3.2 Finite-Time Robust L 2 − L ∞ Filtering for Multi-model Jumping System
43
=
⎡
⎢
⎢
⎢
⎢
⎣
11i 12i 13i 0 15i
∗ 22i 23i 24i 25i
∗
∗ −Q 0
0
∗
∗
∗ −Q 0
∗
∗
∗
∗ −I
⎤
⎥
⎥
⎥
⎥
⎦
,
=
⎡
⎢
⎢
⎢
⎢
⎣
P i A i + A
T
1 P i A
T
1 P i P i A hi 0 0
P i A i
0
P i A hi 0 0
A
T
hi P i
A
T
hi P i
0
0 0
∗
∗
∗
0 0
∗
∗
∗
∗0
⎤
⎥
⎥
⎥
⎥
⎦
,
11i = P i A i + A
T
i P i +
N
j=1 π i j P j + Q − αP i ,
12i = A
T
i P i − C
T
i B
T
Fi P i − A
T
Fi P i ,
13i = P i A hi ,
15i = P i B i ,
22i = P i A Fi + A
T
Fi P i +
N
j=1 π i j P j + Q − αP i ,
23i = −P i B Fi C hi ,
24i = P i A hi ,
25i = P i B i − P i B Fi D i .
Considering Lemma 1.6, can be rewritten as
= V 11 i (t)V 12 + V
T
12
T
i (t)V
T
11 < δ
−1
i V 11 V
T
11 + δ i V
T
12 V 12
(3.64)
where
V 11 (r ) = col [P i M 1i P i M 1i 0 0 0] , V 12 (r ) = [N i 0 N hi 0 0].
Then inequality (3.63) can be derived as
i =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
11i 12i 13i 0
15i
16i
∗ 22i 23i 0
25i
26i
∗
∗ 33i 0
0
0
∗
∗
∗ −Q
0
0
∗
∗
∗ ∗ −γ
2 e
−ηT I 0
∗
∗
∗ ∗
∗
−δ I
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
< 0
(3.65)
where
11i = 11i + δ i N
T
i N i ,
13i = 13i + δ i N
T
i N hi ,
16i = P i M i ,
26i = P i M i ,
33i = −Q + δ i N
T
hi N hi .
Letting X i = P i A Fi , Y i = P i B Fi and defining ˆ
P i = ˜
R
−1/2
i
˜
P i ˜
R
−1/2
i
, ¯
σ ˆ
P =
max i∈ σ max
ˆ
P i
, σ ˆ
P = min i∈ σ min
ˆ
P i
, the following relations are introduced.