6.1 Robust FD Filter Design for Multi-model Jumping System
101
where
i =
⎡
⎢
⎢
⎣
11 12 13 14
∗ −ξ
2 I 0 24
∗
∗ 33 34
∗
∗
∗ −I
⎤
⎥
⎥
⎦ ,
1i =
⎡
⎣
11 12 Q 13
∗ 22 Q 23
∗ ∗ 33
⎤
⎦ ,
11 = P i A i + A
T
i P i +
N
j=1 π i j P j + Q 11 − β P i ,
i =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
P i A i + A
T
i P
T
i 0 0 0 0 0 P i A hi 0 0 0
∗
0 0 0 0 0
0
0 0 0
∗
∗0 0 0 0
0
0 0 0
∗
∗∗0 0 0
0
0 0 0
∗
∗ ∗ ∗ 0 0
0
0 0 0
∗
∗ ∗ ∗ ∗ 0
0
0 0 0
∗
∗ ∗ ∗ ∗ ∗
0
0 0 0
∗
∗ ∗ ∗ ∗ ∗
∗
0 0 0
∗
∗ ∗ ∗ ∗ ∗
∗
∗ 0 0
∗
∗ ∗ ∗ ∗ ∗
∗
∗ ∗ 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
i can be exhibited as:
i = L 1 i (t)L 2 + L
T
2
T
i (t)L
T
1 < λ
−1
i L
T
2 L 2 + λ i L 1 L
T
1 ,
(6.16)
where
L 1 = col [P i M i , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0] ,
L 2 = [N i , 0, 0, 0, 0, 0, N hi , 0, 0, 0] .
By Schur complements, one can easily find that inequality (6.15) can result in
LMIs (6.11).
Recalling to inequality (6.15), one has:
E{r
T
(t)r (t)} + +V ( ˜
x(t), i) < βE{V ( ˜
x(t), i)} + ξ
2
w
T
(t)w(t).
(6.17)
Multiplying inequality (6.17) both sides by e
−βt renders:
[e
−βt V ( ˜
x(t), i)] < E{e
−βt
[ξ
2
w
T
(t)w(t) − r
T
(t)r (t)]}.
(6.18)
Under zero initial conditions, integrating inequality (6.18) from 0 to T yields:
E[e
−βt V ( ˜
x(t), i)] < E
T
0
e
−βt
[ξ
2
w
T
(t)w(t) − r
T
(t)r (t)]dt
.
(6.19)
From inequality (6.19), it has: