Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
When we consider t ∈ [kδt, (k+1)δt[, this requirement means that
We know from (5.37) that in full generality
�
˜ɛ ′ �
S˜ɛ ˜ (+) ≤
Thus
and since
˜ε ′ ˜ S˜ε (T ) ≤
k
5.2 Continuous-discrete Framework
�
˜ɛ ′ � �
S˜ɛ ˜ (kδt) = ˜ɛ ′ �
S˜ɛ ˜
k (+).
�
˜ɛ ′ �
S˜ɛ ˜ (−). (5.44)
k
�
˜ε ′ �
β
S˜ε ˜ (−)e (−αqm+4 α
kT
Lb)(T −kT δt)
,
�
˜ε ′ �
S˜ε ˜ (−) is the end value of the equation (5.31) for t ∈ [(kT − 1)δt; kT δt[, then:
kT
�
˜ε ′ � �
S˜ε ˜ (−) ≤ ˜ε
kT
′ �
β
S˜ε ˜ (+)e(−αqm+4 α
kT −1 Lb)δt .
We can therefore, iteratively, independently from δt, obtain the inequality:
˜ε ′ ˜ S˜ε (T ) ≤ ˜ε ′ ˜S˜ε (0) e (−αqm+4 β
α Lb)T . (5.45)
We now suppose that θ ≥ θ1 for t ∈ [T, T ∗ ], T ∗ ∈ [ ˜ kδt, ( ˜ k + 1)δt[ and use (5.40):
˜ε ′ � �
S˜ε ˜ ∗ ′ β
(T ) ≤ ˜ɛ ˜S˜ε ˜kδt e (−αqmθ1+4 α Lb)(T ∗−˜ kδt)
. (5.46)
As before, independent of δt, we obtain:
where M0 = sup ε
x,z∈χ
′
Sε (0).
˜ε ′ S˜ε ˜ (T ∗ ) ≤
′
˜ɛ ˜S˜ε (T )e (−αqmθ1+4 β
α Lb)(T ∗−T )
,
≤ ˜ε ′ S˜ε ˜ (0) e (−αqm+4 β
α Lb)T
e (−αqmθ1+4 β
α Lb)(T ∗−T )
,
β
4 LbT ∗
α e−αqmθ1(T ∗−T ) ,
≤ M0e −αqmT e
Now, we choose θ1 and γ such that
M0e −αqmT e
(5.47)
β
4 LbT ∗
α e −αqmθ1(T ∗−T )
≤ γ (5.48)
and (5.42) are satisfied simultaneously, which results because e −cte×θ1 < cte
θ 2n−2
1
for θ1 suffi-
ciently large.
We check that the condition 13 2θ1δt θ1. The adaptation function must have the following features:
13 We arbitrarily set θmax =2θ1.
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