Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
3.9 Conclusion
Remark 44 (alternative result)
We propose a modification of the end of the proof that allows �ε(0)� 2 to appear in the
final inequality.
Consider equation (3.21) and replace it with:
�
1 αε∗ γ ≤ 2n−2 min
(2θ1) 4β , α5q2 m
16 K2β 2
�
. (3.26)
Then equation (3.22) becomes:
˜ε ′ ˜ S˜ε (t) ≤
αε ∗
β (2θ1) 2n−2 e−αqm(t−τ) . (3.27)
Now replace the definition of M0 given in equation (5.18) by M0 = max
�
supx,z∈X ε ′
�
Sε (0) , 1 .
Finally consider the very last inequality (3.25). It can also be developed as follows (with
M0 ≥ 1):
�ε (t)� 2 ≤ (2θ1) 2n−2 �˜ε (t)� 2 2n−2
(2θ1)
≤ ˜ε
α
′ S˜ε ˜ (t)
2n−2
(2θ1)
≤ 4 ˜ε
α
′ S˜ε ˜ −αqm(t−τ)
(τ)e
2n−2
(2θ1)
≤ 4
α
2n−2
(2θ1)
≤ 4
α
2n−2
(2θ1)
≤ 4
α
�
˜ε ′ ˜ S˜ε (0) e (−αqm+4 β
α Lb)T e (−αqmθ1+4 β
α Lb)(T ∗ −T ) �
e −αqm(t−τ)
�
β
2
β. �˜ε (0)� e (−αqm+4 α Lb)T
e (−αqmθ1+4 β
α Lb)(T ∗−T ) �
e −αqm(t−τ)
�
β
2
β. �˜ε (0)� M0e (−αqm+4 α Lb)T
e (−αqmθ1+4 β
α Lb)(T ∗−T ) �
e −αqm(t−τ)
From (3.24) we know that the bracketed expression is smaller than γ. Equations (3.26) and
(3.27), θ(0) = 1 lead to:
�ε (t)� 2 2n−2
(2θ1)
≤ 4 β �˜ε (0)�
α
2
�
≤�ε0� 2 ε ∗ e −αqm(t−τ) .
αε ∗
4β (2θ1) 2n−2
Since τ ≤ T ∗ , e −αqm(τ−T ∗ ) ≥ 1 and we can obtain the inequality
3.9 Conclusion
�ɛ(t)� 2 ≤�ε0� 2 ε ∗ e −αqm(t−T ∗ ) .
�
e −αqm(t−τ)
In this chapter, the adaptive high-gain extended Kalman filter was introduced for a multiple
input, single output system. The exponential convergence of the algorithm has been
proven. This proof has been decomposed in a series of significant lemmas:
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