Robust Estimation Design for Unknown Inputs Fuzzy Bilinear Models: Application to Faults Diagnosis - Complex System Modelling and Control Through Intelligent Soft Computations - page 661

P 1 V i M i ¼ ð 95 Þ

J i ¼ TB i

ð Þ

96

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

Proof To prove the convergence of the state estimation error to zero, we consider
the quadratic Lyapunov function ( 27 ), differentiating it along ( 88 ) and using ( 48 ), it
becomes:

C 1 and

X

r

V

e T H i P

2 I

ð e ð t ÞÞ h i ^ ð ½ x ð t ð t if þ PH i þ ec

ð 97 Þ

i¼1

þ e 1 PT T T P PTG i f

2e T g e ð t Þþ ð t Þ ð t Þ

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

X

r

V

e T H i P

2 I

ð e ð t ÞÞ h i ^ ð ½ x ð t ð t if þ PH i þ ec

ð 98 Þ

i ¼ 1

þ e 1 PT T T P PTG i k

e T

g e ð t Þþ 2 lk ð t Þ

Using ( 84 ), the last inequality ( 98 ) becomes 8 i ¼ 1 ... r

X

r

V

e T H i P

2 I

ð e ð t ÞÞ h i ^ ð ½ x ð t ð t if þ PH i þ ec

ð 99 Þ

i¼1

þ e 1 PT T T P PTG i k

2

Þþa 1

2

e T

g e ð t l k ð t Þ þ a

8 ¼ ...

Hence, one has i 1 r

X

r

V

e T H i P

2 I

ð e ð t ÞÞ h i ^ ð ½ x ð t ð t if þ PH i þ ec

ð 100 Þ

i¼1

þ e 1 PT T T P

2 PTG i G i T T P

þ a 1 l g e ð t Þþa

V

The stability condition ð e ð t ÞÞ \ 0 is veri ed if 8 i ¼ 1 ... r

þ e 1 PT T T P

H i P

2 I

þ

PH i þ ec

ð Þ

101

þ a 1

2 PTG i G i T T P

l þ a I 0

\

Then apply the Schur complement to the condition ( 101 ) with change of

variables Zi i ¼ PH i , we get the linear matrix inequality ( 90 ) and with change of

variables Ui i ¼ PL i , V i ¼ PM i , we get the equalities ( 91 ) and ( 92 ). Thus, the proof is

completed.