38
3 Finite-Time Robust Filtering for Multi-model Jumping System
⎧
⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
˙
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.
(3.36)
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 state 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. {r t , t ≥ 0}
is a Markov stochastic process, which takes values in a finite set = {1, 2, ..., N }
with transition rate matrix = {π i j }, i, j ∈ . The transition probability can be
written as
P r {r t+t = j | r t = i} =
π i j t + o((t),
i = j
1 + π ii t + o((t), i = j
(3.37)
where π i j > 0,
N
j=1, j =i π i j = −π ii , t > 0 and lim t→0 (o((t)//t = 0. 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.
In this chapter, the multi-model jumping system (3.36) is assumed to be finite-time
stable, (A i , B i ) is controllable.
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 ] ,
(3.38)
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 .
We construct the following filter:
⎧
⎨
⎩
˙ ˆ
x(t) = A Fi ˆ
x(t) + A hi ˆ
x(t − τ ) + B Fi y(t),
ˆ
z(t) = C Fi ˆ
x(t),
ˆ
x(t) = ˆ
x 0 , r t = r 0 ,
(3.39)
where ˆ
x(t) ∈ R
n is the filter state and ˆ
y(t) ∈ R
q is filter output. A Fi , B Fi , and C Fi
are unknown parameters to be designed.
Letting e(t) = x(t) − ˆ
x(t), ν(t) = z(t) − ˆ
z(t) and ˜
x(t) = col [x(t) e(t)], the
error dynamic multi-model jumping system can be formulated as:
⎧
⎨
⎩
˙ ˜
x(t) = ˜
A i ˜
x(t) + ˜
A hi ˜
x(t − τ ) + ˜
B i ω(t),
ν(t) = ˜
C i ˜
x(t),
˜
x(t) =
x 0 x 0 − ˆ
x 0
,
(3.40)
3 Finite-Time Robust Filtering for Multi-model Jumping System
⎧
⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎩
˙
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.
(3.36)
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 state 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. {r t , t ≥ 0}
is a Markov stochastic process, which takes values in a finite set = {1, 2, ..., N }
with transition rate matrix = {π i j }, i, j ∈ . The transition probability can be
written as
P r {r t+t = j | r t = i} =
π i j t + o((t),
i = j
1 + π ii t + o((t), i = j
(3.37)
where π i j > 0,
N
j=1, j =i π i j = −π ii , t > 0 and lim t→0 (o((t)//t = 0. 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.
In this chapter, the multi-model jumping system (3.36) is assumed to be finite-time
stable, (A i , B i ) is controllable.
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 ] ,
(3.38)
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 .
We construct the following filter:
⎧
⎨
⎩
˙ ˆ
x(t) = A Fi ˆ
x(t) + A hi ˆ
x(t − τ ) + B Fi y(t),
ˆ
z(t) = C Fi ˆ
x(t),
ˆ
x(t) = ˆ
x 0 , r t = r 0 ,
(3.39)
where ˆ
x(t) ∈ R
n is the filter state and ˆ
y(t) ∈ R
q is filter output. A Fi , B Fi , and C Fi
are unknown parameters to be designed.
Letting e(t) = x(t) − ˆ
x(t), ν(t) = z(t) − ˆ
z(t) and ˜
x(t) = col [x(t) e(t)], the
error dynamic multi-model jumping system can be formulated as:
⎧
⎨
⎩
˙ ˜
x(t) = ˜
A i ˜
x(t) + ˜
A hi ˜
x(t − τ ) + ˜
B i ω(t),
ν(t) = ˜
C i ˜
x(t),
˜
x(t) =
x 0 x 0 − ˆ
x 0
,
(3.40)