Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
We can prove that (III) has a lower bound if we can prove that
k∗ �
Gū,d =
i=1
B.1 Bounds on the Riccati Equation
′
φū(˜τk∗ , ˜τi)C Cφ ′
ū(˜τk∗ , ˜τi) (˜τi − ˜τi−1)
is lower bounded, for all subdivisions {˜τi} i∈{0,...,k∗} such that ˜τi+1 − ˜τi ≤ µ and that 6
T∗ ≤ ˜τk∗ ≤ T∗ + µ.
Let us define ψū(t, s) =(φ −1
ū (t, s)) ′
. Since φū is the resolvent of dx
dτ
= −A′ x, therefore
ψū is the resolvent of dx
dτ = Ax, (Cf. Appendix A.1).
Actually7 Gū,d is the Gramm observability matrix of the continuous discrete version of
a system of the form (A.4).
Let us denote Gū(T ) the grammian of (A.4) when the integral is computed from 0 to
T . Let us apply Lemma 76 on Gū(T∗). There is a a>0, independent from ū, such that
aId ≤ Gū(T∗) ≤ Gū(˜τk∗ ).
From Lemma 93 and Remark 94 there is a K>0 such that
Therefore
aId ≤ Gū(˜τk∗ ) − Gū,d + Gū,d ≤ Gū,d + µK˜τk ∗Id
≤ Gū,d + µK(T∗ + µ)Id.
[a − µK(T∗ + µ)] Id ≤ Gū,d,
and µ can be shortened such that (a − µK(T∗ + µ)) > 0, independently from the
subdivision {˜τi}.
We conclude that there exists a α2 > 0 such that α2Id ≤ (III).
Consequenltly α2Id ≤ Sk(+), for all k such that τk ≥ T∗.
We finally state the equivalent of Lemma 88.
Lemma 96
Consider the prediction correction equation (B.1) and the hypothesis of Lemma 77. There
exist two positive constants µ > 0 and α such that αId ≤ S(τ), for all k ∈ N, and all
τ ∈ [τk, τk+1[, for all subdivisions {τi}i∈N such that (τi − τi−1) ≤ µ.
This inequality is then also true in the t time scale.
Lemma 77 is the sum of Lemmas 88 and 96.
6 This second hypothesis implies that k∗ + 1, the number of elements can vary from one subdivision to the
other.
7 Notice that with the ψū notation
Gū,d =
k∗�
i=1
ψ ′
ū(˜τi, ˜τk∗)C ′
Cψū(˜τi, ˜τk∗) (˜τi − ˜τi−1) .
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