Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
unknowns (i.e. Laf1 = Laf2 ), the model becomes19 :
� V
0
�
=
� I Iωr 0 0
0 I 2 ωr ω 2.08
r
4.3 real-time Implementation
⎛
�
⎜
⎝
From a series of experiments we collected enough data to constitute three sets of values:
(V1, ..., VN), (I1, ..., IN) and ((ωr)1, ..., (ωr)N) and found a mean least squares solution
to the equation above.
2. At this stage a nonlinear optimization routine was used20 . The solutions found above
together with L = J = 1, and Laf = Laf1 = Laf2 where taken as the initial values of the
optimization routine. The cost function to minimize is taken as the distance between
measured data and predicted trajectory
� T ∗
�I(v) − Ĩ(v)�2 � T ∗
dv + α2 �ωr(v) − ˜ωr(v)� 2 dv,
K(L, R, Laf1 , J, Laf2 , B, p) ↦→ α1
where
0
− I and ωr are some measured data that excite the dynamical modes of the process
(in a pseudo random binary sequence manner [85]),
− Ĩ and ˜ωr are the predicted trajectory obtained from the model above using the
measured input variables and Ĩ(0) = I(0) and ˜ωr(0) = ωr(0).
− and α1, α2 are weighting factors, that may be used to compensate for the difference
of scale between the current and voltage amplitudes.
3. The situation may arise when the solution found by the optimization routine is a local
minimum. In this case, the solution set of parameters doesn’t match the experimental
data while the search algorithm stops. A solution would be to slightly modify the
output of the algorithm and relaunch the search.
Because of the great number of parameters and the difficulty in analyzing the cost
function, this re-initialization is rather venturesome. It is difficult to determine which
parameters shall be modified, and how. We propose to compare the measured data (I
and ωr) to the predictions ( Ĩ and ˜ωr), using the initial set of parameters, and to find
p1 (resp. p2) such that p1I ≈ Ĩ (resp. p2ωr ≈ ˜ωr). Then we reuse the initial model in
order to determine how to modify the parameters, e.g.
L ˙ Ĩ = V − RĨ − Laf1Ĩ ˜ωr
R
Laf
B
p
L(p1 ˙
I) =V − R(p1I) − Laf1 (p1I)(p2ωr)
(p1L) ˙
I = V − (Rp1)I − (Laf1 p1p2)Iωr.
Although non standard, this method gives us new initial values for a new optimization
search. In practice, this method produced excellent results.
⎞
⎟
⎠ .
19 If we keep Laf1 �= Laf2, the least square solution of the second line is trivially (0, 0, 0)
20 We kept it simple: the simplex search method (see, for example, [88, 104]).
83
0