Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
B.1 Bounds on the Riccati Equation
2. we show that there exists α2 such that for all k ∈ N, such that T∗ ≤ τk, α2Id ≤ Sk(+),
3. we give the result for all times.
As before, the first fact is proven rather directly.
Lemma 89
Consider equation (B.1) and the assumptions of Lemma 77. Let T∗ > 0 be fixed. There
exists α1 > 0 such that
α1Id ≤ Sk(+),
for all τk ≤ T ∗ , k ∈ N, independently from the subdivision {τi}i∈N.
Proof.
We denote P = S −1 . For all τ ∈ [τk−1, τk[, equation (B.1) gives:
� τ
P (τ) = Pk−1(+) +
τk−1
τk−1
dP (v)
dτ dv,
P (τ) = Pk−1(+)
⎡
� �
τ
+ ⎣P A(u)+ ˜b ∗ (z, u)
θ
Computations performed as in Lemma 78 lead to
�′
+
�
A(u)+ ˜ b ∗ (z, u)
θ
|Pk(−)| ≤ (|Pk−1(+)| + |Q| (τk − τk−1)) e 2s(τk−τk−1)
where s>0 is as before. From (B.1) and Lemma 64
�
⎤
P + Q⎦
dv.
�
Pk(+) = Sk(−)+θδtC ′
r−1 �−1 C
= Sk(−) −1
�
Sk(−) −1 + θδtSk(−) −1C ′
r−1CSk(−) −1
�−1 Sk(−) −1
= Sk(−) −1
�
Sk(−) − C ′ � r
θδt + CSk(−) −1C ′� �
−1
C Sk(−) −1
≤ Sk(−) −1 Sk(−)Sk(−) −1
≤ Pk(−).
We consider a subdivision of {τk}k∈N such that τ0 = 0. Since ¯ P0 = P0:
We iterate to obtain
and for all k ∈ N
|P2(+)| ≤| P0| e 2sτ2 + |Q|
|P1(+)| ≤ (|P0| + |Q| τ1) e 2sτ1 .
�
τ1e 2sτ2
�
2s(τ2−τ1)
+(τ2 − τ1) e ,
|Pk(+)| ≤| P0| e 2sτk
�i=k
+ |Q| (τi − τi−1) e 2s(τk−τi)
.
143
i=1
(B.8)
(B.9)
(B.10)