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3 Finite-Time Robust Filtering for Multi-model Jumping System
3.1 Finite-Time Robust H ∞ Filtering for Multi-model
Jumping System
3.1.1 System Description
Consider a probability space ((, F, P) and define the transition probability as:
P i j = P {r t+t = j | r t = i} =
π i j t + o((t),
i = j
1 + π ii t + o((t), i = j
(3.1)
where t > 0 and we have lim t↓0 o((t)//t → 0. π i j ≥ 0,
N
j=1,i = j π i j = −π ii .
{r t , t ≥ 0} is a continuous-time discrete-state Markov process, which takes values
in a finite set = {1, 2, . . . , N } with the transition rate matrix =
π i j , i, j ∈
.
The continue-time multi-model jumping system is defined as:
⎧
⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
˙
x(t) = [A(r t ) + A(t, r t )]x(t) + [A h (r t ) + A h (t, r t )] x(t − τ )
+B(r t )ω(t),
y(t) = C(r t )x(t) + C h (r t )x(t − τ ) + D(r t )ω(t),
z(t) = E(r t )x(t),
x(t) = x 0 , r t = r 0 , t = 0.
(3.2)
where x(t) ∈ R
n is the state, x(t − τ ) ∈ R
n is the time-delayed state, y(t) ∈ R
m is
the measured output, z(t) ∈ R
q is the controlled output, τ > 0 is the constant timedelay, ω(t) ∈ L
m
2 [0, ∞] is the unknown disturbance, x 0 is the initial value and r 0 is
the initial jumping mode.
A(r t ), A(t, r t ), A h (r t ), A h (t, r t ), B(r t ), C(r t ), C h (r t ), D d (r t ), E(r t ) are known
parameters matrices with compatible dimensions.
To simplify the parameters matrices, we denote A(r t ), A(t, r t ), A h (r t ),
A h (t, r t ), B(r t ), C(r t ), C h (r t ), D(r t ), E(r t ) as A i , A i , A hi , A hi , B i , C i , C hi ,
D i , E i , respectively.
The uncertainties A(t, r t ) and A h (t, r t ) have the following relationship:
[A i A hi ] = M i i (t)[N i N hi ] ,
(3.3)
where M i , N i , N hi are constant matrices with compatible dimensions, i (t) is the
unknown time-varying function which satisfies
T
i (t)) i (t) ≤ I .
Remark 3.1 The uncertainties in (3.2) are bounded by the restrictions (3.3) with
i (t)
T
i (t) ≤ I . Due to the environmental noise and the complexity of process,
it is difficult to obtain an accurate mathematical model in practical engineering.
Therefore, the engineering model always almost contains some types of uncertainties.
As a matter of fact, the uncertainties provided in (3.3) have been extensively used
in robust analysis and synthesis of stochastic multi-model jumping system. The
3 Finite-Time Robust Filtering for Multi-model Jumping System
3.1 Finite-Time Robust H ∞ Filtering for Multi-model
Jumping System
3.1.1 System Description
Consider a probability space ((, F, P) and define the transition probability as:
P i j = P {r t+t = j | r t = i} =
π i j t + o((t),
i = j
1 + π ii t + o((t), i = j
(3.1)
where t > 0 and we have lim t↓0 o((t)//t → 0. π i j ≥ 0,
N
j=1,i = j π i j = −π ii .
{r t , t ≥ 0} is a continuous-time discrete-state Markov process, which takes values
in a finite set = {1, 2, . . . , N } with the transition rate matrix =
π i j , i, j ∈
.
The continue-time multi-model jumping system is defined as:
⎧
⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
˙
x(t) = [A(r t ) + A(t, r t )]x(t) + [A h (r t ) + A h (t, r t )] x(t − τ )
+B(r t )ω(t),
y(t) = C(r t )x(t) + C h (r t )x(t − τ ) + D(r t )ω(t),
z(t) = E(r t )x(t),
x(t) = x 0 , r t = r 0 , t = 0.
(3.2)
where x(t) ∈ R
n is the state, x(t − τ ) ∈ R
n is the time-delayed state, y(t) ∈ R
m is
the measured output, z(t) ∈ R
q is the controlled output, τ > 0 is the constant timedelay, ω(t) ∈ L
m
2 [0, ∞] is the unknown disturbance, x 0 is the initial value and r 0 is
the initial jumping mode.
A(r t ), A(t, r t ), A h (r t ), A h (t, r t ), B(r t ), C(r t ), C h (r t ), D d (r t ), E(r t ) are known
parameters matrices with compatible dimensions.
To simplify the parameters matrices, we denote A(r t ), A(t, r t ), A h (r t ),
A h (t, r t ), B(r t ), C(r t ), C h (r t ), D(r t ), E(r t ) as A i , A i , A hi , A hi , B i , C i , C hi ,
D i , E i , respectively.
The uncertainties A(t, r t ) and A h (t, r t ) have the following relationship:
[A i A hi ] = M i i (t)[N i N hi ] ,
(3.3)
where M i , N i , N hi are constant matrices with compatible dimensions, i (t) is the
unknown time-varying function which satisfies
T
i (t)) i (t) ≤ I .
Remark 3.1 The uncertainties in (3.2) are bounded by the restrictions (3.3) with
i (t)
T
i (t) ≤ I . Due to the environmental noise and the complexity of process,
it is difficult to obtain an accurate mathematical model in practical engineering.
Therefore, the engineering model always almost contains some types of uncertainties.
As a matter of fact, the uncertainties provided in (3.3) have been extensively used
in robust analysis and synthesis of stochastic multi-model jumping system. The