2.2 Unbiased H ∞ Filtering for Multi-model Jumping System
21
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
11 P i B i − Y i D i P i A hi L
T
i − C
T
i D
T
Fi 15 16
∗
−γ
2 I
0
−D
T
i D
T
Fi
0
0
∗
∗
−Q
0
35 0
∗
∗
∗
−I
0
0
∗
∗
∗
∗
− λ i I 0
∗
∗
∗
∗
∗ 66
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
< 0
(2.26)
where
11 = P i A i + A
T
i P i − Y i C i − C
T
i Y
T
i + Q + (π ii − β)P i ,
15 =
λ i P i M i N
T
i
0
0
,
16 =
√ π i1 P i , · · · ,
√ π ii−1 P i ,
√ π ii+1 P i , · · · ,
√ π i N P i
,
35 =
0 N
T
hi
0 0
,
66 = −diag
P i , · · · , P i , P i , · · · , P i
,
then the error dynamic multi-model jumping system (2.24) is stochastically stable
and satisfies the given H ∞ performance (2.25). Moreover, the unbiased H ∞ filter is
given by A Fi = A i − B Fi C i , B Fi = P
−1
i Y i , C Fi = L i − D Fi C i , D Fi = D Fi .
Proof Choose the folowing stochastic Lyapunov–Krasovskii functional as:
V (e(t), r ) = e
T
(t)P i e(t) +
t
t−h
e
T
(s)Qe(s)ds,
(2.27)
where P i , Q are given symmetric positive-definite matrix for any i ∈ .
Considering Definition 1.6 and along the trajectories of the filtering error multimodel jumping system (2.24), we have:
V (e(t), i, t) = e
T
(t)
⎛
⎝ P i A i + A
T
i P i +
N
j=1
π i j P j + Q
⎞
⎠ e(t)
+2e
T
(t)P i A hi e(t − h) + 2e
T
(t)P i B i ω(t)
−e(t − h)
T
(t)Qe(t − h)(t).
(2.28)
For the filtering error multi-model jumping system (2.24), we introduce the following cost function:
J = =V (e(t), i) − βE{V (e(t), i)} + E{r
T
(t)r (t)} − γ
2
ω
T
(t)ω(t)
= e
T
(t)
⎛
⎝ P i A i + A
T
i P i +
N
j=1
π i j P j + Q
⎞
⎠ e(t) + 2e
T
(t)P i A hi e(t − h)
+2e
T
(t)P i B i ω(t) − e(t − h)
T
(t)Qe(t − h)(t) − βe
T
(t)P i e(t)
−γ
2
ω
T
(t)ω(t) + e
T
(t)C
T
i C i e(t) + 2e
T
(t)C
T
i D i ω(t) + ω
T
(t)D
T
i D i ω(t)