3.2 Finite-Time Robust L 2 − L ∞ Filtering for Multi-model Jumping System
41
E
ν
T
(T )ν(T )
= E
˜
x
T
(T ) ˜
C
T
i
˜
C i ˜
x(T )
< γ
2 E
˜
x
T
(T ) ˜
P i ˜
x(T )
< γ
2 e
αT
T
0
e
−αt
ω
T
(t)ω(t)dt
< γ
2 e
αT
T
0
ω
T
(t)ω(t)dt
(3.52)
The cost function (3.47) can be guaranteed by setting ¯
γ =
√
e αT γ, that is, inequality
(3.41) holds. On the other hand, integrating inequality (3.49) from 0 to t, t ∈ [0 T ],
we obtain
e
−αt E{V ( ˜
x(t), i)} − E {V { ˜
x(0), r t = r 0 }} <
t
0
e
−αs
ω
T
(s)ω(s)ds
(3.53)
Denote ˆ
P i = ˜
R
−1/2
i
˜
P i ˜
R
−1/2
i
, Q i = ˜
R
−1/2
i
˜
Q ˜
R
−1/2
i
, σ ˆ
P = min i∈ σ min
ˆ
P i
, ¯
σ ˆ
P =
max i∈ σ max
ˆ
P i
, and σ Q i = max i∈ σ max (Q i ). Due to the fact that α > 0, 0 ≤ t ≤
T , we have
E
˜
x
T
(T ) ¯
P i ¯
x(T )
≤ E{V ( ¯
x(t), i)}
< e
at E {V ( ¯
x(0), r 0 )} + e
αt
t
0
e
−αs
ω
T
(s)ω(s)ds
< e
αT
c 1
σ ˆ
P + τ σ Q i
+
α
1 − e
−αT
(3.54)
Considering equality (3.45), we have
E
˜
x
T
(t) ˜
P i ˜
x(t)
≥ σ ˆ
P E
˜
x
T
(t) ˜
R i ˜
x(t)
(3.55)
Then we derive
E
˜
x
T
(t) ˜
R i ˜
x(t)
<
e
αT
c 1
¯
σ ˆ
P + τ σ Q i
+ (/α)
1 − e
−αT
σ ˆ
P
(3.56)
For ∀t ∈ [0 T ], E
˜
x
T
(t) ˜
R i ˜
x(t)
< c 2 . This completes the proof.
Theorem 3.4 Given T > 0, the error dynamic multi-model jumping system (3.40)
is FTB in relation to (c 1 c 2 T ˜
R i ) with ˜
R i = diag {R i R i } and have a prescribed
L 2 − L ∞ performance level if there exist scalars σ 1 > 0 and σ 2 > 0, matrices X i , Y i ,
and C Fi , positive-definite symmetric matrices P i ∈ R
n×n and Q ∈ R
n×n , such that
the following LMIs hold.