Robust Estimation Design for Unknown Inputs Fuzzy Bilinear Models: Application to Faults Diagnosis - Complex System Modelling and Control Through Intelligent Soft Computations - page 645

lead to a bilinear transformation of the nonlinear model without any approximation
error, and in another hand to reduce the number of local models compared to other
methods (Li and Tsai 2007 ).

In the following section, a method is proposed to design a fuzzy bilinear
observer for fuzzy bilinear models subjects to unknown inputs.

3 Robust Estimation in Fuzzy Bilinear Models

In real life, all the states of the system are not always observed bilinear. Hence, we
need to estimate the states by an observer. Here we de ne a fuzzy observer and

make an analysis of the error system, from which we provides a design method of a
fuzzy observer for the fuzzy bilinear model ( 2 ).

3.1 Problem Formulation

In this subsection, the design of a robust observer for fuzzy bilinear models is
developed. In order to estimate the state of the unknown input fuzzy bilinear model
( 2 ), the considered unknown input observer structure has the following form:

R i

: if n 1 t ðÞ is F i1 and ... and n g t ðÞ is F ig

then

z ð t Þ ¼H i z ð t Þþ L i y ð t Þþ J i u ð t Þþ M i y ð t Þ u ð t Þ
^

ð 4 Þ

x ð t Þ ¼ z ð t Þ Ey ð t Þ

The overall FBO can be represented by:

<

Þ ¼ P

r

z _ ð t h i ðnð t ÞÞ ð H i z ð t Þþ L i y ð t Þþ J i u ð t Þþ M i y ð t Þ u ð t Þ Þ

ð 5 Þ

i ¼ 1

:

x ^ ð t Þ ¼ z ð t Þ Ey ð t Þ

n is the estimated state vector. Hi, i ,
M i , L i , J i and E are constant matrices with appropriate dimensions. Our objective is
to determine the gains of the observer ( 5 ) such that the state estimation error e
(t) converges towards zero when t

n is the observer state and x ð t Þ2<

where z ð t Þ2<

!1 . Let de ne

Þ ^

e ð t Þ ¼ x ð t x ð t Þ ð 6 Þ