Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
2.6 On Adaptive High-gain Observers
In this strategy, α2 is a very important parameter as it is used to decide if the sign is
considered to be either constant or if it is constantly changing.
The ideas behind the second strategy come directly from articles like the one of M. S.
Mehra [92] or the book of P. S. Maybeck [90]. References for this method are [69] (sensor
fusion in robotics), [83] (visual motion estimation via camera sensor), and [37, 102] (in flight
orientation). This method aims at estimating Q, the process state noise covariance matrix.
It is viewed as a measure of the uncertainty in the state dynamics between two consecutive
updates of the observer. An observation of Q, denoted Q ∗ is given by the equation (see [37]):
which can be rewritten
Q ∗ = (zk+1 − z −
k+1 )(zk+1 − z −
k+1 )′
+ P −
k+1 − Pk+1 − Qk ,
Q ∗ = (zk+1 − z −
k+1 )(zk+1 − z −
k+1 )′
= (zk+1 − z −
k+1 )(zk+1 − z −
k+1 )′
− Qk))
− (AkPkA ′
k − Qk)).
− (Pk+1 − (P −
k+1
The new value for the matrix Q, denoted ˆ Qk+1 is obtained using a moving average (or low
pass filter) process:
ˆQk+1 = ˆ Qk + 1
�
Q ∗ − ˆ �
Qk .
LQ
LQ is the size of the window that sets the number of updates being averaged. In this pro-
cedure, LQ is a performance parameter that has to be tuned. The quantity (zk+1 − z −
k+1 )=
Kk(yk − hk(z −
k )) plays a key role in this strategy. It is denoted as innovation, and contains
specific information on the quality of the estimation. In our work, we use a modified definition
of innovation, and prove that it is a quality measurement of the estimation error23 .
A third strategy consists of designing of a set of nonlinear observers with different values
for Q and R. They are used in parallel and the final estimated state is chosen among all the
estimates available. A selection criteria has to be defined: minimization of innovation is the
most straightforward method that can be used (see the observer of Subsection 2.6.6). (We
refer the reader to the algorithm of K. J. Bradshaw, I. D. Reid and D. W. Murray [32], and
references therein.) Every estimate is associated with a probability density computed with a
maximum likelihood algorithm. The state estimate is then obtained as a combination of all
the observers’ outputs weighted by their associated probability.
Notice that for nonlinear systems, E [u(x)] is distinct from u(E [x]). An interesting feature
of this strategy, is that the first quantity can be computed quite naturally.
2.6.5 Boutayeb, Darouach and coworkers
In their research M. Boutayeb, M. Darouach and coworkers proposed several types of
observers, including extended Kalman filter based obsevers [30, 31], observers based on the
differential mean value theorems [116, 117], H∞ filtering [13], or for systems facing bounded
23 Cf. Lemma 33 of Chapter 3
32