
However, it is well-recognised that there is no universal
method modelling any type of nonlinearity in practice.
Different nonlinear models have been introduced by
various nonlinear modelling techniques, such as affine
nonlinear systems, switched LPV systems, ANN-based
nonlinear systems and fuzzy systems and so forth
(Takagi and Sugeno 1985; Ying 2000; Lu and Wu 2004;
Popescu and Dumitrache 2005; Fekih, Xu, and
Chowdhury 2007; Wingerden and Verhaegen 2009;
Nishi and Ishitobi 2010). Therefore, in view of the
complexity of nonlinear dynamics, it is very difficult to
solve the fault estimation of nonlinear systems totally
(Mattone and Luca 2006). In addition, the performance
requirements of fault estimation, which is useful for
fault accommodation, are less considered in the
literature (Tsai et al. 2008; Zhang et al. 2010b).
Recent years have witnessed rapidly growing inter-
est in observer-based fault estimation of Takagi–
Sugeno (T-S) fuzzy nonlinear models (Tuan,
Apkarian, Narikiyo, and Yamamoto 2001; Marx,
Koening, and Ragot 2007; Gao, Shi, and Ding 2008;
Chadli, Akhenak, Ragot, and Maquin 2009; Jiang,
Gao, Shi, and Xu 2010; Khedher, Othman, Benrejeb,
and Maquin 2010; Zhang, Jiang, and Staroswiecki
2010a), because it is conceptually simple and provides a
theoretical foundation in describing some complex
nonlinear systems (Takagi and Sugeno 1985; Feng
2006; Zhang and Xia 2009; Zhao, Lam, and Gao 2009).
Unfortunately, many methods adopted the linear
matrix inequality (LMI)-based observer design accord-
ing to the conventional common quadratic Lyapunov
function or multiple fuzzy Lyapunov function. As a
result, some limitations have not been overcome so far.
For example, only limited results touched upon the fault
estimation performance such as the robustness against
disturbances and the responsive speed of fault estima-
tion as well, which is attractive in fault accommodation
and extensively developed for linear systems (Zhang
et al. 2010a,b). Another is the conservatism of the
existing LMI-based design. Particularly for T-S fuzzy
systems with more rules, the conventional method and
too many LMIs constraints are often too fragile to
obtain feasible solutions (Sala, Guerra, and Babuska
2005). Therefore, fault estimation for T-S fuzzy systems
is still challengeable and deserves in-depth study.
Recently, some less conservative strategies for the
stability analysis and control of T-S fuzzy systems
have been reported in the literature (Tuan et al. 2001;
Tanaka, Hori, and Wang 2003; Delmotte, Guerra, and
Kruszewski 2008), which make the LMI-based fault
estimation observer with less conservatism and
improved performance possible.
Motivated by the aforementioned observations, in
this article, the fault estimation problem is investigated
for a class of discrete-time T-S fuzzy systems. A kind of
on-line fault estimator is proposed based on fuzzy
observer. Attentions are focused on the fuzzy observer-
based fault estimator (FOFE) with multiple perfor-
mance constraints and its less conservative design
method. Multiobjective optimisation and H
1
optimi-
sation strategies (Scherer, Gahinet, and Chilali 1997;
Tanaka and Wang 2001a) are adopted to deal with the
performance constraints on the regional poles and the
H
1
robustness performance, so that the resulting
estimator is of stability with the prescribed dynamic
behaviour, the robust margin to neglected modelling
dynamics, the expected robustness against external
disturbances and the desired estimation performance
to fault signals. The technique in Delmotte et al. (2008)
and equivalent transforms of matrix inequalities are
utilised to render the results less conservative and less
computing. For comparison, the conventional multi-
objective optimisation method with common quadratic
Lyapunov function and relaxing strategy is also
presented. All the results are formulated in the LMIs
form.
The remainder of this article is arranged as follows:
Section 2 formulates the problem under consideration.
Some related preliminaries are presented in Section 3.
Two FOFEs design methods are introduced in Sections
4 and 5, respectively. Simulative examples are illustrated
in Section 6 to show the validity of the proposed
approach, and this article is concluded in Section 7.
The notations used throughout this article are fairly
standard. The superscript ‘T’ stands for matrix
transposition, R
n
denotes the n-dimensional
Euclidean space, 0 represents the zero matrix with
appropriate dimensions, the notation P > 0(<0) means
that P is real positive (negative) definite. In symmetric
block matrices or complex matrix expressions, we use
an asterisk (*) to represent a term that is induced by
symmetry and a diag{...} to denote a block-diagonal
matrix. In addition, the 2-norm of vector x(t) is noted
by x
kk
¼
4
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x
T
ðtÞxðtÞ
p
: The quasi-L
2
norm over a finite-
time interval [0, t
d
] is defined as
x
kk
t
d
¼
4
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
X
t
d
i¼0
x
T
ðtÞxðtÞ
v
u
u
t
,or x
kk
2
t
d
¼
4
X
t
d
i¼0
x
T
ðtÞxðtÞ:
2. System description and problem formulation
We consider a class of discrete-time fuzzy nonlinear
systems which can be described by the following T-S
fuzzy model with the ith rule:
Rule i (i ¼ 1, 2, ..., p):
If
1
ðtÞ is M
i1
and and
r
ðtÞ is M
ir
Then
x
i
ðt þ 1Þ¼A
i
x
i
ðtÞþB
i
uðtÞþE
i
!ðtÞþD
i
fðtÞ,
y
i
ðtÞ¼C
i
x
i
ðtÞ,
ð1Þ
630 D. Zhang et al.
Downloaded by [Central South University] at 00:39 26 November 2012