Robust Estimation Design for Unknown Inputs Fuzzy Bilinear Models: Application to Faults Diagnosis - Complex System Modelling and Control Through Intelligent Soft Computations - page 658

or equivalently

C 1 CF ¼ 0 ; F ¼½ F 1 ; F 2 ; ...; F r ð 69 Þ

If the condition ( 70 ) is satis ed,

ð Þ ¼ ð Þ ð Þ

rank CF rank F 70

Then, we get

Þ þ C 1 ¼ X I p CF ð CF ð 71 Þ

Þ þ is the pseudo inverse of CF, and where ð CF X is an arbitrary matrix.

Substituting Eq. ( 71 )in( 64 ) leads to

Þ þ C C 2 T X CI n F ð CF ¼ 0 ð 72 Þ

A suitable choice of E 1 and T satisfying the relation ( 72 )is

C 2 ¼ X C ð 73 Þ

and

T

Þ þ C ¼ I n F ð CF ð 74 Þ

4.2 Design and Stability Analysis

This subsection proposes suf cient linear design conditions to guarantee the global

asymptotic convergence of state estimation error. Therefore, the design problem is
studied for the two cases; when the decision variables are available or unavailable.

4.2.1 Fault Detection Observer Design with Measurable Decision
Variables

The stability of the dynamic estimation error ( 65 ) is given by the following
theorem.

Theorem 3 The residual generator ( 55 ) converges asymptotically to the state of
the fuzzy bilinear model ( 54 ), if the fault f ð t Þ satis es k f ð t kf l , l [ 0 and if there

a

exist a symmetric de nite positive matrix P, matrices Zi, i , V i , U i and positive scalar

such that the following linear conditions hold 8 i ¼ 1 ... r: