Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
The Riccati equation turns into
d ˜ S
dt = ˙ θ
θ N∆−1 S∆ −1 + ˙ θ
3.5 Preparation for the Proof
θ ∆−1S∆−1N − ∆−1 (A + b∗ (z, u)) ′
S∆−1 −∆−1S (A + b∗ (z, u))∆ −1 + ∆−1C ′
= ˙ �
θ
θ N ˜ S + ˜ �
SN − � A∆−1 + b∗ (z, u) ∆−1�′ ∆ ˜ S
R −1
θ C∆−1
−∆ −1 SQθS∆ −1
− ˜ S∆ � A∆−1 + b∗ (z, u) ∆−1� + C ′
�
˙θ
= θ θ2 �
N ˜ S + ˜ � �
SN − A ′ �
S ˜ + SA ˜ + C ′
R−1C − ˜ SQ˜ S
− ˜ S∆b∗ (z, u) ∆−1 − ∆−1b∗′ (z, u) ∆ ˜ �
S
�
F(θ,I)
= θ θ2 �
N ˜ S + ˜ � �
SN − A ′ �
S ˜ + SA ˜ + C ′
R−1C − ˜ SQ˜ S
�
.
The derivative of the Lyapunov function �ε ′ � S�ε is
d˜ɛ ′ �
S˜ɛ ˜
dt = θ − F(θ,I)
= θ
R −1
θ C − ˜ S∆Qθ∆ ˜ S
− 1
θ ˜ S ˜ b ∗ (˜z, u) − 1
θ ˜ b ∗′
(˜z, u) ˜ S
θ2 N˜ɛ + A˜ɛ − ˜ S−1C ′
R−1C˜ɛ + 1
� ��′
˜b
θ (˜z, u) − ˜b (˜x, u) ˜S˜ɛ
+θ˜ɛ ′ � F(θ,I)
θ2 �
N ˜ S + ˜ � �
SN − A ′ �
S ˜ + SA ˜ + C ′
R−1C − ˜ SQ˜ S
− 1
�
˜S ˜
θ b∗ (˜z, u)+ ˜b ∗ ′
(˜z, u) ˜ ��
S ˜ɛ
+θ˜ɛ ′ �
S˜
− F(θ,I)
θ2 N˜ɛ + A˜ɛ − ˜ S−1C ′
R−1C˜ɛ + 1
� ��
˜b
θ (˜z, u) − ˜b (˜x, u)
� ��
˜b (˜z, u) − ˜b (˜x, u) − ˜b ∗ (˜z, u)˜ɛ .
�
−˜ɛ ′
C ′
R −1 C˜ɛ − ˜ɛ ′ ˜ SQ ˜ S˜ɛ + 2
θ ˜ɛ′ ˜ S
We consider that Q ≥ qm Id, and since �ε ′
C ′
R −1 C�ε ≥ 0
(3.10)
(3.11)
d
�
�ε
dt
′ �
S�ε � ≤−θqm�ε ′ � �
S� 2 ′
�ε +2�ε �S ˜b (�z, u) − ˜b (�x, u) − ˜∗ b (�z, u) �ε . (3.12)
The theorem is proven using the inequality (3.12), which requires some knowledge of the
properties of S. We can note that the equality (3.10) has a strong dependency on θ, which
we must remove in order to derive properties for S. Indeed, for the moment we don’t know
which values θ should reach during runtime. To determine these values, we introduce the
time reparametrization dτ = θ (t) dt, or equivalently τ = � t
0 θ (ν) dν. We denote:
− ¯x (τ) = ˜x(t), ¯z (τ) = ˜z(t), and ε (τ) =�ε (t), and
− ¯ θ (τ) =θ(t), ū (τ) =u(t), and S (τ) = � S (t).
We obtain the time derivative with respect to the time scale, τ, using the simple calculation:
Therefore
d¯ɛ
dτ
d¯ε
dτ
= d˜ε (t)
dt
dt
dτ
= 1
θ (t)
d˜ε (t)
.
dt
I)
= −F(θ,
¯θ 2
N ¯ε + A¯ε − ¯ S −1 C ′
R −1 C ¯ε + 1
� �
˜b
¯θ
(¯z,ū) − ˜b (¯x, ū)
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