Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
5.2 Continuous-discrete Framework
is, therefore, also symmetric positive definite. We consider the same function as before (Cf.
Lemma 33):
Λ : L∞ �
[0,d] R n(n+1) �
2 × L∞ [0,d] (Rnu ) −→ R + .
With the same reasoning as in Lemma 33, we deduce the existence of a scalar λ0 d > 0 such that
Gd ≥ λ0 d Id for any u and any matrix B(t) having the structure specified above. Therefore:
�i=d
�
�yi i=0
5.2.5 Preparation for the Proof
� 0
0,x � � 0
− yi 0, ξ �� �2 = � x 0 − ξ 0�′ � 0 0
Gd x − ξ �
≥ λ 0 �
� 0 0
d x − ξ � �2 .
In order to establish the preliminary inequalities needed in the proof, we first recall the
matrix inversion lemma:
Lemma 64 (matrix inversion lemma [67], Section 0.7.4)
If M is symmetric positive definite, and λ > 0 then
(M + λMC ′
CM) −1 = M −1 − C ′
(λ −1 + CMC ′
) −1 C.
The estimation error is denoted ɛ(t) =z(t) − x(t), and we consider the change of variables:
− ˜x = ∆x, ˜z = ∆z and ˜ɛ = ∆ɛ,
− ˜ b(., u) =∆b(∆ −1 ., u), and ˜ S = ∆ −1 S∆ −1 ,
− ˜ b ∗ (., u) =∆b ∗ (∆ −1 ., u)∆ −1 .
As before, the Lipschitz constant of the vector field remains the same in the new system of
coordinates (Cf. Lemma 51).
The error dynamics are given by
in the continuous case, and
˙ɛ = A(u)ɛ +(b(z, u) − b(x, u)) (5.26)
ɛk(+) = ɛk(−) − δtS −1 (+)C ′
r −1
θ Cɛk(−) (5.27)
in the discrete case.
We highlight the relations below as they are useful for the following computations:
where N = diag ({0, 1, . . . , n − 1}).
∆A(u) =θA(u)∆,
∆−1A ′
(u) =θA ′
(u)∆−1 ,
Remark 65
Notice that on intervals of the form [kδt, (k + 1)δt[, the derivative of θ is equal to zero.
108
(5.28)