28.01.2013 Views

Adaptative high-gain extended Kalman filter and applications

Adaptative high-gain extended Kalman filter and applications

Adaptative high-gain extended Kalman filter and applications

SHOW MORE
SHOW LESS

You also want an ePaper? Increase the reach of your titles

YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.

tel-00559107, version 1 - 24 Jan 2011<br />

2.6 On Adaptive High-<strong>gain</strong> Observers<br />

part of the world, early references may be found in the book edited by A. Gelb[58], the two<br />

volumes book of P. S. Maybeck [89, 90], Chapter 10 of the second one in particular, the book<br />

from C. K. Chui <strong>and</strong> G. Chen [43]. Recent papers can be found in the INS/GPS community,<br />

such as [96, 115] or the book from M. S. Grewal [60]. The review article of R. K. Mehra<br />

[92] is warmly advised as an introduction to the topic. We do not expatiate on the subject<br />

since 1) most of the techniques developed in those papers are statistical methods, 2) the main<br />

bottleneck in the analysis is due to the linearization of the model in order to use the Riccati<br />

equation which is specific to the nonlinear case, 3) when switching based methods are used<br />

we have a better time explaining them directly in the nonlinear setting.<br />

In the non linear case a vast majority of strategies are proposed for discrete-time systems.<br />

We describe a subset of those strategies below.<br />

Definition 25<br />

A discrete time system is defined by a set of two equations of the form:<br />

�<br />

xk+1 = f(xk,uk)<br />

yk+1 = h(xk+1)<br />

(2.17)<br />

with the usual notation xk = x(kδt), δt > 0 being the sample time. At least one of the two<br />

functions f <strong>and</strong> h is nonlinear.<br />

The discrete <strong>extended</strong> <strong>Kalman</strong> <strong>filter</strong> associated with this system is given by the set of<br />

equations:<br />

where<br />

P rediction<br />

�<br />

⎧<br />

⎪⎨<br />

Correction<br />

⎪⎩<br />

z −<br />

k+1<br />

P −<br />

k+1<br />

= f(zk,uk)<br />

′<br />

= AkPkA k + Qk,<br />

zk+1 = z −<br />

k+1 + Lk+1(yk+1 − h(z −<br />

k+1 ))<br />

P +<br />

k+1 = (Id − Lk+1C)P −<br />

k+1<br />

Lk+1 = P −<br />

k+1<br />

C ′<br />

(CP −<br />

k+1<br />

′<br />

C + R) −1 ,<br />

− zk denotes the estimated state, <strong>and</strong> P0 is a symmetric positive definite matrix,<br />

− A is the jacobian matrix of f, computed along the estimated trajectory,<br />

− C is the jacobian matrix of h, computed along the estimated trajectory.<br />

The first strategy we present was proposed by M. G. Pappas <strong>and</strong> J. E. Doss in their<br />

article [99] (1988). Although they do not consider the observability issue of the system, it is<br />

nonetheless an underlying concern of their work:<br />

− when the system is at steady state, the system is less observable <strong>and</strong> a slow observer<br />

is required to give an accurate estimate, noise smoothing being an additional derived<br />

benefit,<br />

30

Hooray! Your file is uploaded and ready to be published.

Saved successfully!

Ooh no, something went wrong!