Robust Estimation Design for Unknown Inputs Fuzzy Bilinear Models: Application to Faults Diagnosis - Complex System Modelling and Control Through Intelligent Soft Computations - page 656

4 Robust Fault Detection in Fuzzy Bilinear Models

In this section, the design of a residual generator based on unknown input observer
for fuzzy bilinear models is developed. A more general situation is analyzed since
both unknown input and faults are envisaged. Thus, we consider a fuzzy bilinear
system affected by an actuator fault vector f ð t Þ2<

nf . A residual generator is then
synthesized such that it is sensitive to fault vector f ð t Þ

and insensitive to the

unknown inputs d ð t Þ .

4.1 Problem Formulation

The considered fuzzy bilinear model ( 2 ) subject to unknown inputs d ð t Þ and

affected by a fault vector f ð t Þ is described by the following equation:

8
<
:

0
@

1
A

A i x ð t Þþ B i u ð t Þþ N i x ð t Þ u ð t Þ

Þ ¼ P

r

_

x ð t h i ðnð t ÞÞ þ F i d ð t Þþ G i f ð t Þ

ð 54 Þ

i¼1

y ð t Þ ¼ Cx ð t Þ

where f ð t Þ represents the vector of faults and the Gi i represents matrix with
appropriate dimensions.
The global residual generator is de ned by:

8
<

ðÞ ¼ P

r

zt _ h i nð ð t Þ Þ ð H i z þ L i yt ðÞþ J i ut ðÞþ M i yt ðÞ ut ðÞ Þ

i¼1

^ xt ðÞ

r ðÞ ¼ C 1 y ðÞC 2 z ðÞ

ðÞ ¼ zt ðÞ Ey t ð 55 Þ

:

where z(t) represents the estimated vector, and r(t) being the output signal called the
residual.
The residual generator design is reduced to determine the gain matrices
H i ; M i ; L i ; J i ; E ; C 1 and C 2 such that the state estimate x ð t Þ converges asymptotically

to system state x . Then, to analyze the convergence of the residual generator, let

consider the estimation error from ( 55 )to( 54 ) such that

ð t Þ

Þ ¼Tx e ð t Þ ¼ x ð t Þ^ x ð t ð t Þ z ð t Þ ð 56 Þ

where T ¼ I n þ EC.