Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
Let us set x0 1 = z(t − d), and x02 = x(t − d) then Lemma 33 gives:
�z(t − d) − x(t − d)� 2 ≤ 1
λ0 � t
�y (τ) − y (t − d, z (t − d) , τ)�
d t−d
2 dτ,
or, equivalently,
3.3 Innovation
�z(t − d) − x(t − d)� 2 ≤ 1
λ0 Id(t).
d
This is to say, that up to a multiplicative constant, innovation at time t upper bounds the
estimation error at time t − d.
Remark 34
One could think that the adaptation scheme is likely to react with a delay time d when
the estimation error is large. However, as is explained in Chapter 4, Remark 45, it may not
always be the case in practice.
Proof.
Let x1 (t) =x x 0 1 ,u (t) and x2 (t) =x x 0 2 ,u (t) be the solutions of (3.2) with xi (0) = x 0 i ,
i =1, 2. For any a ∈ [0, 1] ,
b (ax2 + (1 − a) x1,u)
� a ∂
= b (x1,u)+
0 ∂α b (α x2 + (1 − α) x1,u) dα
� a ∂
= b (x1,u)+
∂x b (α x2 + (1 − α) x1,u) dα (x2 − x1) .
Hence for a =1
b (x2,u) − b (x1,u) =
�� 1 ∂b
0 ∂x (α x2
=
�
+ (1 − α) x1,u) dα (x2 − x1)
B (t)(x2 − x1) ,
where B (t) = (bi,j) (i,j)∈{1,..,n} is a lower triangular matrix since
Set ε = x1 − x2, and consider the system:
⎧
⎨
⎩
0
b (x, u) =(b (x1,u) ,b(x1,x2,u) , . . . , b (x, u)) ′
.
˙ε = A (u) x1 + b (x1,u) − A (u) x2 − b (x2,u)
= [A (u)+B (t)] ε
yɛ = C (u) ε = a1 (u) ε1.
It is uniformly observable5 as a result of the structure of B (t). Let us consider Ψ( t), the
resolvent of the system, and the Gramm observability matrix Gd:
� d
Gd = Ψ (v) ′ C ′ CΨ (v) dv.
5
See [57] for instance, or compute the observability matrix
�
φO = C ′
|(CA) ′
| . . . |(CA n ) ′�′
,
and check the full rank condition for all inputs.
0
42