Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
3.4 Main Result
3.4 Main Result
The exponential convergence of the adaptive high-gain extended Kalman filter is expressed
in the theorem below.
Theorem 36
For any time T ∗ > 0 and any ε ∗ > 0, there exist 0 < d < T ∗ and a function F (θ, Id) such
that, for all times t ≥ T ∗ and any initial state couple (x0,z0) ∈ χ 2 :
�x (t) − z (t)� 2 ≤ ε ∗ e −a (t−T ∗ )
where a>0 is a constant (independent from ε ∗ ).
This theorem can be expressed in two different ways: with or without a term �ɛ0� 2 in the
upper bound. The bound of Theorem 36 was presented in [23] and [22] 8 . The expression we
use here should be interpreted to mean that the square of the error can be made arbitrarily
small in an arbitrary small time. For the sake of completeness, we develop the other inequality
in Remark 44.
The proof is a Lyapunov stability analysis which requires several preliminary computations
and additional results. In order to facilitate comprehension, we divide the proof into
several parts:
1. the computation of several preliminary inequalities, in particular the expression of the
Lyapunov function we want to study,
2. the derivation of the properties of the Riccati matrix S,
3. the statement of several intermediary lemmas, among which is the lemma that states
the existence of eligible adaptive functions, and finally
4. the articulation of the proof.
We begin with the computation of some preliminary inequalities in Section 3.5.
3.5 Preparation for the Proof
Remember 9 that θ ≥ 1, for all t ≥ 0.
We denote by z the time dependent state variable of the observer.
The estimation error is ε = z − x.
We consider the change of variables ˜x = ∆x, and
− ˜z = ∆z, and ˜ε = ∆ε,
− ˜ S = ∆ −1 S∆ −1 ,
8 The other bound is
�x (t) − z (t)� 2 ≤�ε0� 2 ε ∗ e −αqm(t−τ) .
9 This is one of the requirements F(θ, Id) have to meet. Existence of such a function is shown in Lemma 42.
44