Robust Estimation Design for Unknown Inputs Fuzzy Bilinear Models: Application to Faults Diagnosis - Complex System Modelling and Control Through Intelligent Soft Computations - page 668

The considered system can be described using the T-S fuzzy bilinear model of
predator-prey model as follows:

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

4

x _ ð t h i ð^ x ð t ÞÞ þ F i d ð t Þþ G i f ð t Þþdð t Þ

i¼ 1

:

y ð t Þ ¼Cx ð t Þ

with A i , B i , N i , F i , C are the same previous matrices and

0 : 5

G 1 ¼ G 2 ¼ G 3 ¼ G 4 ¼

0

The system is subjected of fault f ð t Þ of the following form:

0 : 1 sin 0 : 314t for t 2 ½ 68

ð Þ ¼

f t

0 elsewhere

We can check that the condition ( 70 ) is veri ed. Indeed:

rank ð CF Þ ¼ rank ð F Þ ¼ 1

Then an unknown input fuzzy bilinear fault diagnosis observer given by ( 55 )
exists, and it is given by:

8
<

ðÞ ¼ P

4

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

i ¼ 1

:

^ xt ðÞ ¼ zt ðÞ Ey t ðÞ

rt ðÞ ¼ C 1 yt ðÞC 2 zt ðÞ

We may choose the arbitrary matrix X ¼ 1 to obtain the matrices C 1 and C 2

using ( 73 ) and ( 71 ) and the LMIs ( 90 ) can be ef ciently solved under constraints

( 91 ) and ( 92 ) via numerical approach within the LMI toolbox in order to compute
the gains matrices. Therefore, these inequalities are satis

ed with

0 : 372 0 : 929

10 4 P ¼

0 : 929 2 : 321

The remaining matrices Hi, i , L i , M i , J i can be determined from ( 93 ), ( 94 ), ( 95 ),
( 96 ) respectively.
The robust residual signal response is shown in the following gure.

Figure 11 displays the convergence of the residual corresponding to the fault
signal. One can see that the residual is almost zero throughout the time simulation
run despite the presence of unknown inputs except at time t ¼ 6 s where it appears