Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
2.6 On Adaptive High-gain Observers
consequently means that provided the disturbances vanish, the estimation error converges to
0. This observer is not based on any quality metric of the estimation and the evolution of
the high-gain doesn’t depend of the quality convergence. Therefore the situation may arise
such that the observer has already converged and that the high-gain is still high, which would
therefore amplify the noise.
The work herein, contrary to this section’s observer, is set in the global Lipschitz setting.
Further, it aims to provide an observer for which the high-gain parameter decreases to 1 (or
the the lowest value allowed by the user) when the local convergence 18 of the algorithm can
be used (i.e. the high-gain is not needed anymore).
2.6.3 Ahrens and Khalil
This paper deals with a closed loop control strategy that comprises an observer having a
high-gain switching scheme. We focus on the observer’s definition together with the switching
strategy used, and only give a simplified version of the system the authors consider, refer to
[11] for details.
Definition 22
The simplified version of the system used in [11] is
�
˙x = Ax + Bφ(x, d, u)
y = Cx + v
where
− x ∈ R n is the state variable,
− y ∈ R is the output,
− d(t) ∈ R p is a vector of exogenous signals,
− v(t) ∈ R is the measurement noise, and
− u(t) is the control variable.
The matrices A, B and C are:
⎛
0
⎜ .
⎜
A = ⎜ .
⎜
⎝ 0
1
0
. . .
0
1
. ..
. . .
...
. ..
. ..
0
⎞ ⎛ ⎞
0
0
⎟ ⎜ ⎟
.
⎟ ⎜
⎟ ⎜ .
⎟
⎟ ⎜ ⎟
0
⎟ B = ⎜
⎟ ⎜ .
⎟
⎟ ⎜ ⎟
1 ⎠ ⎝ 0 ⎠
0 . . . . . . . . . 0
1
�
�
C = 1 0 ... ... 0 .
The set of assumptions for this system is:
(2.16)
18 The convergence of the extended Kalman filter can be proven for small initial errors only (see the proof
of Chapter 3).
27