Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
4.2 Simulation
With a high value of λ, θ oscillates as well. In order to give some resilience to θ
in this case, λ should be not be set too high. A value between 1 and 10 seems to
be sufficient.
− ∆T : (in the adaptation function of the observer (4.5)). The smaller∆ T , the
shorter the rising time of θ. We take∆ T =0.1. This is sufficiently small that the
equation F0(θ) remains compatible with the ODE solver 9 .
As explained at the end of the previous subsection (i.e. Subsection 4.2.2), innovation
is considered as a discrete function of time. Therefore we need to set the sampling time
of this process. We denote it δ. It depends on the measurement hardware and is
not a critical parameter. Nevertheless, it seems intuitive that d
δ must be sufficiently
d
high to at least reflect the rank of the observability of the system, i.e. δ ≥ n − 1.
Indeed, innovation is used as a direct measurement of distinguishability. A theoretical
justification of this remark is provided in Section 5.2 of Chapter 5 where an adaptive
continuous-discrete version of our observer is provided.
Parameters d and m2 are closely related to the application, but in a very clear manner.
When d is too small, innovation is not sufficiently large to distinguish between an
increase in the estimation error and the influence of noise. On the other hand, a value
for the innovation that is too high increases the computation time as the prediction is
made on a larger time interval. The value of d has to be chosen using our knowledge of
the time constant of the system (simulations or data samples), i.e., some fraction (e.g.
) of the smallest time constant appears to be a reasonable choice.
1
3
to 1
5
Remark 45
When the system encounters a high perturbation at time t, one would expect a delay
of length d in the adaptation of θ. However, the inequality of Lemma 33 is valid for
any delay 0