154
8 Filtering-Based Robust Fault Detection of Fuzzy Multi-model Jumping System
22 (r ) = P(r )A Fi (r ) + A
T
Fi (r )P(r ) +
N
j=1 π i j P(k) + Q 22 .
Using Schur complements, we have:
S
i=1
h
2
i ii (r ) +
S
i=1
h i
S
i=1
h j
i j (r ) + ji (r )
< 0.
(8.54)
and it can be derived by inequalities (8.48) and (8.49). This completes the proof.
Corollary 8.2 Theorem 8.2 describes the conditions of devising the FDF for the
overall FDF system (8.47). It should be noticed that the coupled LMIs (8.48) and
(8.49) are respect to P(r ), Q 11 , Q 22 , Q 33 , X i (r ), Y i (r ), C Fi (r ), D Fi (r ), Q 12 , Q 13 ,
Q 23 , λ i (r ) and γ
2
. Thus, we can use γ
2 as the optimized value, that is, in order to
obtain an optimal FDF, the attenuation level γ
2 can be reduced to the minimum
possible value while LMIs (8.48) and (8.49) are satisfied. The optimization problem
is represented as:
min P(r ),X i (r ),Y i (r ),C Fi (r ),D Fi (r ),Q,λ i (r ),ρ ρ
s.t. LMIs (8.48 − 8.49) with ρ = γ
2
.
(8.55)
8.2.3 Numeral Example
Considering a tunnel diode circuit [122], which is described by:
⎧
⎪ ⎨
⎪ ⎩
˙
x 1 (t) = −0.1x 1 (t) −
α(i)+α(i)
C
x
2
1 (t)
x 1 (t) + 10x 2 (t),
L ˙
x 2 (t) = −x 1 (t) − Rx 2 (t) + 0.1ω(t),
y(t) = J x(t) + 0.1ω(t),
(8.56)
where x(t) =
x 1 (t)
x 2 (t)
are the states, ω(t) is the unknown input, y(t) is the measured
output and J is the sensor matrix. The parameters in the circuit are C = 20 mF, L =
1 H and R = 10 , J = [10]. Assume that the uncertain mode-dependent parameters α(i) + α(i) are aggregated into three modes shown as α(1) + α(1) =
0.01 ± 10% α(2) + α(2) = 0.02 ± 10% and α(3) + α(3) = 0.03 ± 10%. The
transition rate matrix is defined by:
=
⎡
⎣
−3 1.8 1.2
0.3 −2 1.7
0.3 0.7 −1
⎤
⎦ .
Assuming that x 1 (t) ≤ 3, we can employ the fuzzy model to construct the
following nonlinear circuit model:
8 Filtering-Based Robust Fault Detection of Fuzzy Multi-model Jumping System
22 (r ) = P(r )A Fi (r ) + A
T
Fi (r )P(r ) +
N
j=1 π i j P(k) + Q 22 .
Using Schur complements, we have:
S
i=1
h
2
i ii (r ) +
S
i=1
h i
S
i=1
h j
i j (r ) + ji (r )
< 0.
(8.54)
and it can be derived by inequalities (8.48) and (8.49). This completes the proof.
Corollary 8.2 Theorem 8.2 describes the conditions of devising the FDF for the
overall FDF system (8.47). It should be noticed that the coupled LMIs (8.48) and
(8.49) are respect to P(r ), Q 11 , Q 22 , Q 33 , X i (r ), Y i (r ), C Fi (r ), D Fi (r ), Q 12 , Q 13 ,
Q 23 , λ i (r ) and γ
2
. Thus, we can use γ
2 as the optimized value, that is, in order to
obtain an optimal FDF, the attenuation level γ
2 can be reduced to the minimum
possible value while LMIs (8.48) and (8.49) are satisfied. The optimization problem
is represented as:
min P(r ),X i (r ),Y i (r ),C Fi (r ),D Fi (r ),Q,λ i (r ),ρ ρ
s.t. LMIs (8.48 − 8.49) with ρ = γ
2
.
(8.55)
8.2.3 Numeral Example
Considering a tunnel diode circuit [122], which is described by:
⎧
⎪ ⎨
⎪ ⎩
˙
x 1 (t) = −0.1x 1 (t) −
α(i)+α(i)
C
x
2
1 (t)
x 1 (t) + 10x 2 (t),
L ˙
x 2 (t) = −x 1 (t) − Rx 2 (t) + 0.1ω(t),
y(t) = J x(t) + 0.1ω(t),
(8.56)
where x(t) =
x 1 (t)
x 2 (t)
are the states, ω(t) is the unknown input, y(t) is the measured
output and J is the sensor matrix. The parameters in the circuit are C = 20 mF, L =
1 H and R = 10 , J = [10]. Assume that the uncertain mode-dependent parameters α(i) + α(i) are aggregated into three modes shown as α(1) + α(1) =
0.01 ± 10% α(2) + α(2) = 0.02 ± 10% and α(3) + α(3) = 0.03 ± 10%. The
transition rate matrix is defined by:
=
⎡
⎣
−3 1.8 1.2
0.3 −2 1.7
0.3 0.7 −1
⎤
⎦ .
Assuming that x 1 (t) ≤ 3, we can employ the fuzzy model to construct the
following nonlinear circuit model: