Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
3.8 Proof of the Theorem
3.8 Proof of the Theorem
First of all let us choose a time horizon d (in Id (t)) and a time T such that 0 < d < T < T ∗ .
Set∆ T = T − d. Let λ be a strictly positive number and M = 1
∆T
+ λ
4
as in Lemma 42. Let
α and β be the bounds from Lemma 39.
From the preparation for the proof, inequality (3.12) can be written, using Lemma 39
(i.e. using ˜ S ≥ α Id), and omitting the control variable u
d˜ε ′ ˜ S˜ε (t)
dt
≤−αqmθ˜ε ′ � �
S˜ε ˜ ′
(t) + 2˜ε S˜
˜b (˜z) − ˜b (˜x) − ˜∗ b (˜z)˜ε . (3.18)
From (3.18) we can deduce two inequalities: the first one, local, will be used when ˜ε ′ ˜ S˜ε (t)
is small, whatever the value of θ. The second one, global, will be used mainly when ˜ε ′ ˜ S˜ε (t)
is not in a neighborhood of 0 and θ is large.
Using � �
�� ˜b (˜z) − ˜b (˜x) − ˜∗ �
b (˜z)˜ε � ≤ 2Lb �˜ε� ,
together with α Id ≤ ˜ S ≤ β Id (Lemma 39), (3.18) becomes the “global inequality”
d˜ε ′ ˜ S˜ε (t)
dt
≤
�
−αqmθ +4 β
α Lb
�
˜ε ′ S˜ε ˜ (t) . (3.19)
Because of Lemma 41, we obtain the“local inequality” as follows:
�
�
�˜b (˜z) − ˜b (˜x) − ˜b ∗ �
�
(˜z)˜ε � ≤ Kθ n−1 �˜ε� 2 .
Since 1 ≤ θ ≤ 2θ1, inequality (3.18) implies
Since �˜ε� 3 =
� 3
�˜ε�
2� 2
d˜ε ′ ˜ S˜ε (t)
dt
≤
≤−αqm˜ε ′ S˜ε ˜ (t)+2K (2θ1) n−1 � �
�
�S˜ �
� �˜ε� 3 .
� 1
α ˜ε′ ˜ S˜ε (t)
� 3
2
, the inequality becomes
˜ε ′ S˜ε ˜ (t) ≤−αqm˜ε ′ S˜ε ˜
2K (2θ1)
(t)+ n−1 β
α 3
2
Let us apply 15 Lemma 40 which states that if
then, for any t ≥ τ,
˜ε ′ ˜ S˜ε (τ) ≤
α5q2 m
16 K2 (2θ1) 2n−2 ,
β2 ˜ε ′ ˜ S˜ε (t) ≤ 4˜ε ′ ˜ S˜ε (τ)e −αqm(t−τ) .
�
˜ε ′ � 3
2
S˜ε ˜ (t) . (3.20)
15 This lemma cannot be applied if we use Qθ and R instead of Qθ and Rθ in the definition of the observer
as it is done in [38]. This is due to the presence of a F
θ
for all times.
term that prevents parameters k1 and k2 to be positive
52