Robust Estimation Design for Unknown Inputs Fuzzy Bilinear Models: Application to Faults Diagnosis - Complex System Modelling and Control Through Intelligent Soft Computations - page 649

Let us denote nð

Þ

the estimate of the decision variables dependent estimated
state variables x ð t Þ . The system ( 2 ) can be rewritten as:

t

8
<

0 1

A i xt ðÞþ B i ut ðÞþ F i d ð t Þ

ðÞ ¼ P

r

@ A

xt _ h i ^ ðÞ x ð t þ N i xt ðÞ ut ðÞþDð t Þ

ð Þ

34

:

i¼1

yt ðÞ ¼ Cx t ðÞ

where Dð t Þ acts like a disturbance on the dynamic of the fuzzy bilinear model and is

de ned by:

0
@

1
A

A i xt ðÞþ B i ut ðÞ

X

r

Dð t Þ ¼

1 ð

h i ð x ð t ÞÞ h i ð^ x ð t ÞÞÞ þ N i xt ðÞ ut ðÞþ F i d ð t Þ ð 35 Þ

i¼

The dynamic of the state estimation error becomes:

X

r

e _ ð t Þ ¼ h i ð^ x ð t ÞÞ ð H i e ð t Þþ T Dð t Þ Þ ð 36 Þ

i¼1

Assumption 1 kDð t Þk ck e ð t Þk , i.e. Dð t Þ is Lipschitz in i.e. ð t Þ where c is a positive

scalar.

This assumption, used in previous works (Bergsten et al. 2002 ; Khalil 1996 ),
will be useful to derive design conditions. The following lemma will be also used
(Boyd et al. 1994 ).

Lemma 1 For any matrices X and Y with appropriate dimensions, the following
property holds for any positive scalar e :

X T Y Y T X X T X

þ e 1 Y T Y

þ e ð Þ

37

cient design conditions for fuzzy bilinear
models subjects to unknown inputs ( 2 ) with the decision variable is unmeasurable.

The following theorem gives suf

Theorem 2 For a given 0, the fuzzy bilinear observer ( 5 ) converges asymp-

totically to the state of the fuzzy bilinear model ( 2 ), if there exist a symmetric
de

c [

nite positive matrix P, matrices Wi, i , V i , S, R i and scalar e such that the following

8 ...

linear conditions hold i ¼ 1 r:

T

P þ P þ ec

2 IP

þ SC

0 ð 38 Þ

\

e I

R i ¼ ð P þ SC Þ B i ð 39 Þ