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5 Higher Order Moment Robust Filtering …
and the right side of (5.5) is written as
A i q j (t)dt+
N
i=1
π i j q i (t)dt+B di ω(t)dt
=
∞
p=0
⎧
⎨
⎩
c
A i q j (t)dt+
N
i=1
π i j q i (t)dt+B di ω(t)dt
( p, n)
T
⊗ p
p!
⎫
⎬
⎭
(5.11)
which yields
c dq j (t) ( p, n) = c
A i q j (t)dt+
N
i=1
π i j q i (t)dt+B di ω(t)dt
( p, n)
(5.12)
with the cumulant property c T r ( p) = T
⊗ p C r ( p), where T is the transform matrix,
thus we have,
c dq j (t) ( p, n) = A i
⊗ p c q j (t)dt ( p, n) +
N
i=1
π i j c q i (t)dt ( p, n) + B di
⊗ p c ω(t)dt ( p, n)
(5.13)
Defining s j (t, p) = c q j (t) ( p), S(t, p) = {s 1 (t, p), . . . , s N (t, p)}
T , ω N p (t) =
{ω(t), . . . , ω(t)}
T , Eq. (5.13) can be rewritten as
˙
S(t, p) = A tp S(t, p) + B dt p ω N p (t)
(5.14)
where
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 )
,
B dt p = diag {B ω1 , . . . , B ω N } ⊗ I n w
p−1 ∈ R
(n x
p ×M)×(n ω
p ×N )
,
tp =
π i j
N ×N
. Then, the multi-model jumping system (5.1) is transformed into
the following higher order expression
⎧
⎨
⎩
˙
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)
(5.15)
where
y tp (t) = {y(t), . . . , y(t)}
T
,
z tp (t) = {z(t), . . . , z(t)}
T
,
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 system (5.15) with state realization
described in the following form: