Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
5.2 Continuous-discrete Framework
As in Section 3.5, we need to write the prediction-correction Riccati equations in a different
time scale (τ), so that we can bound the Riccati matrix independently from θ. We consider
dτ = θ(t)dt or equivalently τ = � t
0 θ(v)dv and keep the notation ¯x(τ) = ˜x(t).
⎧
⎨ d
⎩
¯ �
S
dτ = − A(u)+ ˜b ∗ �′ �
(z,u)
S − S A(u)+
θ
˜b ∗ �
(z,u)
− SQS
θ
¯Sk(+) = ¯ Sk(−)+θδtC ′
r−1 (5.38)
C.
Since θ(t) varies within an interval of the form [1, θmax], the instants tk = kδt, k ∈ N are
difficult to track in the τ time scale. For convenience, tk = kδt is denoted τk in the τ time
scale.
With the help of this representation we are able to derive the following lemma:
Lemma 66
Let us consider the prediction correction Riccati equations (5.38), and the assumptions:
�
�
− the functions ai (u (t)), �˜b ∗ i,j (z,u)
�
�
�, are smaller than aM > 0,
− ai (u (t)) ≥ am > 0, i =2, ..., n,
− θ(0) = 1, and
− S(0) is a symmetric positive definite matrix taken from a compact subset of the form
aId ≤ S(0) ≤ bId.
Then, there exists a constant µ, and two scalars 0 < α 0, and ε ∗ as in Theorem 62.
Let us now set a time T such that 0 < T < T ∗ .
Let α and β be the bounds of Lemma 66.
For t ∈ [kδt;(k + 1)δt[, inequality (5.31) can be written (i.e. using αId ≤ ˜ S),
d˜ε ′ ˜ S˜ε (t)
dt
≤−αqmθ˜ε ′ � �
′
S˜ε ˜ (t) + 2˜ε ˜S ˜b (˜z) − ˜b (˜x) − ˜∗ b (˜z)˜ε
110
(5.39)