Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
Definition 30
The adaptive high-gain extended Kalman filter is the system:
⎧
⎪⎨
dz
dt
⎪⎩
= A(u)z + b(z, u) − S−1C ′ R −1
θ (Cz − y(t))
dS
dt = −(A(u)+b∗ (z, u)) ′ S − S(A (u)+b∗ (z, u)) + C ′ R −1
θ C − SQθS
dθ
dt = F(θ, Id (t)).
3.3 Innovation
(3.3)
The functions F and Id will be defined later, in Section 3.3 and Lemma 42. The function
Id is called the innovation. The initial conditions are z(0) ∈ χ, S(0) is symmetric positive
definite, and θ(0) = 1.
The function F has only to satisfy certain requirements stated precisely in Lemma 42.
Therefore, several different choices for an adaptation function are possible.
Roughly speaking, F(θ, Id (t)) should be such that if the estimation z (t) is far from x (t)
then θ (t) increases (high-gain mode). Contrarily, if z (t) is close to x (t), θ goes to 1 (Kalman
filtering mode). As it is clear from the proof of Theorem 36, this observer makes sense only
when θ(t) ≥ 1, for all t ≥ 0. This is therefore another requirement that F(θ, Id) has to meet.
The achievement of this behavior requires that we evaluate the quality of the estimation.
This is the object of the next section.
Remark 31
1. Readers familiar with high-gain observers may notice that the matrices Rθ and Qθ are
not exactly the same as in earlier articles such as [38, 47, 57]. The definitions developed
here can also be substituted into those previous works without consequence.
2. The hypothesis θ(0) = 1 may appear a bit atypical as compared to the results in [38, 57]
for instance.
− For technical reasons, the result of Lemma 39 depends on the initial value of θ,
and θ(0) = 1 has no impact on α and β.
− Secondly, note that in the case of large perturbations, θ will increase. In these
instances, the initial value of θ is of little importance as we show in Lemma 42
that the adaptation function can be chosen in such a way that θ reaches any large
value in an arbitrary small time.
− Finally, in the ideal case of no initial error, θ doesn’t increase which saves us from
useless noise sensitivity due to a large high-gain initial value.
3.3 Innovation
The innovation Id is a measurement of the quality of the estimation. It is different 4 from
the standard concept of innovation, which is based on a linearization around the estimated
4 The same definition of innovation is used for moving horizon observers where the estimated state is the
solution of a minimization problem. The cost function used here represents the proximity to the real state. It
is the innovation we use, ([12, 95]).
In related publications, observability is defined by the inequality of Lemma 33. In our case, the inequality is
a consequence of the observability theory.
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