3.2 Finite-Time Robust L 2 − L ∞ Filtering for Multi-model Jumping System
39
where
˜
A i =
A i + A i
0
A i + A i − B Fi C i − A Fi 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 =
E i − C Fi C Fi
.
Definition 3.3 Given T > 0, the error dynamic multi-model jumping system (3.40)
is FTB and satisfies a predefined L 2 − L ∞ disturbance attenuation index, if there exist
parameters A Fi , B Fi , C Fi and a positive scalar γ such that the following constraint
condition holds for any nonzero ω(t).
E
ν(t)
2
∞
− γ
2
ω(t)
2
2 < 0
(3.41)
where ν(t)
2
∞ = sup t∈[0 T ] ν
T
(t)ν(t), ω(t)
2
2 =
T
0 ω
T
(t)ω(t)dt.
Remark 3.4 In the process of finite-time L 2 − L ∞ filtering, the unknown disturbance ω(t) is energy bounded. The issue of this chapter is to design an appropriate
filter that guarantees finite-time boundedness of the error dynamic multi-model jumping system (3.40) with a prescribed L 2 − L ∞ performance index from the unknown
disturbance ω(t) to the error output ν(t). In a word, the designed finite-time L 2 − L ∞
filter satisfies inequality (3.41) with a disturbance attenuation γ.
3.2.2 Finite-Time Analysis and Design of Jumping L 2 − L ∞
Filter
Theorem 3.3 Given T > 0, the error dynamic multi-model jumping system (3.40)
is FTB in relation to (c 1 c 2 T ˜
R i ) and satisfies a prescribed L 2 − L ∞ performance
index if there exist symmetric positive-definite matrices ˜
P i ∈ R
2n×2n and ˜
Q ∈ R
2n×2n
such that the following matrix inequalities hold,
⎡
⎣
i − α ˜
P i ˜
P i ˜
A hi ˜
P i ˜
B i
∗
− ˜
Q 0
∗
∗
−I
⎤
⎦ < 0
(3.42)
˜
C
T
i
˜
C i < γ
2 ˜
P i
(3.43)
c 1
¯
σ ˆ
P + τ σ Q i
+
α
1 − e
−αT
< e
−αT c 2 σ ˆ
P
(3.44)
39
where
˜
A i =
A i + A i
0
A i + A i − B Fi C i − A Fi 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 =
E i − C Fi C Fi
.
Definition 3.3 Given T > 0, the error dynamic multi-model jumping system (3.40)
is FTB and satisfies a predefined L 2 − L ∞ disturbance attenuation index, if there exist
parameters A Fi , B Fi , C Fi and a positive scalar γ such that the following constraint
condition holds for any nonzero ω(t).
E
ν(t)
2
∞
− γ
2
ω(t)
2
2 < 0
(3.41)
where ν(t)
2
∞ = sup t∈[0 T ] ν
T
(t)ν(t), ω(t)
2
2 =
T
0 ω
T
(t)ω(t)dt.
Remark 3.4 In the process of finite-time L 2 − L ∞ filtering, the unknown disturbance ω(t) is energy bounded. The issue of this chapter is to design an appropriate
filter that guarantees finite-time boundedness of the error dynamic multi-model jumping system (3.40) with a prescribed L 2 − L ∞ performance index from the unknown
disturbance ω(t) to the error output ν(t). In a word, the designed finite-time L 2 − L ∞
filter satisfies inequality (3.41) with a disturbance attenuation γ.
3.2.2 Finite-Time Analysis and Design of Jumping L 2 − L ∞
Filter
Theorem 3.3 Given T > 0, the error dynamic multi-model jumping system (3.40)
is FTB in relation to (c 1 c 2 T ˜
R i ) and satisfies a prescribed L 2 − L ∞ performance
index if there exist symmetric positive-definite matrices ˜
P i ∈ R
2n×2n and ˜
Q ∈ R
2n×2n
such that the following matrix inequalities hold,
⎡
⎣
i − α ˜
P i ˜
P i ˜
A hi ˜
P i ˜
B i
∗
− ˜
Q 0
∗
∗
−I
⎤
⎦ < 0
(3.42)
˜
C
T
i
˜
C i < γ
2 ˜
P i
(3.43)
c 1
¯
σ ˆ
P + τ σ Q i
+
α
1 − e
−αT
< e
−αT c 2 σ ˆ
P
(3.44)