Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
− Consider that the output y(t) is known for all times t ≥ 0.
2.3 Observability Normal Forms
− Compute the successive time derivatives of the outputs until there exists a k>0 such
that the state x(t) can be uniquely computed, for all times, from the equations of
(˙y(t), ¨y(t), ...) where y(t),u(t) are known time varying parameters.
− We have found a k-jets extension of (Σ) that is injective and therefore it is at least
differentially observable.
− Assuming that (Σ) is analytic, then it is L ∞ observable.
An illustration of this method is given in Chapter 4, Section 4.1.2. The process under
consideration is a series-connected DC machine (see also [21], part 3.5).
2.3 Observability Normal Forms
The main significance of the theory is the existence of two distinct situations, which depend
on the number of outputs with respect to the number of inputs. In one case observability
is a generic property 9 , and it is not a generic property in the other case. These two specific
situations are explained in the subsections below.
2.3.1 First case: ny >nu
The first case occurs when the number of outputs is greater than the number of inputs,
i.e. ny >nu. The situation is defined by two theorems. The first states the genericity
property of the set of observable systems in S; the second theorem introduces the (generic)
observability normal form.
Theorem 6 ([57], 4.2.2 and 4.2.4 page 40)
1. The set of systems that are strongly differentially observable of order 2n +1 is residual
in S.
2. The set of analytic strongly differentially observable systems (of order 2n +1) that are
moreover L ∞ -observable is dense in S.
Theorem 7 ([40])
The following is a generic property on S. Set k =2n +1. For all sufficiently smooth u(.),
denote j k u(t) = � u(t), ˙u(t), ..., u (k−1) (t) � . Choose an arbitrarily large, relatively compact 10
open subset Γ of X. Consider also an arbitrary bound on the control and its first k derivatives
(i.e. u, ˙u, .., u (k) ). Then the mappings
Φ Σ
k,j k u : X → R kny
x(t) ↦→ � y(t), ˙y(t), ..., y (k−1) (t) �
9 A subset is said to be generic if it contains a residual subset. A subset is said to be residual if it is a
countable intersection of open dense subsets.
10 A subset is said relatively compact if its closure is compact.
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