Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
2.6 On Adaptive High-gain Observers
part of the world, early references may be found in the book edited by A. Gelb[58], the two
volumes book of P. S. Maybeck [89, 90], Chapter 10 of the second one in particular, the book
from C. K. Chui and G. Chen [43]. Recent papers can be found in the INS/GPS community,
such as [96, 115] or the book from M. S. Grewal [60]. The review article of R. K. Mehra
[92] is warmly advised as an introduction to the topic. We do not expatiate on the subject
since 1) most of the techniques developed in those papers are statistical methods, 2) the main
bottleneck in the analysis is due to the linearization of the model in order to use the Riccati
equation which is specific to the nonlinear case, 3) when switching based methods are used
we have a better time explaining them directly in the nonlinear setting.
In the non linear case a vast majority of strategies are proposed for discrete-time systems.
We describe a subset of those strategies below.
Definition 25
A discrete time system is defined by a set of two equations of the form:
�
xk+1 = f(xk,uk)
yk+1 = h(xk+1)
(2.17)
with the usual notation xk = x(kδt), δt > 0 being the sample time. At least one of the two
functions f and h is nonlinear.
The discrete extended Kalman filter associated with this system is given by the set of
equations:
where
P rediction
�
⎧
⎪⎨
Correction
⎪⎩
z −
k+1
P −
k+1
= f(zk,uk)
′
= AkPkA k + Qk,
zk+1 = z −
k+1 + Lk+1(yk+1 − h(z −
k+1 ))
P +
k+1 = (Id − Lk+1C)P −
k+1
Lk+1 = P −
k+1
C ′
(CP −
k+1
′
C + R) −1 ,
− zk denotes the estimated state, and P0 is a symmetric positive definite matrix,
− A is the jacobian matrix of f, computed along the estimated trajectory,
− C is the jacobian matrix of h, computed along the estimated trajectory.
The first strategy we present was proposed by M. G. Pappas and J. E. Doss in their
article [99] (1988). Although they do not consider the observability issue of the system, it is
nonetheless an underlying concern of their work:
− when the system is at steady state, the system is less observable and a slow observer
is required to give an accurate estimate, noise smoothing being an additional derived
benefit,
30