122
7 Observer-Based Robust Fault Detection for Fuzzy Multi-model Jumping System
N
j=1
π i j
t
t−τ
ˆ
x
T
(τ )Q ˆ
x(τ )dτ =
⎛
⎝
N
j=1
π i j
⎞
⎠
t
t−τ
ˆ
x
T
(τ )Q ˆ
x(τ )dτ
= 0. (7.15)
Thus, it follows that
( ˆ
x(t), r ) X
T
(t) ¯
i j (r )X (t).
(7.16)
where X (t) =
ˆ
x
T
(t) ˆ
x
T
h (t)
T .
From inequality (7.12), we can summarize that V ( ˆ
x(t), r ) < 0. Moreover, if
exist matrix i j (r ) > 0, we have:
( ˆ
x(t), r ) = −X
T
(t)) i j (r )X (t).
(7.17)
Since V ( ˆ
x(t), r ) < 0, we have:
V ( ˆ
x(t), r ) < V
ˆ
x(0), r 0
t=0
= ˆ
x
T
(0)P(r ) ˆ
x(0) +
0
−τ
ˆ
x
T
(τ )Q ˆ
x(τ )dτ . (7.18)
Then, the following relationship is established:
V ( ˆ
x(t), r )
V ( ˆ
x(t), r )
<
−X
T
(t)) i j (r )X (t)
V ( ˆ
x(0), i)
.
(7.19)
For given M 1 = inf −τ δt E
X (δ)
2
, M 2 = sup −τ ζ0 E
ˆ
x(ζ)
2
, σ Q =
σ max (Q), σ P = max r ∈ σ max (P(r )) and σ = min r ∈ σ min
i j (r )
. Then, we get:
X
T
(t)) i j (r )X (t) σ M 1 , V ( ˆ
x(t), r ) < V
ˆ
x(0), r 0
t=0
σ P + hσ Q
M 2 .
(7.20)
Thus, when a scalar σ > 0, we can obtain the following inequality:
V ( ˆ
x(t), r )
V ( ˆ
x(t), r )
<
−X
T
(t)) i j (r )X (t)
V ( ˆ
x(0), i)
−M 1 σ
σ P + hσ Q
M 2
= −σ.
(7.21)
Since M 1 > 0, M 2 > 0, σ Q , σ P and σ are a series of positive scalars, we obtain:
( ˆ
x(t), r ) −σV ( ˆ
x(t), r ).
(7.22)
That is,
E{V ( ˆ
x(t), r )} exp(−σt)V
ˆ
x(0), r 0
.
(7.23)
For a given scalar λ > 0 and let ρ =
σ P + τ σ Q
M 2 , we get:
λE
ˆ
x
T
(t) ˆ
x(t)
E{V ( ˆ
x(t), r )} ρ exp(−σt).
(7.24)