Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
2.6 On Adaptive High-gain Observers
− consider the adaptation function:
�
γ|y(t) − z1(t)|
˙s =
2 for |y(t) − z1(t)| > λ
0 for |y(t) − z1(t)| ≤ λ.
The parameter θ is adapted according to the rule:
− when t =0set θ =1
− when s(t) =θi, set θ = θi (or in other words θ = sup{θi ≤ s(t)}).
(2.12)
It is clear from the definition of the adaptation procedure that the parameter θ cannot
decrease which makes this observer a solution to a tuning problem rather than the noise
reduction problem. The convergence of the observer is established in the following theorem.
This observer comes from a paper published in 1997 (i.e. [35]), we therefore checked
for updates of the strategy in more recent paper. In [36], the authors address λ-tracking
problems 15 . The observer they use is the one described in this section.
Theorem 18 ([35])
Consider the system (2.10) above, together with the assumptions
1. the system exhibits no finite escape time, and
2. the nonlinearity φ(x, u) is bounded.
Then for any λ > 0, γ > 0, β > 0 and any S0:
the total length of time for which the observer output error is larger than λ is finite.
That is to say:
�
∃Tmax < ∞ : dt < Tmax where T = {t|�y(t) − z1(t)� ≥λ}.
T
In another theorem of the same paper (i.e. Theorem 3 of [35]), an upper bound for the
estimation error for high values of t is provided. But in the original article from which this
observer is inspired, [111], the estimation error is not bounded by an exponentially decreasing
function of the time.
The main differences between these works and with the work proposed here is that firstly
that our observer is a Kalman based observer which, implies taking into account the evolution
of the Riccati matrix. The second distinction is that here we address the problem of noise
reduction when the estimation is sufficiently good.
2.6.2 L. Praly, P. Krishnamurthy and coworkers
The second observer with a dynamically sized high-gain parameter proposed that we will
expose in this review is one from L. Praly, K. Krishnamurthy and coworkers. Descriptions of
15 As explained in the article, λ-tracking is used for processes for which we know for certain that asymptotic
stabilization can be achieved only by approximation. The objective is therefore modified into one of stabilizing
the state within a sphere of radius λ centered on the set point.
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