82
5 Higher Order Moment Robust Filtering …
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.
With similar process to the last subsection, the deterministic higher order component form of the multi-model jumping system (5.31) can be deduced as
⎧
⎨
⎩
˙
S(T, p) = A tp S(t, p) + B dt p ω N p (t),
y tp (t) = C 1tp S(t, p) + D dt p ω N p (t),
z tp (t) = C 2tp S(k, p),
(5.33)
where
S(t, p) = {s 1 (t, p), . . . , s N (t, p)}
T
,
s j (t, p) = c q j (t) ( p),
q j (t) = E
x(t)1 {r (t)= j }
,
ω N p (t) = {ω(t), . . . , ω(t)}
T
,
y tp (t) = {y(t), . . . , y(t)}
T
,
z tp (t) = {z(t), . . . , z(t)}
T
,
A tp = diag{A 1
⊗ p
, A 2
⊗ p
, . . . , A N
⊗ p
} + tp
T
⊗ I n x
p ∈ R
(n x
p ×N )×(n x
p ×N )
,
tp =
π i j
N ×N
,
B dt p = diag {B ω1 , . . . , B ω N } ⊗ I n w
p−1 ∈ R
(n x
p ×M)×(n ω
p ×N )
,
C 1tp = diag {C 11 , . . . , C 1N } ⊗ I n
p−1
x
,
D dt p = diag {D ω1 , . . . , D ω N } ⊗ I n
p−1
v
,
C 2tp = diag {C 21 , . . . , C 2N } ⊗ I n
p−1
x
.
Consider a high-order moment filter for the multi-model jumping system (5.33)
with state realization described in the following form:
˙
S f (t, p) = A f S f (t, p) + B f y tp (t),
z f (t) = C f S f (t, p) + D f y tp (t)
(5.34)
where S f (t, p) and z f (t) are the state and output of the filter, respectively. A f , B f ,
C f , and D f are the filter parameters to be determined.
Combining systems (5.33) and (5.34), the high-order moment filtering error
dynamic system is expressed as
˙ ˜
S(t, p) = ˜
A ˜
S(t, p) + ˜
Bω N p (t)
e(t) = ˜
C ˜
S(t, p) + ˜
Dω N p (t)
(5.35)