Digital Signal Processing Reference
In-Depth Information
where
d
dt
δ
PA
−
δ
PC
(
HH
)
−
1
CP
PC
(
HH
)
−
1
C
P
=
A
δ
P
+
δ
−
δ
P
and
d
dt
P
−
1
P
−
1
PP
−
1
P
−
1
A
A
P
−
1
P
−
1
GG
P
−
1
C
(
HH
)
−
1
C
=−
=−
−
−
+
As a result
2Tr
GG
P
−
1
d
dt
g
P
(δ
P
−
1
PP
−
1
P
,δ
P
)
=−
(
δ
δ
P
)
2Tr
HH
)
−
1
CP
P
−
1
PP
−
1
C
(
−
(
δ
δ
P
)
2Tr
GG
P
−
1
P
−
1
PP
−
1
≤−
(
δ
δ
P
)
which proves that the Riccati flow is contracting in the sense of (
3.6
) for the natural
metric of the cone with
is a lower bound on the eigenvalues
of
GG
, that is assumed to be invertible, and
p
max
is an upper bound on the eigenvalues
of
P
.
λ
=
μ/
p
max
where
μ
The contraction property of the Riccati equation is in fact due to the invariances
enjoyed by the natural metric of the cone. This can be understood the following way:
the equation writes
HH
)
−
1
CP
. The two first
terms neither expand nor contract as it is the differential form of the transformation
γ
B
(
d
PA
+
GG
−
PC
(
dt
P
=
AP
+
)
which is an isometry for the distance
d
P
+
(
n
)
. The addi-
P
)
=
(
I
+
τ
A
)
P
(
I
+
τ
A
d
tion of a positive matrix
Q
where
Q
is a positive matrix is neither expanding
nor contracting in the Euclidian space, but it contracts for the natural metric as
g
P
tends to dilate distances when
P
becomes large. Finally
dt
P
=
HH
)
−
1
CP
is a naturally contracting term which is paramount in the theory of the Kalman fil-
ter and observers (correction term). We have the following proposition, refining the
results of [
12
] in the case of time-independent coefficients:
d
PC
(
dt
P
=−
,
,
(
,
)
Proposition 1
Suppose the matrices A
C
G do not depend on the time t ,
A
C
∈
P
+
(
)
is observable and G is full-rank. There exists a unique solution Q
n
of the
algebraic Riccati equation defined by
Φ(
Q
)
=
0
where
Φ
is given by
(
3.3
)
. For any
R
∈
P
+
(
n
)
,letS
R
be the ball center Q and radius d
P
+
(
n
)
(
Q
,
R
)
S
R
={
P
∈
P
+
(
n
),
d
P
+
(
n
)
(
Q
,
P
)
≤
d
P
+
(
n
)
(
Q
,
R
)
}
0
be the lowest eigenvalue of G G
.
Let M
R
=
sup
{
P
2
,
P
∈
S
R
}
<
∞
, and
μ>
d
Let P
1
(
t
),
P
2
(
t
)
be two arbitrary solutions of the Riccati equation
dt
P
=
Φ(
P
)
initialized in S
R
. Then P
1
(
t
),
P
2
(
t
)
∈
S
R
for any t
≥
0
, and the following contraction
result holds
