Robust Estimation Design for Unknown Inputs Fuzzy Bilinear Models: Application to Faults Diagnosis - Complex System Modelling and Control Through Intelligent Soft Computations - page 644

2 General Structure of Fuzzy Bilinear Model

Fuzzy bilinear models based on the T-S fuzzy model with bilinear rule consequence
are de ned by extending the T-S fuzzy ordinary model. It is proved that often

nonlinear behaviors can be approximated by T-S fuzzy bilinear description. This
technique is based on the bilinearization of the nonlinear system around some
operating points and using adequate weighting functions. This kind of T-S fuzzy
model is especially suitable for a nonlinear system with a bilinear term. Moreover,
the fuzzy model is described by if-then rules and used to present a fuzzy bilinear
system. The ith rule of the fuzzy bilinear model for nonlinear systems is represented
by the following form:

R i : if n 1 ð t Þ is F i1 and ... and n g ð t Þ is F ig

then

x ð t Þ ¼A i x ð t Þþ B i u ð t Þþ N i x ð t Þ u ð t Þþ F i d ð t Þ
y ð t Þ ¼Cx ð t Þ

ð 1 Þ

where R i denotes the ith fuzzy rule

8 i ¼ f 1 ; ...; r g , r is the number of if-then rules,

is the

n i t ðÞ are the premise variables assumed to be measurable and F ij n j t ðÞ

n

membership degree of n j t ðÞ in the fuzzy set F ij , x ð t Þ2< is the state vector,

q is the unknown input vector and y p is

u ð t Þ2< is the input vector, d ð t Þ2< ð t Þ2<

n n n 1 n n

the system output. The matrices A i 2< ; B i 2< ; N i 2< ; C 2

<

p n

ne the ith local bilinear model and Fi i 2<

n q

are known matrices and de

represent the in uence matrix of the unknown input.

Then, the overall fuzzy bilinear model can be described as follows:

fl

8
<
:

x ð t Þ ¼ P

r

h i ðnð t ÞÞ A i x ðÞþ B i u ðÞþ N i x ðÞ u ðÞþ F i d ð t Þ ð Þ

ð 2 Þ

i¼1

y ð t Þ ¼Cx ð t Þ

where h i ð:Þ verify the following properties

<
:

P

r

h i ðnð t ÞÞ ¼ 1

i ¼ 1

ð 3 Þ

8 i 2 f 1 ; 2 ; ...; r g

0 h i ðnð t ÞÞ 1

assumed to
depend on measurable variables. It can depend on the measurable state variables, be
a function of the measurable outputs of the system and possibly of the known inputs
(Murray-Smith and Johansen 1997 ; Takagi and Sugeno 1985 ).
Remark 1 Matrices A i , B i , N i , and C can be obtained by using the polytopic
transformation (Tanaka et al. 1998 ). The advantage of this method is in one hand to

The activation functions hið:Þ i ð:Þ depend on the decision vector nð t Þ