Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
− let θ increase when innovation is large,
− have θ decrease toward 1 when innovation is small.
5.2 Continuous-discrete Framework
In the continuous-discrete setting, an extra hypothesis, related to the sampling time, appears
(see Lemma 63 below). As a consequence the beginning of the proof of Subsection 5.2.6 is
performed independently from δt.
5.2.4 Innovation
The lemma for innovation in the continuous-discrete case is:
Lemma 63
Let x0 , ξ0 ∈ Rn �
and u ∈ Uadm. Let us consider the outputs yj 0,x0 � �
and yj 0, ξ0 � of
system (5.21) with initial conditions x0 and ξ0 respectively. Then the following condition
(called persistent observability) holds:
∀d ∈ N ∗ large enough (i.e. d ≥ n − 1), ∃λ 0 d > 0 such that ∀u ∈ L1 b (Uadm)
�
�x0 − ξ 0
�2 ≤ 1
λ0 i=d
d
i=0
� 0
�yi 0,x � � 0
− yi 0, ξ � � 2 .
Proof.
Let x (t) and ξ (t) be the solutions of the first equation of system (5.21) with x 0 and ξ 0
as initial values. We denote the controls by u(t).
As in the proof of Lemma 33 we have:
b(ξ,u) − b(x, u) =B(t)(ξ − x) (5.24)
where B (t) = (bi,j) (i,j)∈{1,..,n} is a lower triangular matrix since b (x, u) is a lower triangular
vector field. Set ε = x − ξ. Then
˙ε = [A (u)+B (t)] ε. (5.25)
We consider the system formed by equation (5.25) and C (uk) εk = a1 (uk) ε1,k as output. Let
us consider Ψ( t), the resolvent (5.25), and the Gramm observability matrix
�i=d
Gd = Ψ (iδt) ′
i=0
C ′
iCiΨ (iδt) .
From the lower triangular structure of B(t), the upper triangular structure of A(u) and the
form of the matrix Ci = C(u(iδt)), we can deduce that Gd is invertible 12 when d ≥ n − 1. It
12 ψ0(.) is the resolvent of system (5.25) (refer to Appendix A.1). The Gramm observability matrix can be
written:
⎛
⎜
Gd = ⎜
⎝
C0
C1Ψ0(kδt)
...
CdΨ0(dδt)
⎞′
⎛
⎟ ⎜
⎟ ⎜
⎟ ⎜
⎠ ⎝
C0
C1Ψ0(kδt)
...
CdΨ0(dδt)
⎞
⎟
⎠
= φ′ φ.
Therefore Gd is invertible provided it is of rank n, which can be achieved for d ≥ n − 1.
107