60
4 Finite-Frequency Robust Filtering for 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.
(4.19)
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. 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
(4.20)
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.
The main objective of robust multiple frequency filtering is to design a filter to
simultaneously satisfy the stochastic stability of the multi-model jumping system
and the multiple frequency performances.
Similar to the last subsection, the derandomisation technique is introduced and
define the indicator function 1 A as:
1 A (ω) =
1,
if ω ∈ A
0,
otherwise
and denote
q i (t) = E
x(t)1 {r (t)=i}
.
(4.21)
Then, the extended system can be obtained in the following form:
⎧
⎨
⎩
˙
q(t) = Aq(t) + B d ˜
ω(t),
˜
y(t) = C 1 q(t) + D d ˜
ω(t),
˜
z(t) = C 2 q(t),
(4.22)
where
q(t) =
q 1
T
(t) · · · q N
T
(t)
T ,
˜
y(t) =
y
T
(t) · · · y
T
(t)
T ,