Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
2.1 Systems and Notations
The work presented in this dissertation follows the framework of the theory of deterministic
observation developed by J-P Gauthier, I. Kupka, and coworkers. This theory has
a very long history the beginning of which can be found in the articles of R. Herman, A.
J. Krener [65], and H. J. Sussmann [110]. In the book [57], which is in itself a summary of
several papers [53–56, 70], J-P. Gauthier and I. Kupka exposed well established definitions
for observability and several important subsequent theorems.
An important result of the theory is that there exist representations of nonlinear systems
that characterize observability (those are denoted by observability normal forms in the literature).
The article [52], from J-P Gauthier and G. Bornard, often referenced, contains early
results.
The construction of high-gain observers, either of the Luenberger (J-P. Gauthier, H.
Hammouri and S. Othman [54]) or of the Kalman (F. Deza, E. Busvelle et al. [47]) style,
rests upon the theory that normal forms are crucial in order to establish the convergence
of such observers. Here convergence means that the estimation error decreases to zero. As
we will see in Chapter 3, and in Theorems 14, 16 and 29, the estimation error decays at an
exponential rate. Those observers are also called exponential observers.
In the present chapter, the main concepts of observability theory in the deterministic
setting are presented together with more recent results such as those from E. Busvelle and
J-P. Gauthier [38–40].
The main contribution of this work is the construction and the proof of convergence of a
high-gain observer algorithm, whose high-gain parameter is time varying. Hence a review of
such adaptive-gain observers is proposed: F. Allgőwer and E. Bullinger (1997), M. Boutayeb
et al. (1999), E. Busvelle and J-P. Gauthier (2002), L. Praly et al.(2004 and 2009) and a
recent paper from H. K. Khalil (2009).
2.1 Systems and Notations
A system in the state space representation is composed of two time dependent equations1 :
⎧
⎨
(Σ)
⎩
dx(t)
dt
y(t)
x(0)
=
=
=
f(x(t),u(t),t)
h(x(t),u(t),t)
x0
where
− x(t) denotes the state of the system, belonging to R n , or more generally to a ndimensional
analytic differentiable manifold X,
− u(t) is the control (or input) variable with u(t) ∈ Uadm ⊂ R nu ,
− y(t) denotes the measurements (or outputs) and takes values in a subset of R ny ,
− f is a u-parameterized smooth nonlinear vector field, and
− the observation mapping h : X × Uadm → R ny is considered to be smooth and possibly
nonlinear.
1 Later on, the time dependency will be omitted for notation simplicity.
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