Robust Estimation Design for Unknown Inputs Fuzzy Bilinear Models: Application to Faults Diagnosis - Complex System Modelling and Control Through Intelligent Soft Computations - page 648

P 1 V i M i ¼ ð 24 Þ

Þ A i P 1 W i C

H i ¼ I n þ EC ð ð 25 Þ

P 1 W i L i ¼ H i E ð 26 Þ

Proof Let us consider the following Lyapunov function

e T P T

V ð e ð t ÞÞ ¼ ð t Þ Pe ð t Þ; P ¼ 0 ð 27 Þ

[

Using ( 17 ), the derivative of the Lyapunov function ( 27 ) is given by

X

r

e

V

T P

e T

ð e ð t ÞÞ ¼ h i ðnð t ÞÞ ð t Þð TA i K i C Þ þ P ð TA i K i C Þ ð t Þ ð 28 Þ

i¼1

Stability condition for the estimation error yields to that the time derivative of
the Lyapunov function should be negative de

ne over ( 3 ). Then, one has:

T P

ð TA i K i C Þ þ P ð TA i K i C Þ \ 0 ð 29 Þ

Taking into account ( 9 ) and considering the variables change:

S ¼ PE ð 30 Þ

W i ¼

ð Þ

PK i 31

we get the LMI ( 18 ). Taking into account ( 9 ) and ( 30 ), equality ( 21 ) is derived from
( 13 ).
Similarly, using the following variable change

R i ¼ PJ i ð 32 Þ

V i ¼ PM i ð 33 Þ

we get equalities ( 19 ) and ( 20 ) from ( 12 ) and ( 11 ) respectively. Which ends the proof.

3.2.2 Observers Design with Unmeasurable Decision Variables

In this section, the design of a robust estimation for fuzzy bilinear models with
unmeasurable premise variables is proposed. This case, reputed very dif cult, is

important in diagnosis method based on observer banks to detect and isolate
actuator and/or sensor faults.