Adaptative high-gain extended Kalman filter and applications
Adaptative high-gain extended Kalman filter and applications
Adaptative high-gain extended Kalman filter and applications
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tel-00559107, version 1 - 24 Jan 2011<br />
B.1 Bounds on the Riccati Equation<br />
B.1 Bounds on the Riccati Equation<br />
This section deals with the properties of the Riccati matrix S. We consider the continuous<br />
discrete framework. The proof follows the ideas of [57] where those properties are investigated<br />
in details for continuous time systems 1 .<br />
Lemma 77<br />
Let us consider the prediction correction equations:<br />
⎧<br />
⎪⎨<br />
⎪⎩<br />
d ¯ S<br />
dτ<br />
�<br />
= − A(ū)+ ¯ �′<br />
b(¯z,ū)<br />
θ<br />
R −1 C<br />
¯Sk(+) = ¯ Sk(−)+θδtC ′<br />
with the notations of Section 5.2.1, <strong>and</strong> the set of assumptions:<br />
¯S − ¯ �<br />
S A(ū)+ ¯ �<br />
b(¯z,ū)<br />
−<br />
θ<br />
¯ SQ¯ S<br />
− Q <strong>and</strong> R are fixed symmetric positive definite matrices,<br />
�<br />
�<br />
− the functions ai (u (t)), �˜b ∗ �<br />
�<br />
i,j (z, u) �, are smaller than aM > 0,<br />
− ai (u (t)) ≥ am > 0,<br />
− θ(0) = 1, <strong>and</strong><br />
(B.1)<br />
− S(0) is a symmetric positive definite matrix taken in a compact of the form aId ≤<br />
S(0) ≤ bId.<br />
Then, there exist a constant µ, <strong>and</strong> two scalars 0 < α
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