6.2 Robust FD Observer Design for Multi-model Jumping System
107
Further, it has:
t
0 e
−βt r
T
eo (t)r eo (t) <
t
0 e
−βt
2
1 ω
T
(t)ω(t)dt.
(6.41)
In view of ∀t ∈ [0 T ], one has:
t
0 r
T
eo (t)r eo (t) < e
βT
2
1
t
0 ω
T
(t)ω(t)dt.
(6.42)
where ¯
1 = 1
√
e βT .
In the absence of unknown disturbances, i.e., ω(t) = 0, and in terms of inequality
(6.29), the error dynamic multi-model jumping system (6.26) is stochastically stable
from Lemma 6.3, which ends the proof.
Theorem 6.3 The error dynamic multi-model jumping system (6.26) is stochastically stable and satisfies condition (6.29), if there exists mode dependent symmetric
positive-definite matrix P i > 0 and mode-dependent matrix ¯
H i such that the following relationship yields:
5 6
∗ 4
< 0,
(6.43)
where
5 =
⎡
⎢
⎢
⎢
⎢
⎣
1i 0 5i 0 P i B f i
∗ 2i 3i 0 7i
∗ ∗ 4i 0
0
∗ ∗ ∗ −P i 0
∗ ∗ ∗ ∗ −
2
1 I
⎤
⎥
⎥
⎥
⎥
⎦
,
6 =
⎡
⎢
⎢
⎢
⎢
⎣
0 1 0 M
T
i P 0
C
T
i
0 1 0 M
T
i P
0 0 0
0
0
0 0 0
0
0
D
T
f i 0 0
0
0
⎤
⎥
⎥
⎥
⎥
⎦
,
7i = P i B f i − P i H i D f i .
Proof For given mode-dependent symmetric positive-definite matrices P i ∈ R
n×n ,
i ∈ , considering the error dynamic multi-model jumping system (6.26), the Lyapunov function is assigned as V ( ˜
e f , i) = ˜
e
T
f P i ˜
e f ≥ 0.
Setting ω(t) = 0, Eq. (6.35) amounts to:
˙ ˜
e f (t) = ˜
A i ˜
e f (t) + ˜
A hi ˜
e f (t − h) + ˜
B f i f (t),
r f (t) = ˜
C i ˜
e f (t) − ˜
C hi ˜
e f (t − h) + D f i f (t).
(6.44)
By setting J 2 = E{{V ( ˜
e f , i) + r
T
f (t)r f (t) −
2
2 f
T
(t) f (t) − βV ( ˜
e f (t), i) and
obey the similar proof in Theorem 6.2, inequality (6.43) can be obtained and
¯
2 = r 2
√
e βT . This ends the proof.