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sensor fault estimation. For T-S fuzzy bilinear models, Saoudi et al. (
2010
) proposed
an observer design method using iterative procedure and extension of this approach
in discrete-time case was developed in Saoudi et al. (
2012b
). By against, in Saoudi
et al.
(
2012a
)
the proposed design is given in LMI
formulation solved
simultaneously.
Despite numerous works available, none of them seem able to de
ne an LMI
formulation for the problem of state estimation for T-S fuzzy models with
unmeasurable premise variables. Few works are dedicated to the use of these
models for state estimation (Ichalal et al.
2009
; Saoudi et al.
2014
) and for fault
estimation (Marx et al.
2007
). In Bergsten et al. (
2002
), the authors proposed the
Thau-Luenberger observer which is an extension of the classical Luenberger
observer. In Yoneyama (
2009
), the authors proposed a
filter estimating the state and
minimizing the effect of disturbances. Recently, other approaches for observer
design, fault diagnosis and fault tolerant control, have been proposed for this class
of systems in Ichalal et al. (
2012
). But, these results were only obtained for ordinary
nonlinear systems, this chapter addresses the state estimation for fuzzy bilinear
models with unmeasurable decision variables.
The chapter deals with fuzzy bilinear observers design for a class of nonlinear
system in the case of measurable decision variables and in the case of unmeasurable
decision variables. The nonlinear system is modeled as a fuzzy bilinear model. This
kind of T-S fuzzy model is especially suitable for a nonlinear system with a bilinear
term. The considered bilinear observer is obtained by a convex interpolation of
unknown input bilinear observers. Based on Lyapunov theory, the synthesis con-
ditions of the given fuzzy observer are expressed in LMI terms for the two cases.
The design conditions lead to the resolution of linear constraints easy to solve with
existing numerical tools. The given observer is then applied for fault diagnosis. So,
this chapter brings some results for the state estimation and fault detection dedicated
to fuzzy bilinear models with measurable and unmeasurable decision variables.
The remainder of the chapter is organized as follows: Sect.
2
presents the general
structure of a fuzzy bilinear model with unknown input for continuous-time. In
Sect.
3
, the proposed structure of fuzzy bilinear observer with measurable and
unmeasurable decision variables and design conditions are developed. Section
4
is
devoted to the problem of fault diagnosis by using unknown input fuzzy bilinear
observer for fuzzy bilinear models. A predator-prey model is provided in Sect.
5
to
show the effectiveness of the proposed approach. A conclusion
finishes the chapter.
Notations. Throughout the chapter, the following useful notations are used:
<
denotes the set of real numbers, I denotes the identity matrix of the appropriate
dimension, X
T
denotes the transpose of the matrix X, X > 0 denotes symmetric
nite matrix, X
1
denotes the Moore-Penrose inverse of X, X
þ
denotes
positive de
denotes symmetric
A
BC
the pseudo inverse of X such that XX
þ
X
¼
X, and
matrix where
ðÞ
¼B
T
.
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