Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
to obtain, for all 0 ≤ τ ≤ τ0, and with �s >0:
�
�P � � ≤ (|P0| + |Q| τ0) e2�sτ0 1 =
≤ � 1
a
√ n + |Q| τ0
� α1
2�sτ0 1 e =
From the inequalities (3.14) and (3.15) we deduce that 12 for all 0 ≤ τ ≤ τ0
We now define
such that for all τ ≥ 0
α1 Id ≤ S (τ) ≤ β1 Id.
α1 .
�
α = min (a,α 1, �α) and β = max b,β 1, � �
β
α Id ≤ S (τ) ≤ β Id.
This relation is therefore true also in the t time scale.
3.7 Technical Lemmas
3.7 Technical Lemmas
(3.15)
Three lemmas are proposed in the following section. The two first are purely technical and
are used in the very last section of the present chapter. They are from [38]. Their respective
proofs are reproduced in Appendix B.2.
The third lemma concerns the adaptation function. It basically shows that the set of
canditate adaptation functions for our adaptive high-gain observer is not empty. The proof
is constructive: we display such a function.
Lemma 40 ([38])
Let {x (t) > 0,t≥ 0} ⊂ R n be absolutely continuous, and satisfying:
12 Trivially, we have
However
This is the reason why we need the relation:
dx(t)
dt ≤−k1x + k2x √ x,
�S� ≤β1 ⇒ S ≤ β1Id.
α1 ≤�S� � α1Id ≤ S.
�P �≤ 1
α1
⇒�P �≤ 1
and then we use the following matrix property (see Appendix B.1):
in order to end up with
α1
(P ≥ Q>0) ⇒ � Q −1 ≥ P −1 > 0 � .
α1Id ≤ S.
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