14
2 Robust Filtering for Multi-model Jumping System
an unbiased filter that satisfies the given H ∞ norm performance index condition
for the multi-model jumping system. Finally, the above two filtering problems are
described as optimization algorithms for solving.
2.1 Robust H ∞ Filtering for Multi-model Jumping Systems
Consider the following multi-model jumping system defined on the probability space
((, F, P):
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
˙
x(t) = [A(r t ) + A(t, r t )]x(t) + [A h (r t ) + A h (t, r t )] x(t − h) + B(r t )ω(t)
y(t) = C(r t )x(t) + C h (r t )x(t − h) + D(r t )ω(t)
z(t) = L(r t )x(t)
x(t) = λ(t), r t = r 0 , t ∈ [−h 0].
(2.1)
where x(t) ∈ R
n is the state, y(t) ∈ R
m is the measure output, z(t) ∈ R
l is the controlled output and ω(t) ∈ L
m
2 [0, ∞] is the the unknown disturbance. 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 ), L(r t ) are known mode-dependent
matrices with appropriate dimensions, h is the time-delay, {r t , t ≥ 0} denotes a
continuous-time discrete state Markov stochastic process with values in a finite set
= {1, 2, . . . , N }. Moreover, λ(t) is a continuous vector-valued initial function
assumed to be continuously differentiable on [−h 0] and r 0 is the initial mode. r t
depends on a homogeneous Markov process with the following transition probability:
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
(2.2)
where π i j ≥ 0 with
N
j=1, j =i π i j = −π ii is the transition probability rates from mode
i at time t to mode j (i = j) at time t + t.
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 ), L(r t ) as
A i , A i , A hi , A hi , B i , C i , C hi , D i , L i , respectively. Moreover, the time-varying
uncertain matrices A i and A hi satisfy:
A i A hi
= M i i (t)
N i N hi
(2.3)
where i (t) is a mode-dependent Lebesgue norm measurable function with i (t) ≤
1; M i , N i and N hi are known mode-dependent matrices.
Proposition 2.1 The multi-model jumping system (2.1) is said to be stochastically
stable, if the following inequality is satisfied:
2 Robust Filtering for Multi-model Jumping System
an unbiased filter that satisfies the given H ∞ norm performance index condition
for the multi-model jumping system. Finally, the above two filtering problems are
described as optimization algorithms for solving.
2.1 Robust H ∞ Filtering for Multi-model Jumping Systems
Consider the following multi-model jumping system defined on the probability space
((, F, P):
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
˙
x(t) = [A(r t ) + A(t, r t )]x(t) + [A h (r t ) + A h (t, r t )] x(t − h) + B(r t )ω(t)
y(t) = C(r t )x(t) + C h (r t )x(t − h) + D(r t )ω(t)
z(t) = L(r t )x(t)
x(t) = λ(t), r t = r 0 , t ∈ [−h 0].
(2.1)
where x(t) ∈ R
n is the state, y(t) ∈ R
m is the measure output, z(t) ∈ R
l is the controlled output and ω(t) ∈ L
m
2 [0, ∞] is the the unknown disturbance. 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 ), L(r t ) are known mode-dependent
matrices with appropriate dimensions, h is the time-delay, {r t , t ≥ 0} denotes a
continuous-time discrete state Markov stochastic process with values in a finite set
= {1, 2, . . . , N }. Moreover, λ(t) is a continuous vector-valued initial function
assumed to be continuously differentiable on [−h 0] and r 0 is the initial mode. r t
depends on a homogeneous Markov process with the following transition probability:
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
(2.2)
where π i j ≥ 0 with
N
j=1, j =i π i j = −π ii is the transition probability rates from mode
i at time t to mode j (i = j) at time t + t.
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 ), L(r t ) as
A i , A i , A hi , A hi , B i , C i , C hi , D i , L i , respectively. Moreover, the time-varying
uncertain matrices A i and A hi satisfy:
A i A hi
= M i i (t)
N i N hi
(2.3)
where i (t) is a mode-dependent Lebesgue norm measurable function with i (t) ≤
1; M i , N i and N hi are known mode-dependent matrices.
Proposition 2.1 The multi-model jumping system (2.1) is said to be stochastically
stable, if the following inequality is satisfied: