Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
2.2 The Several Definitions of Observability
This definition appears as the most natural one 4 , but as injectivity is not a stable property
observability is difficult to manipulate from a topological point of view 5 . In order to render
the notion of observability more tractable, Definition 1 is modified by considering the first
order approximation of the state-output mapping. Let us begin with the definition of the
first order approximation of a system:
Definition 2
− The first (state) variation of (Σ) (or lift of (Σ) on TX) is given by:
⎧
⎪⎨
(TXΣ)
⎪⎩
dx(t)
dt
dξ(t)
dt
= f(x, u)
= Dxf(x,
ˆy =
u)ξ
dxh(x, u)ξ
where
– (x,ξ ) ∈ TX (or R n × R n ) is the state of (TXΣ),
– dxh is the differential of h w.r.t. to x, and
(2.1)
– Dxf is the tangent mapping to f (represented by the Jacobian matrices of h and
f w.r.t. x).
− the state-output mapping of TXΣ is denoted PTXΣ,u. This mapping is in fact the
differential of PXΣ,u w.r.t. x0 (i.e. its first order approximation, TPXΣ,u|x0 ).
The second part of Definition 1 is adapted to this new state-output mapping in a very
natural way.
Definition 3
The system (Σ) is said uniformly infinitesimally observable 6 w.r.t. a class C of
inputs if for each u(.) ∈ C and each x0 ∈ X, all the (x0 parameterized) tangent mappings
TPX Σ,u|x0 are injective.
Since the state-output mapping considered in this definition is linear then the injectivity
property has been topologically stabilized. Finally a third definition of observability has been
proposed by using the notion of k−jets 7 .
4
An equivalent definition, based on the notion of indistinguishability can be found in [19] (Definitions 2
and 3).
� 5 3
e.g. x ↦→ x � is injective, but for all ɛ > 0, � x ↦→ x 3 − ɛx � isn’t.
6 ∞
Infinitesimal observability can also be considered “at a point (u, x) ∈ L × X”, or only “at a point
u ∈ L ∞ ”
7 The k-jets j k u of a smooth function u at the point t = 0 are defined as
j k u =
�
u(0), ˙u(0), ..., u (k−1) �
(0) .
Then for a smooth function u and for each x0 ∈ X, the k-jets j k y =
�
y(0), ˙y(0), . . . , y (k−1) �
(0) is well defined:
this is the k-jets extension of the state-output mapping.
See [9] for details. The book is though not so easy to find. R. Abraham’s webpage can be of help.
13