Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
With ε > 0 being given, define ε ∗ = ε
k
take δ ≤ ε∗
2γm
, where γ = sup
x∈R n
A.4 Bounds on a Gramm Matrix
, where k comes from the preceding lemma. We
�G(x)�, and m = sup �un − u�∞. We get, for all θ ∈ [0,T]:
n
��
�
� θ
�
�
� (un(s) − u(s))G(x(s))ds�
�
ti
≤ mγδ ≤ ε∗
2
By the weak-* convergence of (un), there exists N ∈ N∗ such that for all n > N,
��
�
� ti
� ε∗
�
� (un(s) − u(s))G(x(s))ds�
� ≤
2 .
0
By the lemma, there exists N ∈ N ∗ such that for all n > N, for all t ∈ [0,T], �xn(t)−x(t)� ≤ε,
which proves the sequential continuity.
A.4 Bounds on a Gramm Matrix
This section’s material is taken from [57], Chapter 6, Section 2.4.2. We present a lemma
useful to investigate the properties of the Riccati matrix of extended Kalman filters, both in
the continuous and continuous discrete settings. This lemma is used in Appendix B.
Let Σ be a system12 on Rn , of the form
⎧
where
⎪⎨
(Σ)
⎪⎩
dx
dτ
= A(u)x +
y = C(u)x,
n�
k, l =1
l ≤ k
uk,lek,lx
− A(u) is a an anti-shift matrix whose elements never equals zero, and are bounded,
− C(u) = � α(u) 0 ... 0 � , where α never equals zero and is bounded,
− ei,j is such that ei,jxk = δjkvi, where {vk} denotes the canonical basis of R n .
(A.4)
The term on the right of the “ + ” sign is a lower triangular (n × n) matrix. For (ui,j),
a measurable bounded control function, defined on [0,T], we denote ψu(t, s) the associated
resolvent matrix (see Section A.1 above). We define the Gramm observability matrix of Σ by
� T
Gu =
0
ψ ′
u(v, T )C ′
Cψu(v, T )dv.
The matrix Gu is symmetric and positive semi-definite. The system (A.4) is observable for
all u(.) measurable and bounded.
12 In order to make this system easier to understand, we only describe it as single output. The result of the
lemma is though valid for multi outputs systems.
128