Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
Chapter 6
Conclusion and Perspectives
The work described in this thesis deals with the design of an observer of the Kalman type
for nonlinear systems.
More precisely, we considered the high-gain formalism and proposed an improvement of
the high-gain extended Kalman filter in the form of an adaptive scheme for the parameter at
the heart of the method. Indeed, although the high-gain approach allows us to analytically
prove the convergence of the algorithm in the deterministic setting, it comes with an increased
sensitivity to measurement noise. We propose to let the observer evolve between two end
point configurations, one that rejects noise and one that makes the estimate converge toward
the real trajectory. The strategy we developed here allowed us to analytically prove this
convergence.
Observability theory constitutes the framework of the present study. Thus, we began this
thesis by providing a review and some insight into the main results of the theory of [57]. We
also provided a review of similar adaptive strategies. In this introduction and background
review, we stated that the main concern of the thesis would be theoretically proving that the
observer is convergent.
The observer has been described in Chapters 3 and 5. It was initially described in the
continuous setting, and afterwards extended to the continuous-discrete setting. The adaptive
strategy was also explained in those chapters. This strategy is composed of two elements:
1. a measurement of the quality of the estimation, and
2. an adaptation equation.
The quality measurement is called innovation or innovation for an horizon of length d. It
is slightly different than the usual concept of innovation. The major improvement provided
by our definition is a proof that shows that innovation places an upper bound on the past
estimation error (the delay equals the parameter d above mentioned). This fact is a corner
stone of the overall convergence proof.
The second element of the strategy is the adaptation equation that drives the high-gain
parameter. A differential equation was used in the continuous setting, and a function in
the continuous-discrete setting (i.e. the adaptation is performed at the end of the update
procedure). The sets of requirements for those two applications have been proposed such that
several adaptation functions can be conceived. The set of possible applications isn’t void, as
we demonstrated by actually displaying an eligible function.
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