Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
B.1 Bounds on the Riccati Equation
In the same manner as in Lemma 78 we conclude that for all τk ≤ T∗, k ∈ N:
�
|Pk(+)| ≤ |P0| + |Q|
�
e
2s
2sT∗ = 1
.
Therefore Pk(+) ≤ 1
α1 Id, for all τk ≤ T∗, k ∈ N, for all subdivisions {τi}i∈N.
Equivalently, from matrix fact 3.(c): α1Id ≤ Sk(+).
We now proceed with the proof of the second part of this subsection. We need a different
set of tools than the one used in the preceding section. It is a series of lemmas adapted from
[57]. As we will see, in addition to the proof of the existence of a lower bound for S, we also
prove that it is a symmetric positive definite matrix for all times.
Lemma 90
For any λ ∈ R ∗ , any solution S : [0,T[→ Sm (Possibly, T =+∞) of
we have for all τ ∈ [0,T[:
dS
dτ
S(τ) =e−λτ φu(τ, 0)S0φ ′
u(τ, 0)
� τ
+λ
0
α1
= −A′ (τ)S − SA(τ) − SQS,
e −λ(τ−v) �
φu(τ,v) S(v) − S(v)QS(v)
�
φ
λ
′
u(τ,v)dv
where φu(τ,s) is such that: � dφu(τ,s)
dτ = −A ′
(τ)φu(τ,s),
φu(s, s) = Id.
(B.11)
Remark 91
Notice that S(τ) is not λ-dependent. This latter scalar is used only to provide us with a
convenient way to express S(τ).
Proof.
Let us consider an equation of the form
φu(τ,s) denotes the resolvent of the system dx
dτ
d
Λ(τ) =−A′ Λ(τ) − Λ(τ)A + F (τ). (B.12)
dτ
dφu(τ,s)
= −A′ x:
dτ = −A ′
φu(τ,s),
φu(s, s) = Id.
We search for a solution of the formΛ( τ) =φu(τ,s)h(τ)φ ′
u(τ,s).
dΛ
dτ = � d
dτ φu(τ,s) � h(τ)φ ′
�
d
u(τ,s)+φu(τ,s)h(τ) dτ φ′
�
u(τ,s)
= −A ′
φu(τ,s)h(τ)φ ′
u(τ,s) − φu(τ,s)h(τ)φ ′
+φu(τ,s) � d
dτ h(τ)� φ ′
u(τ,s)A(τ)
= −A ′
Λ(τ) − Λ(τ)A + φu(τ,s) � d
dτ h(τ)� φ ′
u(τ,s),
144
+φu(t, s) � d
dτ h(τ)� φ ′
u(τ,s)
u(τ,s)