Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
Abstract
Keywords - nonlinear observers, nonlinear systems, extended Kalman filter,
adaptive high-gain observer, Riccati equation, continuous-discrete observer, DCmotor,
real-time implementation.
The work concerns the “observability problem” — the reconstruction of a dynamic
process’s full state from a partially measured state— for nonlinear dynamic systems.
The Extended Kalman Filter (EKF) is a widely-used observer for such
nonlinear systems. However it suffers from a lack of theoretical justifications and
displays poor performance when the estimated state is far from the real state, e.g.
due to large perturbations, a poor initial state estimate, etc. . .
We propose a solution to these problems, the Adaptive High-Gain (EKF).
Observability theory reveals the existence of special representations characterizing
nonlinear systems having the observability property. Such representations
are called observability normal forms. A EKF variant based on the usage of a
single scalar parameter, combined with an observability normal form, leads to an
observer, the High-Gain EKF, with improved performance when the estimated
state is far from the actual state. Its convergence for any initial estimated state
is proven. Unfortunately, and contrary to the EKF, this latter observer is very
sensitive to measurement noise.
Our observer combines the behaviors of the EKF and of the high-gain EKF. Our
aim is to take advantage of both efficiency with respect to noise smoothing and
reactivity to large estimation errors. In order to achieve this, the parameter that
is the heart of the high-gain technique is made adaptive. Voila, the Adaptive
High-Gain EKF.
A measure of the quality of the estimation is needed in order to drive the adaptation.
We propose such an index and prove the relevance of its usage. We provide a
proof of convergence for the resulting observer, and the final algorithm is demonstrated
via both simulations and a real-time implementation. Finally, extensions
to multiple output and to continuous-discrete systems are given.