Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
5.2 Continuous-discrete Framework
We claim that there exists τ ≤ T ∗ such that ˜ε ′ ˜ S˜ε (τ) ≤ γ. Indeed, if ˜ε ′ ˜ S˜ε (τ) > γ for all
τ ≤ T ∗ then because of Lemma 50:
γ < ˜ε ′ S˜ε ˜ 2 2 β
(τ) ≤ β �˜ε (τ)� ≤ β �ε (τ)� ≤
λ0 Id (τ + d) .
d
Therefore, Id (τ + d) ≥ γ1 for τ ∈ [0,T ∗ ] and hence Id (τ) ≥ γ1 for τ ∈ [d, T ∗ ], so we have
θ (t) ≥ θ1 for t ∈ [T, T ∗ ], which results in a contradiction (˜ε ′ ˜ S˜ε (T ∗ ) ≤ γ) because of (5.17)
and (5.19).
Finally, for t ≥ τ, using (5.16)
and the theorem is proven.
�ε (t)� 2 ≤ (2θ1) 2n−2 �˜ε (t)� 2
2n−2
(2θ1)
≤ α
≤ ε∗e−αqm(t−τ) ,
˜ε ′ ˜ S˜ε (t)
(5.20)
Remark 58
Just as in the single output case, an alternative result can be derived. Refer to Remark
44, Chapter 3.
5.2 Continuous-discrete Framework
In the present section we develop an adaptation of the observer to continuous discrete
systems. In such systems, the evolution of the state variables is described by a continuous
process, and the measurement component of the system by a discrete function. When we
cannot use the quasi-continuity assumption of measurements, as we did before, this version
is the one to use.
In engineering sciences, when pure discrete filters are used, the discrete model needed is
sometimes obtained from a continuous formulation. The model
is transformed into:
˙x = f(x(t),u(t),t)
x(t ∗ + δt) =x(t ∗ )+f (x(t ∗ ),u(t ∗ ),t ∗ ) δt,
where δt represents the sampling time of the process. This equation represents nothing
more than Euler’s numerical integration method. In other words, this modeling technique is
a special case of the continuous-discrete framework. The mechanization equation of inertial
navigation systems 7 is an example of such modeling, details of which can be found in [10, 114]
for example.
Although we directly consider a single output continuous-discrete normal form, we want
to draw the reader’s attention to the problem of the preservation of observability under sampling.
In [15], the authors prove that for a continuously observable 8 system, observability is
7
The mechanization process merges the data coming from 3-axis accelerometers to those coming from a
gyroscope.
8
Uniformly observable for a class of input functions, and uniformly infinitesimally observable in the sense
of the definitions given in Chapter 2.
103