems solutions can sometimes be very di4cult to obtain. The focus of this work is to determine if in such cases fuzzy
dynamic system models may provide an alternative approach that more easily leads us to a good solution. In this paper,
we present a fuzzy estimator whose system model is a fuzzy dynamic system. We show that for the linear, Gaussian case
the fuzzy estimator produces the same result as the Kalman 9lter. More importantly, we show that the fuzzy estimator
can succeed for some non-Gaussian, nonlinear systems. Finally, we illustrate the application of the fuzzy estimator on a
non-linear, non-Gaussian, time-varying rocket launch problem where we show that it performs better than the extended
Kalman 9lter. From a broad perspective this paper essentially shows how to build on Zadeh’s seminal ideas in fuzzy sets,
logic, and systems and use Kalman’s seminal ideas on optimal estimators to construct a novel fuzzy estimator for non-linear
estimation problems. While this seems to reconcile some of the fundamental ideas of Zadeh and Kalman it is unfortunate that
the fuzzy estimator can be very computationally complex to implement for practical applications. c 2001 Elsevier Science
B.V. All rights reserved.
Introduction
In control system and estimator design, the establishment
of a system model is quite fundamental.
Many traditional approaches assume that models are
available in the form of ordinary di#erential equations
or discrete-time di#erence equations. In the
cases where there exists uncertainty the models are
generally assumed to be in the form of stochastic dif-
if the process is highly complex. Even when models
are available it is sometimes di4cult to propagate the
uncertainty through the system dynamics according to
the axioms of probability theory, especially for nonlinear
systems. Therefore, in this article, we examine
the use of an alternative yet practical framework
of fuzzy dynamic system models for dealing with
systems exhibiting uncertain behavior.
Linear Gaussian case
In this section we consider linear systems with
Gaussian-shaped membership functions. We will
show that the fuzzy estimator described above will
produce a result that is analogous to Kalman 9lter in
the sense that the “center” and “spread” of the membership
functions propagate the same as the mean and
covariance of the Kalman 9lter.
Non-linear non-Gaussian case
Using fuzzy composition on discretized membership
functions it is possible to implement a fuzzy
estimator for practically any shaped membership function
and any non-linear system. However, this is done
with a signi9cant increase in computational complexity.
The reason is due to the fact that we are propagating
all elements of the discretized membership
function instead of a few parameters of the function
such as the “center” and “spread” as was done for the
linear Gaussian case.
The goal of this section is to identify membership
functions that maintain their basic shape when propagated
in a fuzzy dynamic system when the underlying
dynamics are linear (such as the case of the Gaussianshaped
membership functions) or non-linear. We will
show in this section that the uniform shaped membership
function exhibits this property.