116 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 40, NO. 1, FEBRUARY 2010
Set-Membership Fuzzy Filtering for Nonlinear
Discrete-Time Systems
Fuwen Yang, Senior Member, IEEE, and Yongmin Li, Senior Member, IEEE
Abstract—This paper is concerned with the set-membership
filtering (SMF) problem for discrete-time nonlinear systems. We
employ the Takagi–Sugeno (T-S) fuzzy model to approximate the
nonlinear systems over the true value of state and to overcome the
difficulty with the linearization over a state estimate set rather
than a state estimate point in the set-membership framework.
Based on the T-S fuzzy model, we develop a new nonlinear SMF
estimation method by using the fuzzy modeling approach and the
S-procedure technique to determine a state estimation ellipsoid
that is a set of states compatible with the measurements, the
unknown-but-bounded process and measurement noises, and the
modeling approximation errors. A recursive algorithm is derived
for computing the ellipsoid that guarantees to contain the true
state. A smallest possible estimate set is recursively computed
by solving the semidefinite programming problem. An illustrative
example shows the effectiveness of the proposed method for a class
of discrete-time nonlinear systems via fuzzy switch.
Index Terms—Convex optimization, linear set-membership
filtering (SMF), nonlinear SMF, unknown-but-bounded noise,
Takagi–Sugeno (T-S) fuzzy model.
I. INTRODUCTION
T
HE filtering problem for nonlinear systems remains chal-
lenging and has been attracting considerable research
interests over the past four decades. Since the time evolution
of the probability density of the state vector conditional on the
measurements cannot directly be calculated in most nonlinear
cases [2], various approximation methods have been developed
in the literature [1], [4], [16], [24], [32], [36]. For nonlinear
systems with Gaussian noises, the extended Kalman filtering
(EKF) method was used for state estimation, which applied the
linear Kalman filtering theory by linearization of the nonlinear
systems around the current estimate [16], [24]. However, the
EKF may bring large errors in the true posterior mean and
covariance and even diverge if the linearization error is not
sufficiently small. These drawbacks have been overcome by
unscented Kalman filtering (UKF) by using a deterministic
Manuscript received September 17, 2008; revised January 27, 2009 and
February 17, 2009. First published July 21, 2009; current version published
October 30, 2009. This work was supported in part by the Engineering
and Physical Sciences Research Council (EPSRC) of U.K. under Grant
EP/C007654/1 and in part by the National Nature Science Foundation of China
under Grant 60874059 and Grant 60604027. This paper was recommended by
Associate Editor H. Gao.
F. Yang is with the School of Information Science and Engineering, East
China University of Science and Technology, Shanghai 200237, China. He was
with the Department of Information Systems and Computing, Brunel Univer-
sity, UB8 3PH Middlesex, U.K. (e-mail: fwyang@ecust.edu.cn).
Y. Li is with the Department of Information Systems and Computing, Brunel
University, UB8 3PH Middlesex, U.K.
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSMCB.2009.2020436
sampling approach to capture the mean and covariance esti-
mates with a minimal set of sample points [36]. Recently, a
gain-constrained UKF has been developed for nonlinear sys-
tems [47]. For nonlinear systems with non-Gaussian noises,
a Gaussian sum approach has been proposed for state esti-
mation by density approximation [1]. In this algorithm, the
conditional densities are approximated by a sum of Gaussian
density functions [32]. An alternative is particle filtering, which
is also known as sequential Monte Carlo method [4], which is
a sophisticated estimation technique based on simulation. The
basic idea of the particle filter is to use a number of independent
random variables called particles, which are directly sampled
from the state space, to represent the posterior probability
and update the posterior by involving the new observations
according to the Bayesian rule. However, its computation is
very demanding.
The above nonlinear filtering approaches require the system
noises, including process noise and measurement noise in a
stochastic (Gaussian or non-Gaussian) framework, and then
provide a probabilistic state estimation [10], [39]–[41]. The
probabilistic nature of the estimates leads to the use of mean
and variance to describe the state spreads (distributions). These
spreads cannot guarantee that the state is included in some re-
gion, because they are not hard bounds. However, in many real-
world applications, such as target tracking, system guidance,
and navigation, 100% confidence is required for state estima-
tion. This has motivated the development of an ellipsoidal state
estimation. The idea of the ellipsoidal state estimation is to
provide a set of state estimates in state space, which always
contain the true state of the system by assuming hard bounds
on the noise signals (unknown but bounded noises) instead of
stochastic descriptions on the system noises [3], [14], [27].
The actual estimate is a set in state space rather than a single
vector. These methods are, therefore, known as set-membership
or set-valued state estimation (filtering) [3], [27], [38]. We
adopt the name set-membership filtering (SMF) in this paper
as it is easy to distinguish between a set estimation and a point
estimate.
Most publications on SMF deal with linear systems [7], [9],
[13], [14], [18], [19], [22], [23], [25], [28]. Only a few consider
nonlinear systems [20], [26], as it is not straightforward to use
the EKF method where the nonlinear dynamics are linearized
around a state estimate point by a first-order Taylor series
approximation. In the set-membership framework, linearization
should best fit the nonlinear functions over a state estimate set
rather than a state estimate point. An approximation method
over the entire estimate set has been proposed by minimizing
the weighted squared errors between the function values and the
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