Digital Signal Processing Reference
In-Depth Information
the identity. The matrices
A
(
t
)
,
G
(
t
)
,
C
(
t
)
and
H
(
t
)
have the appropriate dimensions
and it is assumed that
A
is invertible for all
t
.
The classical equations of the Kalman filter define a recursion for
(
t
)
x
ˆ
(
t
)
= E[
x
(
t
)
|{
y
(
s
)
}
0
≤
s
≤
t
]
the best estimate of the true state
x
, using the conditional error covariance
matrix
P
(
t
)
d
dt
ˆ
PC
(
HH
)
−
1
C
PC
(
HH
)
−
1
y
x
=
(
A
−
)
ˆ
x
+
(3.2)
d
dt
P
PA
+
GG
−
PC
(
HH
)
−
1
CP
=
Φ
t
(
P
)
=
AP
+
(3.3)
The mapping
Φ
t
defines the continous matrix-valued Riccati differential Equation,
and for each
P
∈
P
+
(
n
)
,
Φ
t
(
P
)
is a tangent vector to
P
+
(
n
)
at
P
, i.e.
Φ
t
(
P
)
∈
T
P
P
+
(
n
)
.
3.2.1 Natural Metric of P
+
(
n
)
The geometry of the
n
-dimensional cone of symmetric positive definite matrices
P
+
(
)
(
)
n
has been well-studied in the literature. The group
GL
n
acts transitively on
this set via the following action
APA
γ
A
:
P
+
(
n
)
→
P
+
(
n
),
P
→
(3.4)
for any
A
∈
GL
(
n
)
.If
P
is the covariance matrix of a gaussian vector of zero
mean
x
, then
γ
A
(
P
)
is the covariance matrix of the transformed vector
Ax
.If
P
,
Q
∈
P
+
(
n
)
are two arbitrary points of the cone there always exists
A
∈
GL
(
n
)
such
that
Q
=
γ
A
(
P
)
. This property makes
P
+
(
n
)
a so-called homogeneous space under
the
GL
(
n
)
group action. The isotropy subgroup is the subgroup of
GL
(
n
)
stabilizing
AI A
=
the identity matrix i.e.,
{
A
∈
GL
(
n
),
I
}=
O
(
n
)
. As a general property of
homogeneous spaces, the following identification holds
P
+
(
n
)
=
GL
(
n
)/
O
(
n
)
The isotropy subgroup being compact, there exists a
GL
(
n
)
-invariant Riemannian
metric on
P
+
(
called the natural metric [
11
]. This metric is defined as the usual
scalar product at the identity
n
)
g
I
(
X
1
,
X
2
)
=
Tr
(
X
1
X
2
)
