Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
5.1 Multiple Inputs, Multiple Outputs Case
The proof of convergence is quite long. For the sake of simplicity, we divide the proof into
several parts:
1. the innovation lemma,
2. a study of the properties of the Riccati matrix S,
3. the calculation of some preliminary inequalities (i.e. the preparation for the proof),
4. the three technical lemmas,
5. and the main body of the proof.
Points 2, 4 and 5 are essentially the same as in the single output case. Only points 1 and 3
require modifications for the multiple output case. The modifications are explained here.
5.1.3.1 Lemma on Innovation
Lemma 50 (Lemma on innovation, multiple output case)
Let x0 1 ,x02 ∈ Rn and u ∈ Uadm. Let us consider the outputs y � 0,x0 1 , ·� and y � 0,x0 2 , ·�
of system (5.1) with initial conditions respectively x0 1 and x02 . Then the following property
(called persistent observability) holds:
Proof.
∀d >0, ∃λ0 d > 0 such that ∀u ∈ L1 b (Uadm)
� d
�y � 0,x 0 1, τ � − y � 0,x 0 2, τ � � 2 ny R dτ (5.3)
�x 0
1 − x 0
2� 2 ≤ 1
λ 0 d
0
Let x1 (t) =x x 0 1 ,u (t) and x2 (t) =x x 0 2 ,u (t) the solutions of (5.1) with xi (0) = x 0 i
A few computations lead to
b (x2,u) − b (x1,u)=B (t)(x2 − x1)
� 1 ∂b
where B(t) =
0 ∂x (αx2 + (1 − α)x1,u)dα.
Set ε = x1 − x2. Let us consider the system:
�
˙ε = [A (u)+B (t)] ε
yɛ = C (u) ε = a1 (u) ε1.
, i =1, 2.
It is uniformly observable1 due to the structure of B (t). We define the Gramm observability
matrix, Gd, using Ψ( t), the resolvent2 of this system:
� d
Gd = Ψ (v) ′ C ′ CΨ (v) dv.
1 See [57], for example. Alternatively, one could compute the observability matrix
φO =
and check the full rank condition for any input.
2 Cf. Appendix A.1
0
�
C ′
|(CA) ′
| . . . |(CA n ) ′�′
,
96