Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
3.3 Innovation
Since �B (t)� ≤Lb each bi,j(t) can be interpreted as a bounded element of L∞ [0,d] (R). We
�
identify L∞ [0,d] (R)
� n(n+1)
2
to L∞ �
[0,d] R n(n+1) �
2 and consider the function:
Λ : L ∞ [0,d]
�
R n(n+1)
2
�
× 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]
�
R n(n+1)
2
�
× L ∞ [0,d] (Rnu )
with the weak-* topology6 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
continuous7 .
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.
Since
y � 0,x 0 1, τ � − y � 0,x 0 2, τ � = CΨ (τ) x 0 1 − CΨ (τ) x 0 2,
then � �
�y 0
0,x1, τ � − y � 0,x 0 2, τ �� �2 = � � 0
CΨ (τ) x1 − CΨ (τ) x 0 �
�
2
2 ,
and finally
� d �
�y
0
� 0,x 0 1, τ � − y � 0,x 0 2, τ �� �2 dτ = � x0 1 − x0 � ′ �
2 Gd x0 1 − x0 �
2
≥ λ0 �
�x d
0 1 − x0 �
�
2
2 .
�
(3.6)
Remark 35
As is clear from the proof, we could have used both linearizations along the trajectories x1
and x2 in order to define I. However, that definition would lead to the same inequality. In
addition our definition is more practical to implement.
The solution of the Riccati equation can also not be used to obtain an information equivalent
to innovation. Here, to compute our innovation, we make an exact prediction (without
the correction term that could disturb the estimation).
6 The definition of the weak-* topology is given in Appendix A.
7 This property is explained in Appendix A.
43