54
4 Finite-Frequency Robust Filtering for Multi-model Jumping System
system to a deterministic one makes it possible to utilize GKYP lemma to design
the finite frequency filter. The second advantage lies in the inner connection between
A 2 , A i , i = 1, 2, · · · N and transition rates for multi-model jumping system (4.1)
and (4.5) are expressed in (4.5). This explains the fact that the whole multi-model
jumping system may be unstable while all subsystems are stable, or some subsystems
may be unstable while the whole system is stable.
Then, for the obtained extended deterministic system (4.5), consider the following
filter:
⎧
⎨
⎩
˙
q F (t) = A F q F (t) + B F ˜
y(t),
˜
z F (t) = C F q F (t) + D F ˜
y(t),
q F (0) = 0,
(4.6)
where q F (t) ∈ R
nN is the filter state and ˜
z F (t) ∈ R
r N is the filter output. A F , B F ,
C F and D F are the filter gain matrices to be determined.
Defining the filtering error by e(t) = ˜
z(t) − ˜
z F (t), and combining (4.5) and (4.6),
we have the following filtering error dynamic system:
˙ ˜
q(t) = ˜
A ˜
q(t) + ˜
B ˜
ω(t),
˜
e(t) = ˜
C ˜
q(t) + ˜
D ˜
ω(t),
(4.7)
where
˜
q(t) =
q(t)
q F (t)
,
˜
A =
A
0
B F C 1 A F
,
˜
B =
B d
B F D d
,
˜
C =
C 2 −D F C 1 −C F
,
˜
D = −D F D d .
To formulate the finite-frequency filtering problem, we need the following definitions.
Definition 4.1 The filtering error dynamic system (4.7) (setting ω(t) ≡ 0) is said
to be stochastically stable, if for any initial x 0 and mode r 0 , the following relation
holds:
lim
t→∞
˜
q(t, ˜
q 0 , r 0 ) = 0.
(4.8)
Definition 4.2 For a given constant γ, the extended filtering error dynamic system
(4.7) is said to satisfy a finite frequency H ∞ performance index γ, if the following
inequality:
G ˜
e ˜
w ( jω ω ω)
ω ω ω∈
∞
< γ,
4 Finite-Frequency Robust Filtering for Multi-model Jumping System
system to a deterministic one makes it possible to utilize GKYP lemma to design
the finite frequency filter. The second advantage lies in the inner connection between
A 2 , A i , i = 1, 2, · · · N and transition rates for multi-model jumping system (4.1)
and (4.5) are expressed in (4.5). This explains the fact that the whole multi-model
jumping system may be unstable while all subsystems are stable, or some subsystems
may be unstable while the whole system is stable.
Then, for the obtained extended deterministic system (4.5), consider the following
filter:
⎧
⎨
⎩
˙
q F (t) = A F q F (t) + B F ˜
y(t),
˜
z F (t) = C F q F (t) + D F ˜
y(t),
q F (0) = 0,
(4.6)
where q F (t) ∈ R
nN is the filter state and ˜
z F (t) ∈ R
r N is the filter output. A F , B F ,
C F and D F are the filter gain matrices to be determined.
Defining the filtering error by e(t) = ˜
z(t) − ˜
z F (t), and combining (4.5) and (4.6),
we have the following filtering error dynamic system:
˙ ˜
q(t) = ˜
A ˜
q(t) + ˜
B ˜
ω(t),
˜
e(t) = ˜
C ˜
q(t) + ˜
D ˜
ω(t),
(4.7)
where
˜
q(t) =
q(t)
q F (t)
,
˜
A =
A
0
B F C 1 A F
,
˜
B =
B d
B F D d
,
˜
C =
C 2 −D F C 1 −C F
,
˜
D = −D F D d .
To formulate the finite-frequency filtering problem, we need the following definitions.
Definition 4.1 The filtering error dynamic system (4.7) (setting ω(t) ≡ 0) is said
to be stochastically stable, if for any initial x 0 and mode r 0 , the following relation
holds:
lim
t→∞
˜
q(t, ˜
q 0 , r 0 ) = 0.
(4.8)
Definition 4.2 For a given constant γ, the extended filtering error dynamic system
(4.7) is said to satisfy a finite frequency H ∞ performance index γ, if the following
inequality:
G ˜
e ˜
w ( jω ω ω)
ω ω ω∈
∞
< γ,