32
3 Finite-Time Robust Filtering for Multi-model Jumping System
Definition 3.2 Given T > 0, the error dynamic multi-model jumping system (3.5)
with ω(t) is stochastically FTB in relation to (c 1 c 2 T ˜
R i ) if conditions (3.6) and
(3.7) hold.
Remark 3.2 If we set ω(t) = 0, finite-time boundedness can be transformed as
finite-time stability. In addition, it is different between the concept of Lyapunov
stability and finite-time boundedness. Lyapunov stability is largely known to the
control characteristic in an infinite time-interval. Thus, an FTB stochastic multimodel jumping system may not be Lyapunov stochastically stable and vice versa.
Based on the above consideration, we describe the problem as: Given a scalar
γ > 0, we devise a robust filter in (3.4) such that the estimation error ν(t) satisfies:
(i) The error dynamic multi-model jumping system (3.5) is stochastically FTB.
(ii) Under zero initial conditions, the robust finite-time H ∞ performance index is
satisfied if the following constraint holds for all nonzero ω k :
E
T
0
ν
T
(t)ν(t)dt
≤ γ
2
T
0
ω
T
(t)ω(t)dt
(3.8)
Then the robust filter (3.4) is called as a robust finite-time H ∞ filter of uncertain
multi-model jumping system (3.2) with a disturbance attenuation γ. Define
ω(t) 2,T =
T
0
ω
T
(t)ω(t)dt,
(3.9)
ν(t) 2,E = E
T
0
ν
T
(t)ν(t)dt
(3.10)
Then we have
δ r,
T
=
sup
ω(t)∈L 2 (0 T )
,
ν(t) 2,E
ω(t) 2,T
(3.11)
where δ ν, is the specified boundedness of the induced L 2 norm of the operator
from the unknown disturbance to the error output ν(t). In this case, equality (3.11)
satisfies δ ν, < γ, i.e, γ-disturbance attenuation signifies γ-sub-optimal stochastic
finite-time estimation.
Remark 3.3 In this chapter, the presentation of multi-model jumping system (3.2)
within a finite time interval has generated a lot of interest. The definition can be
explained by the elliptic domains. In (3.7), E
˜
x
T
(0) ˜
R i ˜
x(0)
≤ c 1 involves all the
permitable initial states. In the process of finite-time H ∞ filtering, we assume that
ω(t) is an unknown disturbance with bounded energy. We aim to design a filter to
guarantee the finite-time boundedness of the multi-model jumping system (3.2) with
H ∞ performance from the unknown disturbance ω(t) to the error output ν(t). In a
3 Finite-Time Robust Filtering for Multi-model Jumping System
Definition 3.2 Given T > 0, the error dynamic multi-model jumping system (3.5)
with ω(t) is stochastically FTB in relation to (c 1 c 2 T ˜
R i ) if conditions (3.6) and
(3.7) hold.
Remark 3.2 If we set ω(t) = 0, finite-time boundedness can be transformed as
finite-time stability. In addition, it is different between the concept of Lyapunov
stability and finite-time boundedness. Lyapunov stability is largely known to the
control characteristic in an infinite time-interval. Thus, an FTB stochastic multimodel jumping system may not be Lyapunov stochastically stable and vice versa.
Based on the above consideration, we describe the problem as: Given a scalar
γ > 0, we devise a robust filter in (3.4) such that the estimation error ν(t) satisfies:
(i) The error dynamic multi-model jumping system (3.5) is stochastically FTB.
(ii) Under zero initial conditions, the robust finite-time H ∞ performance index is
satisfied if the following constraint holds for all nonzero ω k :
E
T
0
ν
T
(t)ν(t)dt
≤ γ
2
T
0
ω
T
(t)ω(t)dt
(3.8)
Then the robust filter (3.4) is called as a robust finite-time H ∞ filter of uncertain
multi-model jumping system (3.2) with a disturbance attenuation γ. Define
ω(t) 2,T =
T
0
ω
T
(t)ω(t)dt,
(3.9)
ν(t) 2,E = E
T
0
ν
T
(t)ν(t)dt
(3.10)
Then we have
δ r,
T
=
sup
ω(t)∈L 2 (0 T )
,
ν(t) 2,E
ω(t) 2,T
(3.11)
where δ ν, is the specified boundedness of the induced L 2 norm of the operator
from the unknown disturbance to the error output ν(t). In this case, equality (3.11)
satisfies δ ν, < γ, i.e, γ-disturbance attenuation signifies γ-sub-optimal stochastic
finite-time estimation.
Remark 3.3 In this chapter, the presentation of multi-model jumping system (3.2)
within a finite time interval has generated a lot of interest. The definition can be
explained by the elliptic domains. In (3.7), E
˜
x
T
(0) ˜
R i ˜
x(0)
≤ c 1 involves all the
permitable initial states. In the process of finite-time H ∞ filtering, we assume that
ω(t) is an unknown disturbance with bounded energy. We aim to design a filter to
guarantee the finite-time boundedness of the multi-model jumping system (3.2) with
H ∞ performance from the unknown disturbance ω(t) to the error output ν(t). In a