Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
B.1 Bounds on the Riccati Equation
3. if xk(+) ≥ 2b
a , and xk(+) ≥ B(T ∗ ) then , Lemma 80 tells us that xk+1 ≤ ψxk (τk+1 −τk).
From Lemma 84, we know that it is a decreasing function, and according to Remark
85 we have xk+1(+) ≤ ψxk (0) ≤ β2.
Let us now define k such that τk−1 0, we have x(T ∗ ) ≤ B(T ∗ ) by definition. From equation (B.7) and Lemma 80:
− if x(T ∗ ) ≤ 2b
a therefore xk(−) ≤ 2b
a ≤ B(T ∗ ),
− if x(T ∗ ) ≥ 2b dx
a , then dτ is negative according to equation (B.7), x is decreasing and
xk(−) ≤ B(T ∗ ).
In both cases xk(+) ≤ B(T ∗ )+µr = β2
We finally conclude that, there are two positive constants β2 and µ such that Sk(+) ≤ β2Id
for all τk ≥ T ∗ , k ∈ N. It is true for all subdivision {τi}i∈N, such that τi − τi−1 ≤ µ, ∀i.
Remark 87
The constraint τi − τi−1 ≤ µ, interpreted in the original time scale, gives θiδt ≤ µ, ∀i ∈ N.
We conclude this subsection giving the upper bound property for all times τ.
Lemma 88
Consider the prediction correction equation (B.1) and the hypothesis of Lemma 77. There
exist two positive constants µ > 0 and β such that S(τ) ≤ βId, for all k ∈ N, and all
τ ∈ [τk, τk+1[, for all subdivision {τi}i∈N such that (τi − τi−1) ≤ µ.
This inequality is then also true in the t time scale.
Proof.
Let us define ˆ β = max (β2, β1). Lemmas 86 and 78 give us Sk(+) ≤ β2Id, for all k ∈ N.
Then Lemmas 79 and 80 implies the existence of ˆ β ∗ such that S(τ) ≤ ˆ βId for any τ ∈
[τk, τk+1[, and any k ∈ N.
We define β as the maximum value between ˆ β ∗ and ˆ β.
B.1.2 Part Two: the Lower Bound
As in the previous subsection, the proof is separated into three parts. Let T∗ be a fixed
positive scalar, possibly distinct from T ∗ (Cf. previous subsection).
1. We prove that there is α1 such that for all k ∈ N, such that τk ≤ T∗, α1Id ≤ Sk(+),
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