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The derivative of
V
along the solutions of system (8.6) is
H
)+
1
−
H
)
T
˙
k
A
tr(
A
T
˙
V
=
A
)
−
tr((
I
1
H A
k
A
tr(
A
T
−
H
)
T
−
H
)
T
H
Ad
H
−
1
α
=
−
tr((
I
−
(
I
)
−
β
)
.
(3),tr(
B
T
G
)=tr(
B
T
∈
∈
sl
Knowing that for any matrices
G
SL
(3) and
B
P
(
G
)) =
,
P
B
(
G
)
, one obtains:
1
k
A
β
.
V
=
(
H
T
(
I
−
H
))
α
−
A
(
H
T
(
I
−
H
)) +
P
,
Ad
H
−
1
,
P
Introducing the expressions of
α
and
β
(8.7) in the above equation yields
V
=
(
H
T
(
I
−
H
))
2
−
k
H
||
P
||
.
(8.9)
The derivative of the Lyapunov function is negative semidefinite, and equal to zero
when
(
H
T
(
I
−
H
)) = 0. The dynamics of the estimation error is autonomous,
i.e.
P
it is given by
H
=
H
A
+
k
H
−
H
))
˙
(
H
T
(
I
P
(8.10)
˙
A
=
k
A
(
H
T
(
I
−
H
))
P
.
Therefore, we deduce from LaSalle theorem that all solutions of this system con-
verge to the largest invariant set contained in
(
H
,
A
)
(
H
T
(
I
−
H
)) = 0
.
We now prove that, for system (8.10), the largest invariant set
E
contained in
{
|
P
}
(
H
,
A
)
(
H
T
(
I
−
H
)) = 0
{
|
P
}
is equal to
E
s
∪
E
u
. We need to show that the solutions
(
H
,
A
)
(
H
T
(
I
−
H
)) = 0
of system (8.10) belonging to
{
|
P
}
for all
t
consist of all fixed
points of
E
s
∪
E
u
. Note that
E
s
=(
I
,
0) is clearly contained in
E
. Let us thus consider
˙
such a solution (
H
(
t
)
,
A
(
t
)). First, we deduce from (8.10) that
A
(
t
) is identically zero
−
H
(
t
))) is identically zero on the invariant set
E
and therefore
A
is constant. We also deduce from (8.10) that
H
is solution to the equation
(
H
T
(
t
)(
I
since
P
˙
H
=
H A
.
Note that at this point one cannot infer that
H
is constant. Still, we omit from now
on the possible time-dependence of
H
to lighten the notation.
Since
(
H
T
(
I
−
H
)) = 0, we have that
P
−
H
)=
1
H
(
I
3
trace(
H
(
I
−
H
))
I
(8.11)
which means that
H
is a symmetric matrix. Therefore, it can be decomposed as:
H
=
UDU
(8.12)
where
U
SL
(3) is a diagonal matrix which con-
tains the three real eigenvalues of
H
. Without loss of generality let us suppose that
the eigenvalues are in increasing order:
∈
SO
(3) and
D
= diag(
λ
1
,
λ
2
,
λ
3
)
∈
λ
1
≤
λ
2
≤
λ
3
.
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