Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
trajectory.
Definition 32
For a “forgetting horizon” d>0, the innovation is:
� t
Id (t) =
t−d
3.3 Innovation
�y (t − d, x (t − d) , τ) − y (t − d, z (t − d) , τ)� 2 dτ (3.4)
where y (t0,x0, τ) denotes the output of the system (3.2) at time τ with x (t0) =x0.
Hence y (t − d, x (t − d) , τ) denotes y(τ), the output of the process. Notice that y (t − d, z (t − d) , τ)
is not the output of the observer.
For a good implementation, it is important to understand the significance of this definition.
Figure 3.1 illustrates the situation at time t. Innovation is obtained as the square of the
L 2 distance between the black (plain) and the red (dot and dashed) curves. They respectively
represent the output of the system on the time interval [t−d, t], and the prediction performed
with z(t − d) as initial state.
y(t)
z(0)
x(0)
0
0
t-d
Output traj.
Estimated output
y(t-d,z(t-d),t)
t
time
Figure 3.1: The Computation of Innovation.
The importance of innovation in this construction is explained by the following lemma.
As we will see in Section 3.8, this is the cornerstone of the proof.
Lemma 33
Let x0 1 ,x02 ∈ Rn , and u ∈ Uadm. Let us consider the outputs y � 0,x0 1 , ·� and y � 0,x0 2 , ·�
of system (3.2) with initial conditions respectively x0 1 and x02 . Then the following property
(called persistent observability) holds:
∀d >0, ∃λ 0 d > 0 such that ∀u ∈ L1 b (Uadm)
�x 0
1 − x 0
2� 2 ≤ 1
λ 0 d
� d
�y � 0,x 0 1, τ � − y � 0,x 0 2, τ � � 2 dτ. (3.5)
0
41