Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
3.8 Proof of the Theorem
Consequently, provided there is a real γ such that
�
1 αε∗ γ ≤ 2n−2 min
(2θ1) 4 , α5q2 m
16 K2β 2
�
, (3.21)
then ˜ε ′ ˜ S˜ε (τ) ≤ γ implies, for any t ≥ τ,
˜ε ′ ˜ S˜ε (t) ≤
αε ∗
(2θ1) 2n−2 e−αqm(t−τ) . (3.22)
Note that excluding the change of variables, we have arrived at the end result.
From (3.19):
˜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 (3.19) 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 ) ,
M0 = sup
x,z∈X
ε ′ Sε (0) . (3.23)
β
4 LbT ∗
α e −αqmθ1(T ∗−T )
≤ γ (3.24)
and (3.21) to be satisfied simultaneously, which is possible since e−cte×θ1 cte <
θ 2n−2
1
enough. Let us chose a function F as in Lemma 42 with∆ T = T − d and γ1 = λ0 dγ β .
We claim that there exists τ ≤ T ∗ such that ˜ε ′ ˜ S˜ε (τ) ≤ γ.
Indeed, if ˜ε ′ ˜ S˜ε (τ) > γ for all τ ≤ T ∗ because of Lemma 33:
γ < ˜ε ′ S˜ε ˜ 2 2 β
(τ) ≤ β �˜ε (τ)� ≤ β �ε (τ)� ≤
λ0 Id (τ + d) .
d
for θ1 large
Therefore, Id (τ + d) ≥ γ1 for τ ∈ [0,T ∗ ] and hence Id (τ) ≥ γ1 for τ ∈ [d, T ∗ ]. Thus, we have
θ (t) ≥ θ1 for t ∈ [T, T ∗ ] which provides a contradiction (i.e. ˜ε ′ ˜ S˜ε (T ∗ ) ≤ γ) thanks to (3.8)
and (3.24).
Finally, for t ≥ τ, using (3.22)
which proves the theorem.
�ε (t)� 2 ≤ (2θ1) 2n−2 �˜ε (t)� 2
2n−2
(2θ1)
≤ α
≤ ε∗e−αqm(t−τ) 53
˜ε ′ ˜ S˜ε (t)
(3.25)