62
4 Finite-Frequency Robust Filtering for Multi-model Jumping System
Considering the diverse demands in practice, besides the single frequency range
performance, different performance indices are required in different frequency ranges
to optimize the system performance. For instance, the low-frequency band requires
high gains to ensure the strong anti-interference ability in the system, while the highfrequency band requires low gains to reduce the influence of model uncertainty on the
system performance. Therefore, in this subsection, filter with multiple performances
are designed over multiple frequency ranges for the extended error system (4.24).
The multiple performances here can be mathematically expressed as follows:
(1) Low frequency band constraint
|G ˜
e ˜
ω ( jω ω ω)| =
˜
e 2
˜
ω 2
< γ,
(4.25)
(2) High frequency band index
|G ˜
e ˜
ω ( jω ω ω)| =
˜
e 2
˜
ω 2
< ρ.
(4.26)
Thus, the objectives of this section can be formulated as
(1) Given scalars γ > 0, ρ > 0, the extended filtering error dynamic system (4.24)
with ω(t) ∈ L 2 is stable and satisfies the multiple frequency H ∞ performance
defined in (4.25) and (4.26), respectively;
(2) Obtain the solution of the filter A F , B F , C F and D F .
To formulate the multiple-frequency filtering problem, we recall the followings
which will be used to develop our main results in sequel.
Definition 4.3 The filtering error dynamic system (4.24) is said to be stable, if the
following condition holds:
lim
t→∞
E { ˜
q(t)} = 0.
(4.27)
Definition 4.4 For given parameters γ > 0, ρ > 0, the extended filtering error
dynamic system (4.24) is said to be stable and satisfies the predefined multiple
frequency performances, if there exists a filter in form of equation (4.23) so that
conditions (4.25)–(4.27) hold.
4.2.2 Filter Design Restricted to Multiple Range Frequency
Performances
Theorem 4.4 For predefined finite frequency indices γ, ρ, and given frequencies l
and h , if there exist symmetric matrices P l =
P l1 P l2
∗ P l3
> 0,P h =
P h1 P h2
∗ P h3
>
4 Finite-Frequency Robust Filtering for Multi-model Jumping System
Considering the diverse demands in practice, besides the single frequency range
performance, different performance indices are required in different frequency ranges
to optimize the system performance. For instance, the low-frequency band requires
high gains to ensure the strong anti-interference ability in the system, while the highfrequency band requires low gains to reduce the influence of model uncertainty on the
system performance. Therefore, in this subsection, filter with multiple performances
are designed over multiple frequency ranges for the extended error system (4.24).
The multiple performances here can be mathematically expressed as follows:
(1) Low frequency band constraint
|G ˜
e ˜
ω ( jω ω ω)| =
˜
e 2
˜
ω 2
< γ,
(4.25)
(2) High frequency band index
|G ˜
e ˜
ω ( jω ω ω)| =
˜
e 2
˜
ω 2
< ρ.
(4.26)
Thus, the objectives of this section can be formulated as
(1) Given scalars γ > 0, ρ > 0, the extended filtering error dynamic system (4.24)
with ω(t) ∈ L 2 is stable and satisfies the multiple frequency H ∞ performance
defined in (4.25) and (4.26), respectively;
(2) Obtain the solution of the filter A F , B F , C F and D F .
To formulate the multiple-frequency filtering problem, we recall the followings
which will be used to develop our main results in sequel.
Definition 4.3 The filtering error dynamic system (4.24) is said to be stable, if the
following condition holds:
lim
t→∞
E { ˜
q(t)} = 0.
(4.27)
Definition 4.4 For given parameters γ > 0, ρ > 0, the extended filtering error
dynamic system (4.24) is said to be stable and satisfies the predefined multiple
frequency performances, if there exists a filter in form of equation (4.23) so that
conditions (4.25)–(4.27) hold.
4.2.2 Filter Design Restricted to Multiple Range Frequency
Performances
Theorem 4.4 For predefined finite frequency indices γ, ρ, and given frequencies l
and h , if there exist symmetric matrices P l =
P l1 P l2
∗ P l3
> 0,P h =
P h1 P h2
∗ P h3
>