2.1 Robust H ∞ Filtering for Multi-model Jumping Systems
15
[A i + A i ]
T P i + P i [A i + A i ] +
N
j=1 π i j P j + Q P i [A hi + A hi ]
[A hi + A hi ]
T P i
−Q
< 0
(2.4)
where P i , i ∈ are mode-dependent positive-define symmetric matrices and Q is a
positive-define symmetric matrix.
Proof We can prove Proposition 2.1 by Definition 1.1 and Lemma 1.2.
Consider the following filter:
⎧
⎨
⎩
˙
x F (t) = A F (r t )x F (t) + A h (r t )x F (t − h) + B F (r t )y(t),
z F (t) = C F (r t )x F (t) + D F (r t )y(t),
x F (0) = δ(t), r t = r 0 , t ∈ [−h 0]
(2.5)
where x F (t) ∈ R
n is the filter state, z F (t) ∈ R
l is the filter output and δ(t) is an
initial function. A F (r t ), B F (r t ), C F (r t ) and D F (r t ) are the filter gain matrices to be
designed.
Then, we can get the following error dynamic multi-model jumping system by
letting e(t) = x(t) − x F (t), and r (t) = z(t) − z F (t):
˙ ˜
x(t) = ˜
A i ˜
x(t) + ˜
A hi ˜
x h (t) + ˜
B i w(t),
r (t) = ˜
C i ˜
x(t) + ˜
C hi ˜
x h (t) + ˜
D i w(t),
(2.6)
where ˜
x(t) =
x
T
(t) e
T
(t)
T ,
˜
x h (t) =
x
T
(t − h) e
T
(t − h)
T ,
˜
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 − B Fi C hi A hi
,
˜
B i =
B i
B i − B Fi D i
,
˜
C i =
L i − C Fi − D Fi C i C Fi
,
˜
C hi =
−D Fi C hi 0
,
˜
D i = −D Fi D i .
Definition 2.3 The filter (2.5) is called as a robust H ∞ filter of the multi-model
jumping system (2.1) if there exist filter parameters A f i , B f i , C f i , D f i , i ∈ , such
that the filtering error dynamic multi-model jumping system (2.6) is stochastically
stable and the following inequality:
J = E
∞
0
r
T
(t)r (t)dt
− γ
2 E
∞
0
ω
T
(t)ω(t)dt
< 0
(2.7)
holds for ω(t) = 0, where γ > 0 is a given H ∞ -gain value.