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 />
2.6 On Adaptive High-<strong>gain</strong> Observers<br />
their observer construction can be found in articles like [16, 78, 79, 103] 16 . In the following<br />
we present the observer defined in [16], as it complies with the latest updates of their theory.<br />
Definition 19<br />
The system dynamics are:<br />
⎧ ⎛<br />
⎞ ⎛<br />
0 a2(y) 0 ... 0<br />
⎜ 0 0 a3(y) ... 0<br />
⎟ ⎜<br />
⎪⎨ ⎜<br />
˙x = ⎜<br />
.<br />
⎟ ⎜<br />
⎜ .<br />
.. .<br />
⎟ ⎜<br />
⎟ x + ⎜<br />
⎜<br />
⎟ ⎜<br />
⎝ 0 ... ... ... an(y) ⎠ ⎝<br />
⎪⎩<br />
0 ... ... ... 0<br />
y = x1 + δy(t).<br />
It is also denoted �<br />
where<br />
− y is the measured output,<br />
− the functions ai are locally Lipschitz,<br />
f1(u, y)<br />
f2(u, y, x2)<br />
f3(u, y, x2,x3)<br />
. . .<br />
fn(u, y, x2, ..., xn)<br />
˙x = A(y)x + b(y, x2, .., xn, , u)+∆( t)<br />
y = x1 + δy(t)<br />
⎞<br />
⎛<br />
⎟ ⎜<br />
⎟ ⎜<br />
⎟ ⎜<br />
⎟ + ⎜<br />
⎟ ⎜<br />
⎠ ⎝<br />
δ1(t)<br />
δ2(t)<br />
δ3(t)<br />
. . .<br />
δn(t)<br />
⎞<br />
⎟<br />
⎠<br />
(2.13)<br />
(2.14)<br />
− u is a vector in R nu representing the known inputs <strong>and</strong> a finite number of their derivatives,<br />
− the vector ∆(t) represents the unknown inputs, <strong>and</strong><br />
− δy is the measurement noise.<br />
The construction of the observer is based on some additional knowledge concerning the<br />
system:<br />
1. a function α of the output y such that 0 < ρ
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