Robust Estimation Design for Unknown Inputs Fuzzy Bilinear Models: Application to Faults Diagnosis - Complex System Modelling and Control Through Intelligent Soft Computations - page 647

1. when the decision variable of the weighting function depends of a measured
variable.
2. when this variable depends of an unmeasured variable.

3.2.1 Observers Design with Measurable Decision Variables

The design of the fuzzy bilinear observer ( 5 ) is reduced to satisfy the constraints
( 10 ) ( 13 ) by taking into the stability of the observing error ( 14 ). In order to

establish the gains matrices of the FBO, the substitution of ( 9 ) into ( 10 ) yields to:

-

H i ¼ TA i K i C ð 15 Þ

with

K i ¼ H i E þ L i ð 16 Þ

From the dynamic state estimation error ( 14 ) and using ( 15 ), it becomes

X

r

_ e ð t Þ ¼ h i ðnð t ÞÞðð TA i K i C Þ e ð t ÞÞ ð 17 Þ

i¼1

cient linear conditions and the gains deter-
mination of the unknown input fuzzy bilinear observer ( 2 ) with measurable decision
variables.

The following result gives the suf

Theorem 1 If there exist a symmetric de nite positive matrix P, and matrices Wi, i ,

V i , S, R i such that the following linear conditions hold 8 i ¼ 1 ... r

T

ð ð P þ SC Þ A i W i C Þ þ ð P þ SC Þ A i W i C 0 ð 18 Þ

\

R i ¼ ð P þ SC Þ B i ð 19 Þ

V i C ¼ ð P þ SC Þ N i ð 20 Þ

ð P þ SC Þ F i ¼ 0 ð 21 Þ

then the state estimation of the fuzzy bilinear observer ( 5 ) converges globally and
asymptotically to the state of the fuzzy bilinear model ( 2 ). The observer gains are
determined by:

P 1 S E ¼ ð 22 Þ

J i ¼ P 1 R i ð 23 Þ