7.1 Robust FDO Design for Fuzzy Multi-model Jumping System
123
Taking the limitation as T → ∞, then it has:
lim
T →∞
E
T
0
ˆ
x
T
(t) ˆ
x(t)dt | ˆ
x(0) = η(0), r 0
lim
t→∞
ρ
λ
(1 − exp(−σt))
=
ρ
λ
< ∞.
(7.25)
Thus, the stochastic stability of the error dynamic multi-model jumping system
(7.8) can be obtained. This completes the proof.
Let f (t) = 0, the error dynamic multi-model jumping system (7.8) is reconstructed as:
˙ ˆ
x(t) = ˆ
A i j (r ) ˆ
x(t) + ˆ
A hi j (r ) ˆ
x h (t) + ˆ
B di j (r )ω(t),
r d (t) = ˆ
C i j (r ) ˆ
x(t) + ˆ
C hi j (r ) ˆ
x h (t) + ˆ
D di (r )ω(t).
(7.26)
In order to reduce the influence of ω(t) to residual, the H ∞ filtering problem in
this chapter can be summarized as designing an appropriate parameter matrix H i (r )
to make the error dynamic multi-model jumping system (7.26) meet the following
condition when fault f (t) = 0,
E
∞
0
r
T
d (t)r d (t)dt
λ
2 E
∞
0
ω
T
(t)ω(t)dt
f (t)=0
.
(7.27)
Theorem 7.1 For a given λ > 0, the error dynamic multi-model jumping system
(7.26) is stochastically stable and satisfies the condition (7.27), if there exist positive matrices P 1 (r ) = P
T
1 (r ), P 2 (r ) = P
T
2 (r ), Q 11 = Q
T
11 , Q 22 = Q
T
22 , matrix Q 12 ,
parameter matrix H i (r ) and scalars α i j (r ) satisfying the following LMIs:
i j (r ) =
⎡
⎢
⎢
⎢
⎢
⎣
1i j (r ) ) 2i j (r ) ) 3i j (r ) ) 4i j (r )
) 5i j (r )
∗
6i j (r )
0
7i j (r )
0
∗
∗
8i j (r ) ) 9i j (r )
0
∗
∗
∗
−2I
M yi (r ) + M yj (r )
∗
∗
∗
∗
−
α i j (r ) + α ji (r )
I
⎤
⎥
⎥
⎥
⎥
⎦
< 0, (7.28)
where
1i j (r ) =
11i j (r ) + 11 ji (r )
Q 12
∗
12i j (r ) + 12 ji (r )
,
11i j (r ) =P 1 (r )A i (r ) + A
T
i (r )P 1 (r ) +
N
r =1
π rk P 1 (k) + Q 11
+ α i j (r )N
T
i (r )N i (r ),