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 />
!<br />
!" #<br />
Figure 4.24: Connections Diagram.<br />
4.3 real-time Implementation<br />
Figure 4.25: A view on the testbed (<strong>and</strong>, on the left, the friction tool).<br />
observability normal form is then<br />
⎛<br />
˙x1<br />
⎝ ˙x2<br />
˙x3<br />
⎞<br />
⎠ =<br />
⎛<br />
⎝ 0 − Laf<br />
0<br />
1<br />
L<br />
0<br />
0<br />
− 1<br />
0 0<br />
⎛<br />
J<br />
0<br />
⎞ ⎛<br />
⎠ ⎝<br />
⎜<br />
+ ⎜<br />
⎝<br />
�<br />
1<br />
L u(t) x2<br />
x1<br />
x2<br />
x3<br />
⎞<br />
⎠<br />
1<br />
L<br />
(u(t) − Rx1)<br />
�<br />
Laf2x31 − Bx2 − p x2.08<br />
2<br />
x1.08 �<br />
�<br />
1<br />
u(t) x3<br />
⎟<br />
�<br />
⎠<br />
x2x3 − Laf1 − Rx3<br />
x1 x1 .<br />
x1 − Rx2 − Laf1 x2 �<br />
2 + x1<br />
1<br />
J<br />
1<br />
L<br />
This model needs to be calibrated, i.e., we need to find appropriate values for the parameters<br />
R, L, B, J, Laf1 ,Laf2 <strong>and</strong> p. We find the values following a st<strong>and</strong>ard approach. First,<br />
we applied a linear method in order to identify some of the parameters. Second, we used<br />
a nonlinear optimization technique to obtain more accurate values for the complete set of<br />
parameters.<br />
1. From data samples (voltage, current <strong>and</strong> speed) an initial estimation is obtained via<br />
the least mean squares technique.<br />
Let us consider the initial model (not in normal form) at steady state. Suppose that<br />
L = J = 1 <strong>and</strong> that the load torque (Tl) is null. With the parameters B, Laf ,p as<br />
82<br />
⎞
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