Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
5.1 Multiple Inputs, Multiple Outputs Case
The dynamics of the error after the change of coordinates are given by:
d˜ε
dt
= θ
and the Riccati equation becomes
d ˜ S
dt
= θ
� F(θ,I)
θ 2
�
F(θ, I)
−
θ2 N ˜ε + A˜ε − ˜ S −1 C ′
R −1 C ˜ε + 1
� �
˜b (˜z, u) − ˜b (˜x, u)
θ
�
, (5.8)
�
N ˜ S + ˜ SN
�
−
�
A ′ ˜ S + ˜ SA
These two equations are used to compute the derivative of �ε ′ � S�ε:
�
+ C ′
R−1C − ˜ SQ˜ S − 1
θ ˜ S˜b ∗ (˜z, u) − 1
θ ˜b ∗′ (˜z, u) ˜ �
S .
(5.9)
d
�
�ε
dt
′ �
S�ε � ≤−θqm�ε ′ � �
S� 2 ′
�ε +2�ε �S ˜b (�z, u) − ˜b (�x, u) − ˜∗ b (�z, u) �ε . (5.10)
The proof of the theorem comes from the stability analysis of this last equation. This analysis
requires the use of four lemmas, which we state below.
5.1.3.3 Intermediary Lemmas
The proofs of Lemmas 54 and 57 can be found in Sections 3.6 and 3.7 respectively. Lemmas
55 and 56 are proven in Appendix B.2.
Lemma 54 (Bounds for the Riccati equation)
Let us consider the Riccati equation (5.8). We suppose that
�
�
− the functions ai (u (t)), ��b ∗ i,j (z,u)
�
�
�,
−
�
�
�
� F(θ,I)
�
�
�
�are smaller than aM > 0 and if ai (u (t)) >am > 0
θ 2
− S (0) = S0 is symmetric definite positive, taken in a compact of the form aId ≤ S0 ≤
bId, and
− θ(0) = 1
Then there exist two constants 0 < α 0,t≥ 0} ⊂ R n be absolutely continuous, and satisfying:
dx(t)
dt ≤−k1x + k2x √ x,
for almost all t>0, for k1, k2 > 0. Then, if x (0) < k2 1
4k2 , x(t) ≤ 4 x (0) e
2
−k1t .
Lemma 56 (Technical lemma two)
Consider �b (˜z) − �b (˜x) − �b ∗ (˜z)˜ε as in the inequality (3.12) (omitting to write u in ˜ �
b) and
�
suppose θ ≥ 1. Then ��b (˜z) − �b (˜x) − �b ∗ �
�
(˜z)˜ε � ≤ Kθn−1 �˜ε� 2 , for some K>0.
100