Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
5.2 Continuous-discrete Framework
with qm > 0 such that qmId < Q (and omitting to write the control variable u).
From (5.39) we can deduce two bounds: the first bound, the local bound, will be useful
when ˜ε ′ ˜ S˜ε (t) is small independent of the value of θ. The second bound, the global bound,
will be useful mainly when ˜ε ′ ˜ S˜ε (t) is not in the neighborhood of 0.
Global bound: Starting from:
�
�
�˜b (˜z) − ˜b (˜x) − ˜b ∗ �
�
(˜z)˜ε � ≤ 2Lb �˜ε� ,
together with α Id ≤ ˜ S ≤ β Id (Lemma 66), (5.39) becomes
d˜ε ′ ˜ S˜ε (t)
dt
≤
�
−αqmθ +4 β
α Lb
�
˜ε ′ S˜ε ˜ (t) . (5.40)
Local bound: Using Lemma 56
�
�
�˜b (˜z) − ˜b (˜x) − ˜b ∗ �
�
(˜z)˜ε � ≤ Kθ n−1 �˜ε� 2 ,
which because 1 ≤ θ ≤ 2θ1, implies that
The fact that �˜ε� 3 =
d˜ε ′ ˜ S˜ε (t)
dt
� 3
�˜ε�
2� 2
≤−αqm˜ε ′ S˜ε ˜ (t)+2K (2θ1) n−1 � �
�
�S˜ �
� �˜ε� 3 .
≤
� 1
α ˜ε′ ˜ S˜ε (t)
� 3
2
, allows us to write
˜ε ′ S˜ε ˜ (t) ≤−αqm˜ε ′ S˜ε ˜
2K (2θ1)
(t)+ n−1 β
α 3
2
Let us apply Lemma 55: If there exists ξ such that
then for any kδt ≤ ξ ≤ t ≤ (k + 1)δt
If γ ∈ R such that
then ˜ε ′ ˜ S˜ε (ξ) ≤ γ implies
˜ε ′ ˜ S˜ε (ξ) ≤
α5q2 m
16 K2 (2θ1) 2n−2 ,
β2 ˜ε ′ ˜ S˜ε (t) ≤ 4˜ɛ ′ ˜S˜ε (ξ)e −αqm(t−ξ) .
�
˜ε ′ � 3
2
S˜ε ˜ (t) . (5.41)
�
1 αε∗ γ ≤ 2n−2 min
(2θ1) 4β , α5q2 m
16 K2β 2
�
, (5.42)
˜ε ′ ˜ S˜ε (t) ≤
αε ∗
β (2θ1) 2n−2 e−αqm(t−ξ) . (5.43)
Given any arbitrary value of δt, there exists kT ∈ N such that T ∈ [kT δt;(kT + 1)δt[. From
the global bound (5.40), with θk ≥ 1, for all k ∈ N:
˜ε ′ ′ β
S˜ε ˜ (T ) ≤ ˜ε ˜S˜ε (kT δt)e (−αqm+4 α Lb)(T −kT δt)
.
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