Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
3.6 Boundedness of the Riccati Matrix
Proof.
We denote by |.|, the Frobenius norm of matrices: |A| = � T race(A ′ A). Recall that for
two symmetric semi-positive matrices A, B such that 0 ≤ A ≤ B we have11 :0≤ |A| ≤| B|.
Choose τ0 > 0 and apply Lemma 38 to obtain �α and � β such that �α Id ≤ S (τ) ≤ � β Id for
all τ ≥ τ0. In order to extend the inequality for 0 ≤ τ ≤ τ0, we start from:
� τ dS (v)
S (τ) = S0 +
0 dτ dv
⎡
� �
τ
= S0 + ⎣− A(u)+
0
�b ∗ (z,u) F(θ, I)
−
θ θ 2 N
�′
S
�
−S A(u)+ �b ∗ (z,u) F(θ,I)
−
θ θ 2 �
N + C ′
R−1 �
C − SQS dv.
As θ (0) = 1, then S (0) = S (0) = S0, which together with SQS > 0 (symmetric semipositive)
leads to
⎡
� �
τ
S (τ) ≤ S0 + ⎣− A(u)+
0
�b ∗ (z,u) F(θ, I)
−
θ θ 2 N
�′
S
and
�
� � τ �
�S (τ) � ≤ |S0| + 2
0
with AM = sup (|A(u(τ))|) and
[0;τ0]
−S
�
A(u)+ � b ∗ (z,u)
θ
F(θ,I)
−
θ 2 �
N + C ′
R−1 �
C dv,
�
�
AM + B + �
F(θ, I)
�
θ 2
�
�
�
� |N|
�
��S � �
� �
+ �C ′
R −1 �
�
C�
dv
�
�
��b ∗ �
�
(z,u) � ≤ B. Then
�
�S � �
� �
≤ |S0| + �C ′
R −1 � � τ
�
C�
τ0 + 2s � �
�S� dv,
with s = aM |N| + AM + B. Applying Gronwall’s lemma gives us for all 0 ≤ τ ≤ τ0,
�
�S � � �
� �
≤ |S0| + �C ′
R −1 � �
�
C�
τ0 e 2sτ
� �
�
≤ |S0| + �C ′
R −1 � �
�
C�
τ0 e 2sτ0
�
≤ b √ �
�
n + �C ′
R −1 � �
�
C�
τ0 e 2sτ0
= β1.
In the same manner we denote P = S −1 , and use the equation
dP
dt
= P
�
A(u)+ � b ∗ (z,u)
θ
−
F(θ, I)
θ 2 N
�′ �
+ A(u)+ �b ∗ (z,u)
−
θ
11 See Appendix B.1 for details of establishing this fact.
48
0
F(θ, I)
θ 2 N
�
P
(3.14)
− PC ′
R −1 CP + Q