3.1 Finite-Time Robust H ∞ Filtering for Multi-model Jumping System
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unknown function i (t) in (3.3) can also be written as i (t) = i (t, x(t)), that is,
we have | i (t, x(t))| ≤ 1.
The filter is constructed as
⎧
⎨
⎩
˙ ˆ
x(t) = A Fi ˆ
x(t) + B Fi y(t)
ˆ
z(t) = C Fi ˆ
x(t)
ˆ
x(t) = ˆ
x 0 , r t = r 0 , t = 0,
(3.4)
where ˆ
x(t) ∈ R
n is the filter state, ˆ
z(t) ∈ R
q is the filer output. A Fi , B Fi and C Fi
are the filter parameters which need to be designed for each value i ∈ .
Letting e(t) = x(t) − ˆ
x(t), ν(t) = z(t) − ˆ
z(t), the error dynamic multi-model
jumping system can be formulated as
⎧
⎨
⎩
˙ ˜
x(t) = ˜
A i ˜
x(t) + ˜
A hi ˜
x h + ˜
B i ω(t),
ν(t) = ˜
C i ˜
x(t),
˜
x(t) = [x 0 x 0 − ˜
x 0 ], r t = r 0 , t = 0,
(3.5)
where
˜
x(t) =
x
T
(t) e
T
(t)
T , ˜
x h = ˜
x(t − τ ),
˜
A i =
A i + A i
0
A i + A i − A Fi − B Fi C i A Fi
,
˜
A hi =
A hi + A hi
0
A hi + A hi − B Fi C hi 0
,
˜
B i =
B i
B i − B Fi D i
,
˜
C i =
E i − C Fi −C Fi
.
Assumption 3.1 The external disturbance ω(t) is bounded by the following restriction in the finite-time interval (0 T ):
T
0
ω
T
(t)ω(t)dt ≤
(3.6)
where > 0 is a scalar.
Definition 3.1 Given T > 0, the error dynamic multi-model jumping system (3.2)
is stochastically finite-time stable (FTS) if there exist matrix ˜
R i ∈ R
2n×2n
> 0, two
scalars c 1 > 0, c 2 > 0, such that
E
˜
x
T
(0) ˜
R i ˜
x(0)
≤ c 1 ⇒ E
˜
x
T
(t) ˜
R i ˜
x(t)
< c 2 , t ∈ [0 T ]
(3.7)