Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
5.1.3.2 Preparation for the Proof
5.1 Multiple Inputs, Multiple Outputs Case
First, we remind the reader of the change of variables that we performed in Subsection 3.5.
− ɛ = x − z is the estimation error,
− ˜x = ∆x,
− ˜ b(., u) =∆b(∆ −1 ., u),
− ˜ b ∗ (., u) =∆b(∆ −1 ., u)∆ −1 .
The definition of the matrix ∆ gives us the following property.
Lemma 51
1. The vector field ˜ b(˜x, u) has the same Lipschitz constant as b(x, u).
2. The matrix ˜ b ∗ (˜x, u) has the same bound as the Jacobian of b(x, u)
Remark 52
This lemma is valid for both the definition of Chapter 3 and the definition of Section 5.1.2
provided above. The proof is given in [57], pg. 215. We reproduce the proof for the multiple
output case since it allows us to justify the definition of ∆.
Proof.
Recall that θ(t) ≥ 1.
1. Consider a component of ˜b(., u) of the form ˜b k i (., u) with i ∈ {1, ..., ny} and k ∈
{1, ..., ni − 1}. From the change of variables ˜b(., u) =∆b(∆−1 ., u) we have:
˜k 1
bi (x, u) =
n∗ − ni + k − 1 b
�
θ n∗−ni 1
xi , θ n∗−ni+1 2
xi , ...,θ n∗ �
−ni+k−1 k
xi ,u .
We denote Lb as the Lipschitz constant of b(., u) w.r.t. the variable x:
�
�
�˜b k i (x, u) − ˜b k �
�
i (z, u) �
=
≤
1
θn∗ �
�b −ni +k−1
� θn∗−nix1 i , ...,θ n∗−ni+k−1xk i ,u �
−b � θn∗−niz1 i , ...,θ n∗−ni+k−1zk i ,u �� �
Lb
θn∗ �
�
−ni +k−1
� θn∗−nix1 i , ...,θ n∗−ni+k−1xk i ,u �
− � θn∗−niz1 i , ...,θ n∗−ni+k−1zk i ,u �� �
Lb
≤
θn∗−ni +k−1 θn∗−ni+k−1 � � � x1 i , ..., xki ,u� − � z1 i , ..., zk i ,u��� �
= Lb
� � x1 i , ..., xki ,u� − � z1 i , ..., zk i ,u��� .
(5.5)
Therefore the Lipschitz constant of b(., .) is the same in the two coordinate systems.
Consider now an element of the form ˜b ni
i (., u). Such an element can be a function of
the full state. First of all we note that when n ∗ = max
i∈{1,...,ny} ni we have:
� � ∆ −1 x − ∆ −1 z � �≤θ n∗ −1 �x − z�.
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