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5 Higher Order Moment Robust Filtering …
Finally, two simulation examples are provided to demonstrate the validity and
superiority of the developed approach. In summary, the proposed approach can not
only cover the mean error and mean square error estimations as special cases, but
also lead to less conservative filtering results.
5.1 Higher Order Moment Robust Filter Design for
Multi-model Jumping System
5.1.1 System Description
For a given probability space ((, F, P), we consider the following continuous-time
multi-model jumping system:
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
˙
x(t) = A(r t )x(t) + B d (r t )ω(t),
y(t) = C 1 (r t )x(t) + D d (r t )ω(t),
z(t) = C 2 (r t )x(t),
x(t) = x 0 , r t = r 0 , t = 0.
(5.1)
where x(t) ∈ R
n is the state, y(t) ∈ R
m is the measured output, z(t) ∈ R
r is the signal to be estimated, ω(t) ∈ L
m
2 [0, ∞] is the unknown disturbance with known finitefrequency ranges. A(r t ), B d (r t ), C 1 (r t ), C 2 (r t ), D d (r t ) are known mode-dependent
matrices with appropriate dimensions, x 0 , r o are the initial state and the initial mode.
The random form process {r t , t ≥ 0} is a continuous-time discrete-state Markov
stochastic process taking values in a finite set = {1, 2, . . . , N } 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
(5.2)
where t > 0 and lim t↓0 o((t)//t → 0. π i j ≥ 0 is the transition probability rates
from mode i at time t to mode j (i = j) at time t + t, and
N
j=1, j =i π i j = −π ii .
The transition rate matrix is denoted by =
π i j , i, j ∈
.
For presentation convenience, we denote A(r t ), B d (r t ), C 1 (r t ), C 2 (r t ), D d (r t ), as
A i , B di , C 1i , C 2i , D di , respectively.
To investigate the high-order moment filtering issue, the multi-model jumping
system (5.1) must be transformed into a higher-order component form including the
mode jumping rule; therefore, the indicator function 1 A is defined by the set A ∈ R:
1 A (ω) =
1
ifω ∈ A
0
otherwise
(5.3)
Define