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 />
which, in combination with Lemma 80, gives Tr(S) ≤ max � Tr(S0), 2b<br />
�<br />
a <strong>and</strong><br />
|S| = � Tr(S 2 ) ≤ � Tr(S) 2 �<br />
= Tr(S) ≤ max Tr(S0), 2b<br />
Choose λ > |Q| max � Tr(S0), 2b<br />
�<br />
a <strong>and</strong> apply Lemma 90:<br />
S(τ) =e −λτ φu(τ, 0)S0φ ′<br />
� τ<br />
u(τ, 0) + e<br />
0<br />
−λ(τ−v) φu(τ,v)<br />
Let τ be equal to θ then S(θ) = (I)+(II) with<br />
(I) is definite posittive �<br />
(II) depends on<br />
0<br />
a<br />
�<br />
.<br />
�<br />
S(τ) − S(τ)QS(τ)<br />
�<br />
φ<br />
λ<br />
′<br />
u(τ,v)dv.<br />
(I) = e−λθφu(θ, 0)S0φ ′<br />
u(θ, 0),<br />
� θ<br />
(II) = e −λ(θ−v) �<br />
φu(θ,v) S − SQS<br />
�<br />
φ<br />
λ<br />
′<br />
u(θ,v)dv.<br />
S − SQS<br />
λ<br />
�<br />
, which we can rewrite √ �<br />
S Id −<br />
itive definite for 0 < τ
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