Robust Estimation Design for Unknown Inputs Fuzzy Bilinear Models: Application to Faults Diagnosis - Complex System Modelling and Control Through Intelligent Soft Computations - page 660

V

The stability condition ð e ð t ÞÞ \ 0 is veri ed if:

PH i þ a 1

2 PTG i G i T T P

H i T P

þ l þ a I 0 ð 87 Þ

\

Then apply the Schur complement to the condition ( 87 ) with change of variables
Z i ¼ PH i , we get the linear matrix inequality ( 75 ) and with change of variables

U i ¼ PL i , V i ¼ PM i , we get the equalities ( 76 ) and ( 77 ). Thus, the proof is

completed.

4.2.2 Fault Detection Observer Design with Unmeasurable Decision
Variables

For the case where the decision variable of the weighting function depends of an
unmeasured state variable, the equation of the observing error e ð t Þ and residual r ð t Þ

becomes

X

r

Þþ TG i f Þþ T

_ ð Þ ¼

h i ð^

ð ÞÞð ð ð Dð ÞÞ ð Þ

e t x t H i e t t t 88

i¼1

r ð t Þ ¼ C 2 e ð t Þ ð 89 Þ

The following theorem gives linear conditions to design fault detection observer
design with unmeasurable decision variables.

Theorem 4 For a given c [ 0, the residual generator ( 55 ) converges asymptoti-

cally to the state of the fuzzy bilinear system ( 54 ), if the fault f ð t Þ satis es

k f ð t Þkl , l [ 0 and if there exist a symmetric de nite positive matrix P, matrices

Z i , V i , U i and real parameters e and a such that the following linear conditions hold

8 i ¼ 1 ... r:

2
4

3
5 \

IPT PTG i Z i þðec

2

Z i þ þ aÞ l

e I 0 0 ð 90 Þ

a I

Z i T PTA i ¼ þ U i C 0 ð 91 Þ

PTN i ¼

V i C 0 ð 92 Þ

The observer gains are determined by:

P 1 Z i

H i ¼ ð 93 Þ

P 1 U i L i ¼

ð Þ

94