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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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