Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
3.7 Technical Lemmas
Therefore θ(t) reaches θ1 at time t 0. The function µ is such that14 :
⎧
⎨ 1 if I ≥ γ1
µ (I) = ∈ [0, 1]
⎩
0
if γ0 ≤ I ≤ γ1
if I ≤ γ0.
We claim that all properties are satisfied.
If I ≥ γ1, F (θ, I) =F0 (θ) ensuring Property 3, (refer to the beginning of the proof).
Conversely, if I ≤ γ0, F (θ, I) =λ (1 − θ) then Property 4 is fulfiled. Moreover, because
F (θ, I) is Lipschitz, Property 1 is verified. Let us check property 2 :
�
�
�
F (θ, I)
� θ2 �
�
�
� ≤
�
�F0
�
(θ)
� θ2 �
�
�
� +
�
�
�
λ (1 − θ)
� θ2 �
�
�
� . (3.17)
The first term satisfies:
�
�
− θ ≤ θ1, � F0(θ)
�
�
� = 1
θ 2
θ 2
�
�
− � F0(θ)
�
�
� = 1
∆T
∆T
, and
� �2 θ−2θ1
θ ≤ 1
∆T if θ ≥ θ1 (and θ < 2θ1).
The second term satisfies:
�
�
�
λ (1 − θ)
� θ2 �
�
�
�
− 1 1
� = λθ = λ
θ2 4 −
θ2 4 − θ +1
θ2 �
⎛
= λ ⎝ 1
4 −
� � ⎞
θ
2
2 − 1
⎠ ≤
θ
λ
4 .
Property 2 is satisfied because of (3.17) with M = 1
∆T
13 When 1 < θ0 ≤ θ1 the solution to equation (5.11) is:
And for θ1 < θ0 < 2θ1 the solution of (5.11) is:
+ λ
4 .
�
∆T θ0
θ(t) =
when θ(t) ≤ θ1
∆T −θ0t
θ(t) = 2θ1 −
when θ(t) > θ1.
θ(t) =2θ1 −
∆T θ0θ1
θ0θ1t+(2θ0−θ1)∆T
∆T (2θ1 − θ0)
∆T + t(2θ1 − θ0) .
The conclusion remains the same: if θ0 < θ1, θ1 is reached in a time smaller than∆ T , and remains below 2θ1
in all cases.
14 Such a function is explicitely defined in Section 4.2.1 of Chapter 4.
51