Robust Estimation Design for Unknown Inputs Fuzzy Bilinear Models: Application to Faults Diagnosis - Complex System Modelling and Control Through Intelligent Soft Computations - page 657

The dynamic estimation error is then described by:

Þ ¼T e _ ð t x _ ð t Þ_ z ð t Þ ð 57 Þ

which is equivalent to

0 1

Þþ TA i H i T

H i e ð t ð L i C Þ x ð t Þ

X

r

@

þ TN i Þþ TB i

A

ð M i C Þ x ð t Þ u ð t ð J i Þ u ð t Þ

e _ ð t Þ ¼ h i ðnð t ÞÞ ð 58 Þ

þ TF i d Þþ TG i f ð t ð t Þ

i¼ 1

From ( 55 ), and by using the estimation error ( 56 ), the general expression of the
residual vector can be written as:

C 2 T

r ð t Þ ¼ C 2 e ð t Þþ C 1 C ð Þ x ð t Þ ð 59 Þ

If the following conditions hold true 8 i ¼ 1 ... r:

TA i

H i T L i C ¼ 0 ð 60 Þ

TN i

M i C ¼ 0 ð 61 Þ

TB i

J i ¼ 0 ð 62 Þ

TF i ¼

0 ð 63 Þ

C 1 C C 2 T ¼ 0

ð 64 Þ

Then the equation of the observing error e ð t Þ and residual r ð t Þ becomes

X

r

Þþ TG i f e _ ð t Þ ¼ h i ðnð t ÞÞð H i e ð t ð t ÞÞ ð 65 Þ

i¼1

r ð t Þ ¼ C 2 e ð t Þ ð 66 Þ

Multiplying ( 64 )byF i , we get:

C 1 CF i C 2 TF i ¼

0 ð 67 Þ

Taking into account the constraint ( 63 ), Eq. ( 67 ) yields:

C 1 CF i ¼ 0 ; i ¼ 1 ; ...; r ð 68 Þ