where A
i
, A
1i
, B
0i
, B
i
, B
1i
, E
i
, C
i
, C
1i
, D
0i
, D
i
, D
1i
, F
i
are known constant
matrices with compatible dimensions, and ΔA
i
ðtÞ, ΔA
1i
ðtÞ, ΔB
0i
ðtÞ,
ΔB
1i
ðtÞ, ΔE
i
ðtÞ, ΔC
i
ðtÞ, ΔC
1i
ðtÞ, ΔD
0i
ðtÞ, ΔD
1i
ðtÞ, ΔF
i
ðtÞ represent the
parameter uncertainties of the system, which are assumed to be of
the form
ΔA
i
ðtÞ ΔA
1i
ðtÞ ΔB
0i
ðtÞ ΔB
1i
ðtÞ ΔE
i
ðtÞ
ΔC
i
ðtÞ ΔC
1i
ðtÞ ΔD
0i
ðtÞ ΔD
1i
ðtÞ ΔF
i
ðtÞ
"#
¼
M
1i
M
2i
"#
G
i
ðtÞ½N
1i
N
2i
N
3i
N
4i
N
5i
; ð10Þ
where M
1i
, M
2i
, N
1i
, N
2i
, N
3i
, N
4i
, N
5i
are known constant matrices
with compatible dimensions, and G
i
(t) are unknown Lebesgue
measurable matrix functions satisfying
G
i
ðtÞ
T
G
i
ðtÞr I; 8 iA S: ð11Þ
The parameter uncertainties
ΔA
i
ðtÞ, ΔA
1i
ðtÞ, ΔB
0i
ðtÞ, ΔB
1i
ðtÞ, ΔE
i
ðtÞ,
ΔC
i
ðtÞ, ΔC
1i
ðtÞ, ΔD
0i
ðtÞ, ΔD
1i
ðtÞ, ΔF
i
ðtÞ are said to be admissible if
both (10) and (11) hold.
Remark 2. The system parameter uncertainties originate from
modeling errors, measurement errors, linearization approxima-
tions, and so on. In fact, it is reasonable and practical that the
model of the system to be controlled almost always contains some
types of uncertainties such as norm-bounded uncertainties or
polyhedral uncertainties [29–32].
Throughout this paper, the nominal system of
Σ is assumed to
be stable, and Markovian process fr
t
g is independent of the
Brownian motion
ϖðtÞ. Our fault detection schemes are concerned
with the construction of a residual generator. For the SMJSs repre-
sented by (1)–(3), we consider the following mode-dependent
fault detection filter designed as the following form:
ð
Σ
c
Þ : dx
c
ðtÞ¼A
ci
x
c
ðtÞ dt þ B
ci
dyðtÞ;
r
c
ðtÞ¼C
ci
x
c
ðtÞ; ð12Þ
where x
c
ðtÞA R
n
is the state vector of the fault detection filter,
r
c
ðtÞA R
l
is the so-called residual signal, and A
ci
; B
ci
; C
ci
are the filter
parameters to be designed.
In order to improve the performance of the fault detection
system, we introduced the weighting fault signal f
w
ðtÞ which
satisfies f
w
ðsÞ¼WðsÞf ðsÞ, where the matrix W(s) presents a given
stable weighting function matrix, f
w
ðsÞ and f ðsÞ denote, respec-
tively, the Laplace transforms of f
w
ðtÞ and f ðtÞ. One minimal state-
space realization of f
w
ðsÞ and f ðsÞ can be
ð
Σ
w
Þ :
_
x
w
ðtÞ¼A
w
x
w
ðtÞþB
w
f ðtÞ;
f
w
ðtÞ¼C
w
x
w
ðtÞþD
w
f ðtÞ;
x
w
ð0Þ¼0; ð13Þ
where x
w
ðtÞA R
k
is the state vector, and A
w
; B
w
; C
w
; D
w
are constant
matrices.
Remark 3. The given stable weighting function matrix W(s)
ensures that the subsystem
Σ
w
is stable. The basic idea behind
the formulation is to design the fault detection filter such that the
influence of the disturbances is limited to a degree and simulta-
neously the difference between the generated residual signal and
the faults to be detected is minimized over a frequency range
which is expressed in terms of the weighting function matrix.
The introduction of weighting fault signal system
Σ
w
results in the
residual signal is robust to disturbances and simultaneously sensi-
tive to faults when the H
1
performance index (19) is guaranteed
[5,24,38]. However, for the problem of robust fault detection filter
design, many researchers usually used two performance indexes
to reflect the sensitivity of residual to faults and the robustness
of residual to disturbances [8,14,17,33]. The two performance
indexes lead to the proof that is very tedious and difficult.
Follow-up researches on the comparison of effectiveness between
the weighting fault signal method and the other robust fault
detection methods including two performance indexes method
should be propelled in the future work.
Denoting e
c
ðtÞ9r
c
ðtÞf
w
ðtÞ and augmenting the model of ðΣÞ
to include the states of ð
Σ
c
Þ and ðΣ
w
Þ, then the overall dynamics of
the fault detection system is governed by ð
~
Σ
c
Þ:
d
ξðtÞ¼½
~
A
i
ðtÞξðtÞþ
~
A
1i
ðtÞHξðt τ
1i
ðtÞÞþ
~
B
i
ðtÞvðtÞ dt
þ
~
E
i
ðtÞHξðt τ
2i
ðtÞÞ dϖ;
e
c
ðtÞ¼r
c
ðtÞf
w
ðtÞ¼
~
C
i
ðtÞξðtÞþ
~
D
i
ðtÞvðtÞ; ð14Þ
where
ξðtÞ¼
xðtÞ
x
c
ðtÞ
x
w
ðtÞ
2
6
4
3
7
5
; vðtÞ¼
uðtÞ
ωðtÞ
f ðtÞ
2
6
4
3
7
5
;
~
A
i
ðtÞ¼
A
i
þΔA
i
ðtÞ 00
B
ci
ðC
i
þΔC
i
ðtÞÞ A
ci
0
00A
w
2
6
4
3
7
5
; H ¼½I 00;
~
A
1i
ðtÞ¼
A
1i
þΔA
1i
ðtÞ
B
ci
ðC
1i
þΔC
1i
ðtÞÞ
0
2
6
4
3
7
5
;
~
B
i
ðtÞ¼
B
0i
þΔB
0i
ðtÞ B
i
B
1i
þΔB
1i
ðtÞ
B
ci
ðD
0i
þΔD
0i
ðtÞÞ B
ci
D
i
B
ci
ðD
1i
þΔD
1i
ðtÞÞ
00B
w
2
6
4
3
7
5
;
~
E
i
ðtÞ¼
E
i
þΔE
i
ðtÞ
B
ci
F
i
þΔF
i
ðtÞ
0
2
6
4
3
7
5
;
~
C
i
ðtÞ9
~
C
i
¼½0 C
ci
C
w
;
~
D
i
ðtÞ9
~
D
i
¼½00D
w
;
Δ
~
A
i
ðtÞ¼
ΔA
i
ðtÞ 00
B
ci
ΔC
i
ðtÞ 00
000
2
6
4
3
7
5
; Δ
~
A
1i
ðtÞ¼
ΔA
1i
ðtÞ
B
ci
ΔC
1i
ðtÞ
0
2
6
4
3
7
5
;
Δ
~
B
i
ðtÞ¼
ΔB
0i
ðtÞ 0 ΔB
1i
ðtÞ
B
ci
ΔD
0i
ðtÞ 0 B
ci
ΔD
1i
ðtÞ
000
2
6
4
3
7
5
;
Δ
~
E
i
ðtÞ¼
ΔE
i
ðtÞ
B
ci
ΔF
i
ðtÞ
0
2
6
4
3
7
5
;
~
A
i
ðtÞ¼
~
A
i
þΔ
~
A
i
ðtÞ;
~
A
1i
ðtÞ¼
~
A
1i
þΔ
~
A
1i
ðtÞ;
~
B
i
ðtÞ¼
~
B
i
þΔ
~
B
i
ðtÞ;
~
E
i
ðtÞ¼
~
E
i
þΔ
~
E
i
ðtÞ: ð15Þ
Before proceeding further, we first introduce the following
definitions, which will play key roles in deriving our main results
in the sequel.
Definition 1 (Mao and Yuan [15]). For nonlinear stochastic differ-
ential equations with Markovian switching
dxðtÞ¼f ðxðtÞ; t; r
t
Þ dt þ gðxðtÞ; t; r
t
Þ dϖðtÞ; ð16Þ
where f : R
n
R
þ
S-R
n
, g : R
n
R
þ
S-R
nm
, t Z t
0
Z 0. The
trivial solution of Eq. (16) is said to be stochastically stable if for
any
εA ð0; 1Þ, ρ4 0 and t
0
Z 0, there exists a δ ¼ δðε; ρ; t
0
Þ4 0, such
that
Pfjxðt; t
0
; x
0
; iÞj o ρ; for all t Z t
0
gZ 1ε ð17Þ
for any ðx
0
; iÞ A S
δ
S. Additionally, if we also have
P lim
t-1
xðt; t
0
; x
0
; iÞ¼0
¼ 1; ð18Þ
G. Zhuang et al. / ISA Transactions 53 (2014) 1024–10341026