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 />
2.4 Single Output Normal Form<br />
the single-output assumption for simplicity <strong>and</strong> clarity of the exposure only. Up to a few<br />
modifications, the theorems can be proven in the multiple output case. Indeed, there is no<br />
unique normal form when ny > 1, therefore the definition of the observer has to be changed<br />
according to each specific case. In Chapter 5 a block wise generalization of the MISO normal<br />
form is considered, <strong>and</strong> the differences between the single output <strong>and</strong> the multiple output<br />
case are explained.<br />
As usual the system is represented by a set of two equations:<br />
− an ordinary differential equation that drives the evolution of the state,<br />
− an application that models the sensor measurements.<br />
Those two equations are of the form:<br />
�<br />
dx<br />
dt<br />
y<br />
=<br />
=<br />
A (u) x + b (x, u)<br />
C (u) x,<br />
where<br />
− x (t) ∈ X ⊂ R n , X compact,<br />
− y (t) ∈ R,<br />
− u(t) ∈ Uadm ⊂ R nu bounded.<br />
The matrices A (u) <strong>and</strong> C (u) are defined by:<br />
⎛<br />
0<br />
⎜<br />
A(u) = ⎜<br />
.<br />
⎝<br />
a2 (u)<br />
0<br />
0<br />
a3 (u)<br />
. ..<br />
· · ·<br />
. ..<br />
. ..<br />
0<br />
⎞<br />
0<br />
⎟<br />
. ⎟<br />
0 ⎟<br />
an (u) ⎠<br />
0 · · · 0<br />
C (u) = � a1 (u) 0 · · · 0 �<br />
(2.7)<br />
with 0
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