Robust Estimation Design for Unknown Inputs Fuzzy Bilinear Models: Application to Faults Diagnosis - Complex System Modelling and Control Through Intelligent Soft Computations - page 651

where W i C.

Then, the derivative of the Lyapunov function is negative if

P ¼ ð P þ SC Þ A i

T

T 2 I

þ e 1

P þ P þ ec ð P þ SC Þð P þ SC Þ 0 ð 51 Þ

\

By using the Schur complement on inequality ( 51 ), we get ( 38 ).
Taking into account ( 9 ) and ( 30 ), equality ( 41 ) is derived from ( 13 ).
Similarly, using the following variable change

R i ¼ PJ i ð 52 Þ

V i ¼ PM i ð 53 Þ

we get equalities ( 39 ) and ( 40 ) from ( 12 ) and ( 11 ) respectively. Which ends the
proof.

3.3 Illustrative Example

In this subsection, we apply the proposed method to design a fuzzy bilinear
observer for a continuous fuzzy bilinear models subjects to unknown inputs and
unmeasurable premise variables. Consider FBM ( 2 ) with the following data:

2
4

3
5 ;

2
4

3
5

6 10 40 1

A 1 ¼ A 2 ¼ 5 10

01 2

5 20

00 1

2
4

3
5 ;

2
4

3
5

1
0
0

0 : 6

B 1 ¼ B 2 0

1

2
4

3
5

0 : 500

N 1 ¼ N 2 ¼ 0 0 0

:

0 0 0 5

2
4

3
5

0
0 F 1 ¼ F 2 ¼ : 5

0

0 : 2 10

C ¼

0 0 : 11

The case study is envisaged for fuzzy bilinear models subjects to unknown
inputs where the decision variable is unmeasurable. Let consider the weighting