Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
2.2 The Several Definitions of Observability
The inputs are time functions defined on open intervals of the form [0,T[ (with the
possibility that T =+∞). The functions u(.) are assumed to be measurable and bounded
almost everywhere on any compact sub-interval of [0,T[. The corresponding function set is
L ∞ (Uadm).
The outputs are also functions of time defined on open intervals of the form [0,T(u)[.
This notation takes into account that for a given input function, defined for a maximum time
T , the system might become unstable (i.e. explode toward infinity). The explosion time is
likely to be less than T . Therefore we have T (u) ≤ T . Output functions are also measurable
and bounded almost everywhere on any compact sub-interval of [0,T(u)[. The corresponding
function set is L ∞ (R ny ).
The set of all systems of the form (Σ) is denoted S = {Σ =(f, h)}. The genericity (or
non-genericity) property of observable systems is considered with respect to the set S.
The topologies associated to those sets are
− the C ∞ Whitney topology for the set S (see, e.g. [66]) 2 . Two important features of that
topology are
1. it is not metrizable and,
2. it has the Baire property 3 ,
− either the topology of uniform convergence or the weak-∗ topology for the sets L ∞ (Uadm)
and L ∞ (R ny ),
− we will also use the topology of the euclidean norm when dealing with subspaces of R q ,
q = n, nu or ny.
2.2 The Several Definitions of Observability
Observability is the notion that translates the property of a system to permit the reconstruction
of the full state vector from the knowledge of the input and output variables. In
other words: considering any input function u(.), can any two distinct initial states x 1 0 , x2 0 be
distinguished from one another?
Definition 1
− The state-output mapping of the system (Σ) is the application:
PXΣ,u : X → L ∞ (R ny )
x0 ↦→ y(.)
− A system (Σ) is uniformly observable (or just observable) w.r.t. a class C of inputs
if for each u(.) ∈ C, the associated state-ouput mapping is injective.
2 A basic neighborhood of a system Σ = (f, h) in the C j Whitney topology is determined by a set of
functions ɛ(z) > 0, and formed by the systems ˜ Σ =( ˜ f,˜g) ∈ S such that the derivatives, up to the order j, of
(f − ˜ f, h − ˜ h), w.r.t all the variables, have their norm at point z =(x, u) less than ɛ(z).
3 Baire property: a countable intersection of open dense subsets is dense.
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