Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
Since 1 ≤ θ ≤ 2θ1 inequality (5.12) implies
Since �˜ε� 3 =
� 3
�˜ε�
2� 2
d˜ε ′ ˜ S˜ε (t)
dt
≤
5.1 Multiple Inputs, Multiple Outputs Case
≤−αqm˜ε ′ S˜ε ˜ (t)+2K (2θ1) n−1 � �
�
�S˜ �
� �˜ε� 3 .
� 1
α ˜ε′ ˜ S˜ε (t)
� 3
2
, it becomes
˜ε ′ S˜ε ˜ (t) ≤−αqm˜ε ′ S˜ε ˜
2K (2θ1)
(t)+ n−1 β
α 3
2
Let us apply 6 Lemma 55, 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) . (5.14)
Provided there exists a real γ, such that
�
1 αε∗ γ ≤ 2n−2 min
(2θ1) 4 , α5q2 m
16 K2β 2
�
, (5.15)
then ˜ε ′ ˜ S˜ε (τ) ≤ γ implies, for any t ≥ τ,
From (5.13)
˜ε ′ ˜ S˜ε (t) ≤
αε ∗
(2θ1) 2n−2 e−αqm(t−τ) . (5.16)
˜ε ′ ˜ S˜ε (T ) ≤ ˜ε ′ ˜ S˜ε (0) e (−αqm+4 β
α Lb)T ,
and if we suppose θ ≥ θ1 for t ∈ [T, T ∗ ], T ∗ >T, using (5.13) again:
where
Now, we choose θ1 and γ for
˜ε ′ ˜ S˜ε (T ∗ ) ≤ ˜ε ′ ˜ S˜ε (0) e (−αqm+4 β
α Lb)T e (−αqmθ1+4 β
α Lb)(T ∗ −T )
≤ M0e −αqmT e
M0e −αqmT e
β
4 LbT ∗
α e−αqmθ1(T ∗−T ) ,
(5.17)
M0 = sup
x,z∈X
ε ′ Sε (0) . (5.18)
β
4 LbT ∗
α e −αqmθ1(T ∗−T )
≤ γ (5.19)
and (5.15) to be satisfied simultaneously, which is possible since e−cte×θ1 cte <
θ 2n−2 for θ1
1
sufficiently large. Let us choose a function F as in Lemma 57 with∆ T = T − d and γ1 = λ0 dγ β .
6 This lemma cannot be applied if we use Qθ and R instead of Qθ and Rθ in the definition of the observer
as in [38]. This is due to the presence of a F
θ
times.
term that prevents parameters k1 and k2 to be positive for all
102