Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
5.1 Multiple Inputs, Multiple Outputs Case
Lemma 57 (adaptation function)
For any ∆T >0, there exists a positive constant M(∆T ) such that:
− for any θ1 > 1,and
− any γ1 > γ0 > 0,
there is a function F (θ, I) such that the equation
˙θ = F (θ, I (t)) , (5.11)
for any initial value 1 ≤ θ (0) < 2θ1, and any measurable positive function I (t), has the
properties:
1. (5.11) has a unique solution θ (t) defined for all t ≥ 0, and this solution satisfies 1 ≤
θ (t) < 2θ1,
�
�
2. � F(θ,I)
�
�
� ≤ M,
θ 2
3. if I (t) ≥ γ1 for t ∈ [τ,τ + ∆T ] then θ (τ + ∆T ) ≥ θ1,
4. while I (t) ≤ γ0, θ (t) decreases to 1.
5.1.3.4 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 ∗ .
Let F be a function as in Lemma 57 with∆ T = T − d, and M such that fact 2 of Lemma 57
is true. Let α and β be the bounds from Lemma 54.
From the preparation of the proof, inequality (5.10) can be written, using Lemma 54 (i.e.
using ˜ S ≥ α Id)
d˜ε ′ ˜ S˜ε (t)
dt
≤−αqmθ˜ε ′ � �
S˜ε ˜ ′
(t) + 2˜ε S˜
˜b (˜z) − ˜b (˜x) − ˜∗ b (˜z)˜ε . (5.12)
From (5.12) we can deduce two inequalities: the first one, global, will be used mainly
when ˜ε ′ ˜ S˜ε (t) is not in a neighborhood of 0 and θ is large. The second one, local, will be used
when ˜ε ′ ˜ S˜ε (t) is small, whatever the value of θ.
Using � �
�� ˜b (˜z) − ˜b (˜x) − ˜∗ �
b (˜z)˜ε � ≤ 2Lb �˜ε� ,
together with α Id ≤ ˜ S ≤ β Id (Lemma 54), (5.12) becomes the “global inequality”
d˜ε ′ ˜ S˜ε (t)
dt
≤
�
−αqmθ +4 β
α Lb
�
˜ε ′ S˜ε ˜ (t) . (5.13)
Thanks to Lemma 56 we get the“local inequality” as follows:
�
�
�˜b (˜z) − ˜b (˜x) − ˜b ∗ �
�
(˜z)˜ε � ≤ Kθ n−1 �˜ε� 2 .
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