Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
5.1 Multiple Inputs, Multiple Outputs Case
Since �B (t)� ≤Lb, each bi,j(t) can be considered as a bounded element of L∞ [0,d] (R).
�
We identify L∞ [0,d] (R)
�p with L∞ [0,d] (Rp ) where3 We consider the function:
p =
ny �
i=1
ni(ni − 1)
2
+ nyn.
Λ : L ∞ [0,d] (Rp ) × L ∞ [0,d] (Rnu ) −→ R +
(bi,j) (j≤i)∈{1,..,n},u↩ → λmin (Gd)
where λmin (Gd) is the smallest eigenvalue of Gd. Let us endow L ∞ [0,d] (Rp ) × L ∞ [0,d] (Rnu )
with the weak-* topology4 and R has the topology induced by the uniform convergence. The
weak-* topology on a bounded set implies uniform continuity of the resolvent, hence Λ is
continuous5 .
Since control variables are supposed to be bounded,
Ω1 =
�
L ∞ [0,d]
�
R n(n+1)
2
�
; �B� ≤Lb
and
�
Ω2 = u ∈ L ∞ [0,d] (Rn �
);�u� ≤Mu
are compact subsets. Therefore Λ (Ω1 × Ω2) is a compact subset of R which does not contain
0 since the system is observable for any input. � Thus Gd is never singular. Moreover, for Mu
sufficiently large,
includes L∞ [0,d] (Uadm).
�
u ∈ L∞ [0,d] (Rn );�u� ≤Mu
Hence, there exists λ0 d such that Gd ≥ λ0 d Id for any u and any matrix B(t) as above. We
conclude that � d �
�y � 0,x 0 1, τ � − y � 0,x 0 2, τ �� �2 dτ ≥ λ 0 �
�x d
0 1 − x 0 �
�
2
2 . (5.4)
0
3 The matrix B(t) can be divided into ny parts of dimensions (ni × n), i ∈ {1, ..., ny}. For each of those
parts, the last line may be full (i.e. derivation of the vector field elements of the form bi,n i (x, u) w. r. t. the
state x). It gives a maximum of nyn elements.
For each one of the ny parts, the lower triangular part of a square of dimension (ni −1×ni −1) may contain
non zero elements. That makes ni(ni − 1)/2 elements.
Therefore the maximum number of non null elements of the matrix B(t) is:
ny �
i=1
ni(ni − 1)
2
+ nyn.
4 The definition of the weak-* topology is given in Appendix A.
5 This property is explained in Appendix A.
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