Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
B.1 Bounds on the Riccati Equation
where AM = sup
[0;T ∗ �
�
(|A(u(τ)|), �
]
�b ∗ �
�
(z, u) � ≤ Lb and s = AM + Lb.
Gronwall’s lemma gives
|S(τ)| ≤| Sk−1(+)|e 2s(τ−τk−1)
. (B.3)
Therefore, with c = |C ′
RC|,
|Sk(+)| ≤| Sk−1(+)|e 2s(τk−τk−1) + c (τk − τk−1) . (B.4)
Consider a subdivision {τk}k∈N such that τ0 = 0. From equation (B.4):
|S1(+)| ≤| S0|e 2sτ1 + cτ1.
Since we set θ(0) = 1 then ¯ S0 = S0 and there is no ambiguity in the notation. We iterate to
obtain
|S2(+)| ≤| S0|e 2sτ2 + cτ1e 2s(τ2−τ1) + c(τ2 − τ1),
and for all k ∈ N
|Sk(+)| ≤| S0|e2sτk �i=k
+ c (τi − τi−1) e 2s(τk−τi)
i=1
≤ |S0|e2sτk + ce2sτk �i=k
i=1
(τi − τi−1) e −2sτi .
Since e−2sτ is a decreasing function of τ, the sum on the right hand side of the inequality is
smaller than the integral of e−2sτ over the interval [0, τk]. Therefore
|Sk(+)| ≤| S0|e2sτk + ce2sτk � τk
e
0
−2sτ dτ
≤ |S0|e2sτk + c
�
2sτk
2s e − 1 � .
From this last equation we conclude that for any subdivision and any k ∈ N such that τk ≤ T ∗ :
�
|Sk(+)| ≤ |S0| + c
�
2sT ∗
e = β1.
2s
In order to prove the result for times greater than T ∗ , consider a symmetric semi positive
matrix S. We have S ≤ |S|Id = � Tr(S 2 )Id, and according to fact 2.(b): S ≤ Tr(S)Id.
Therefore, investigations on the upper bound are done in the form of investigations on the
trace of S.
Lemma 79
If S : [0,T[→ Sn is a solution to dS
dτ
where:
= −A′ S − SA − SQS then for almost all τ ∈ [0,T[,
d
dτ Tr(S) ≤−a (Tr (S(τ))) 2 +2bTr (S(τ)) ,
a = λmin(Q)
n
b = sup τ Tr(A ′
(τ)A(τ)) 1
2 .
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