Robust Estimation Design for Unknown Inputs Fuzzy Bilinear Models: Application to Faults Diagnosis - Complex System Modelling and Control Through Intelligent Soft Computations - page 659

PTG i Z i þ a Z i þ I l

0 ð 75 Þ

\

a I

Z i T PTA i ¼ þ U i C 0 ð 76 Þ

V i C PTN i ¼ 0 ð 77 Þ

The observer gains are determined by:

P 1 Z i

H i ¼ ð 78 Þ

P 1 U i L i ¼ ð 79 Þ

P 1 V i M i ¼ ð 80 Þ

J i ¼ TB i ð 81 Þ

where T, C 2 are given in ( 74 ), ( 71 ), ( 73 ), respectively.

Proof In order to establish the stability of the estimation error e

C 1 and

ð Þ

, let us consider
the following Lyapunov function ( 27 ). Using ( 65 ), the derivative of the Lyapunov
function ( 27 ) is given by

t

X

r

e

V

e T H i P 2e T PTG i f

ð e ð t ÞÞ ¼ h i ðnð t ÞÞ ð t Þ þ PH i ð t Þþ ð t Þ ð t Þ ð 82 Þ

i ¼ 1

If k f ð t kf l , then the derivative of V ð e ð t ÞÞ becomes:

ÞÞ P

r

V

PTG i k

e T H i P e T

ð e ð t h i nð ð t Þ Þ ½ ð t Þð þ PH i Þ e ð t Þþ 2 lk ð t Þ

ð 83 Þ

i¼1

For any positive scalar a , we have the following inequality:

PTG i ka 1 PTG i k

2

e T

2

e T

ð 84 Þ

2 lk ð t Þ l k ð t Þ þ a

Hence, one obtains

ÞÞ P

r

V

PTG i k

2

e T H i P Þþa 1

2

e T

ð e ð t h i nð ð t Þ Þ ½ ð t Þð þ PH i Þ e ð t l k ð t Þ þ a

ð 85 Þ

i¼1

V ð e ð t ÞÞ P

r

2 PTG i G i T T P Þ e ð t Þþa

Þ ½e T ð t Þð H i P þ PH i þ a 1

h i nð t Þ ð l

ð 86 Þ

i¼1