Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
For t ∈ [kδt;(k + 1)δt[,
and
d ˜ S
dt
d˜ɛ
dt
� �
= θ − A(u)+ 1
θ ˜b ∗ �′
(z, u)
5.2 Continuous-discrete Framework
�
= θ A(u)˜ɛ + 1
� �
˜b(˜z, u) − b(˜x, u)
θ
�
, (5.29)
˜S − ˜ �
S A(u)+ 1
θ ˜b ∗ �
(z, u) − ˜ SQ˜ �
S . (5.30)
We now consider the Lyapunov function ˜ɛ ′ ˜ S˜ɛ and use identities (5.29, 5.30) to obtain the
equality below:
�
d ˜ɛ ′ �
S˜ɛ ˜ �
2
dt = θ θ ˜ɛ′ � �
S˜
˜b(˜z, u) − ˜b(˜x, u) − ˜b ∗ (˜z, u)˜ɛ − ˜ɛ ′ �
SQ ˜ S˜ɛ ˜ .
(5.31)
Similarly, at time kδt,
�
˜ɛk(+) = Id − θδt ˜ S −1 ′
k (+)C r−1 �
C ˜ɛk(−), (5.32)
and,
˜Sk(+) = ˜ Sk(−)+θδtC ′
r −1 C. (5.33)
As we did for the differential equations, we use (5.32) and (5.33) to compute the Lyapunov
function at time kδt:
�
˜ɛ ′ �
S˜ɛ ˜ (+) = ˜ɛ′ k
k (−)
�
Id − θδt ˜ S −1
k
�
From (5.33), we replace
(+)C ′
×
= ˜ɛ ′
k (−)
�
˜Sk(+) − 2θδtC ′
r−1C r−1 �′
C
Id − θδt ˜ S −1
k
˜Sk(+)
(+)C ′
r −1 C
�
˜ɛk(−)
+(θδt) 2 C ′
r −1 C ˜ Sk(+) −1 C ′
r −1 C
�
θδtC ′
r−1 � �
C with ˜Sk(+) − ˜ �
Sk(−) :
�
˜ɛ ′ �
S˜ɛ ˜ (+) = ˜ɛ′
k
From equation (5.33) we write:
S −1
k
k (−)
= ˜ɛ ′
k (−)
�
˜Sk(−) ˜ Sk(+) −1 �
Sk(−) ˜
� ˜Sk(−) −1 ˜ Sk(+) ˜ Sk(−) −1
−1
(−)Sk(+)Sk (−) =S−1
k (−)+θδt
and compute [S −1
k
This results in:
�
˜ɛ ′ �
S˜ɛ ˜
k (+) = ˜ɛ′ � Sk(−) ˜ − C ′
( r
θδt + C ˜ S −1
=
k
�
˜ɛ ′ �
S˜ɛ ˜ (−) − ˜ɛ′ k (−)
�
C ′
( r
r S−1
k
(−)C ′
˜ɛk(−)
� −1
˜ɛk(−).
�
˜ɛk(−).
(5.34)
(5.35)
CS −1
k (−), (5.36)
(−)Sk(+)S −1
k (−)]−1 by using Lemma 64 with λ = θδt
r and M = ˜ S −1
k (−).
k
109
′
(−)C ) −1 �
C ˜ɛ
θδt + C ˜ S −1
k
(−)C ′
) −1 C
�
˜ɛk(−).
(5.37)