96
6 Robust Fault Detection for Multi-model Jumping System
6.1 Robust FD Filter Design for Multi-model Jumping
System
6.1.1 System Description
We consider the following multi-model jumping system defined on the probability
space ((, F, P):
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
˙
x(t) = [A(r t ) + A(t, r t )]x(t) + [A h (r t ) + A h (t, r t )] x(t − h) + B(r t )u(t)
+B d (r t )ω(t) + B f (r t ) f (t),
y(t) = C(r t )x(t) + C h (r t )x(t − h) + D d (r t )ω(t) + D f (r t ) f (t),
x(t) = x 0 , r t = r 0 ,
(6.1)
where x(t) ∈ R
n is the state, y(t) ∈ R
m is the measured output, u(t) ∈ R
r is the
controlled input, f (t) ∈ R
p is the detected faults, ω(t) ∈ L
m
2 [0, ∞] is the unknown
disturbance, x 0 and r 0 is the initial state and the initial mode.
A(r t ), A(t, r t ), A h (r t ), A h (t, r t ), B(r t ), B d (r t ), B f (r t ), C(r t ), C h (r t ), D d (r t ),
D f (r t ) are known mode-dependent matrices with appropriate dimensions, and r t
represents a continuous-time discrete state Markov stochastic process with values in
the finite set .
For presentation simplification, A(r t ), A(t, r t ), A h (r t ), A h (t, r t ), B(r t ), B d (r t ),
B f (r t ), C(r t ), C h (r t ), D d (r t ), D f (r t ) are replaced by A i , A i , A hi , A hi , B i , B di ,
B f i , C hi , C hi , D di , D f i , respectively. For the sake of clarity, we assume that the
time-delay multi-model jumping system (6.1) is stochastically stable, (C i , A i ) is
observable and B f i is a full rank matrix.
First, we consider the following filter:
⎧
⎨
⎩
˙
x F (t) = A F (r t )x F (t) + B F (r t )y(t),
r F (t) = C F (r t )x F (t) + D F (r t )y(t),
x F (0) = 0,
(6.2)
where x F (t) ∈ R
n is the filter state and r F (t) ∈ R
m is the filter output. A F (r t ), B F (r t ),
C F (r t ) and D F (r t ) are the filter gain matrices to be solved.
Remark 6.1 To detect the faults, we have to obtain the estimation of the faults. In
this paper, we utilize a stable appropriate weighting matrix function W f (s) [119] to
recognize the faults and promote the system performance. Generally speaking, the
transfer function W f (s) is established as a full rank vector or a diagonal identity
matrix. Referring to the latest works, we give the reference residual model as:
r f (s) = W f (s) f (s),
(6.3)
where W f (s) is the minimal realization and satisfies:
6 Robust Fault Detection for Multi-model Jumping System
6.1 Robust FD Filter Design for Multi-model Jumping
System
6.1.1 System Description
We consider the following multi-model jumping system defined on the probability
space ((, F, P):
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
˙
x(t) = [A(r t ) + A(t, r t )]x(t) + [A h (r t ) + A h (t, r t )] x(t − h) + B(r t )u(t)
+B d (r t )ω(t) + B f (r t ) f (t),
y(t) = C(r t )x(t) + C h (r t )x(t − h) + D d (r t )ω(t) + D f (r t ) f (t),
x(t) = x 0 , r t = r 0 ,
(6.1)
where x(t) ∈ R
n is the state, y(t) ∈ R
m is the measured output, u(t) ∈ R
r is the
controlled input, f (t) ∈ R
p is the detected faults, ω(t) ∈ L
m
2 [0, ∞] is the unknown
disturbance, x 0 and r 0 is the initial state and the initial mode.
A(r t ), A(t, r t ), A h (r t ), A h (t, r t ), B(r t ), B d (r t ), B f (r t ), C(r t ), C h (r t ), D d (r t ),
D f (r t ) are known mode-dependent matrices with appropriate dimensions, and r t
represents a continuous-time discrete state Markov stochastic process with values in
the finite set .
For presentation simplification, A(r t ), A(t, r t ), A h (r t ), A h (t, r t ), B(r t ), B d (r t ),
B f (r t ), C(r t ), C h (r t ), D d (r t ), D f (r t ) are replaced by A i , A i , A hi , A hi , B i , B di ,
B f i , C hi , C hi , D di , D f i , respectively. For the sake of clarity, we assume that the
time-delay multi-model jumping system (6.1) is stochastically stable, (C i , A i ) is
observable and B f i is a full rank matrix.
First, we consider the following filter:
⎧
⎨
⎩
˙
x F (t) = A F (r t )x F (t) + B F (r t )y(t),
r F (t) = C F (r t )x F (t) + D F (r t )y(t),
x F (0) = 0,
(6.2)
where x F (t) ∈ R
n is the filter state and r F (t) ∈ R
m is the filter output. A F (r t ), B F (r t ),
C F (r t ) and D F (r t ) are the filter gain matrices to be solved.
Remark 6.1 To detect the faults, we have to obtain the estimation of the faults. In
this paper, we utilize a stable appropriate weighting matrix function W f (s) [119] to
recognize the faults and promote the system performance. Generally speaking, the
transfer function W f (s) is established as a full rank vector or a diagonal identity
matrix. Referring to the latest works, we give the reference residual model as:
r f (s) = W f (s) f (s),
(6.3)
where W f (s) is the minimal realization and satisfies: