Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
− statistical methods [115],
− genetic algorithms based observers [97],
− Neural networks based observers [109],
− fuzzy logic approach [72].
2.6 On Adaptive High-gain Observers
Those observers are based on empirical methods. As a result, very little can be demonstrated
or proven with respect to their convergence properties.
2.6.1 E. Bullinger and F. Allőgower
In a 1997 paper, E. Bullinger and F. Allgőwer proposed a high-gain observer having a
varying high-gain parameter. Their observer is inspired by the structure proposed by A.
Tornambè in [111]. It is a Luenberger like observer for a system of the form (2.7) except that
αi(u) = 1 for all i ∈ 1, ..., n. Only the last component of the vector field b(x, u) is not equal to
zero (see definition below). The control variable u may be of the form u = � u, ˙u, u (2) , . . . , u (n)�
which is not one of the assumptions of system (2.7). As it appears below, the main difference
between this observer and a classic high-gain is that the influence of the vector field to the
model dynamic is neglected.
Definition 17
Consider a single input, single output system of the form
Then define a high-gain obsever
⎧
⎪⎨
⎪⎩
x1 ˙ = x2
x2 ˙ = x3
.
xn−1 ˙ = xn
xn ˙ = φ(x, u)
y = x1
(2.10)
˙z = Az − ∆K(z1 − y) (2.11)
with ∆K defined in the same manner as for the observer (2.8):
− ∆ = diag �� θ,θ 2 , . . . ,θ n�� , and
� �
��
− A − K 1 0 . . . 0 is Hurwicz stable.
Then:
− select a strictly increasing sequence of elements of R: {1, θ1, θ2, ...},
− choose λ > 0, a small positive scalar, and
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