Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
4.2 Simulation
again provided that I>0 (i.e. x1 > 0). This leads us to the expression Tl = x3 . Recall that
x1
Tl
˙ = 0, then
Thus the application:
x3 ˙ = − Laf x2x3
L x1
+ u(t) x3
−
L x1
R
L x3 = b3(x1,x2,x3,u). (4.4)
R ∗+ × R × R → R ∗+ × R × R
(I,ω r,Tl) ↩→ (I, Iωr, ITl)
is a change of coordinates that puts the system (4.1), into the observer canonical form2 defined by equations (4.2), (4.3), (4.4).
The inverse application is:
�
(x1,x2,x3) ↩→ x1, x2
,
x1
x3
�
.
x1
Computations of the coefficients of the matrix b ∗
that appears in the Riccati equation of the
observer — Cf. next section — are left to the reader.
4.2 Simulation
4.2.1 Full Observer Definition
We now recall the equations of the adaptive high-gain extended Kalman filter. As we want
to minimize the computational time required to invert the matrix S, let us define P = S−1 .
The identity dP dS−1
dt = dt
dS = S−1
dt S−1 allows us to rewrite the observer as:
⎧
⎪⎨
⎪⎩
dz
dt = A(u)z + b(z, u)+PC′ R −1
dP
dt
θ (Cz − y(t))
= P (A(u)+b∗ (z, u)) ′ +(A (u)+b∗ (z, u))P − PC ′ R −1
dθ
dt
θ CP + Qθ
= µ(Id)F0(θ) + (1 − µ(Id))λ(1 − θ)
(4.5)
where
− Rθ = θ −1 R,
− Qθ = θ∆ −1 Q∆ −1 ,
− ∆θ = diag �� 1, θ,θ 2 , . . . ,θ n−1�� ,
− F0(θ) =
� 1
∆T θ2 ifθ ≤ θ1
1
∆T (θ − 2θ1) 2 if θ >θ 1
,
− µ(I) = � 1+e −β(I−m)� −1 is a β and m parameterized sigmoid function (Cf. Figure
4.10),
2 One could ask about the compact subset required by Theorem 36. In the present situation, a compact
subset would be a collection of three closed and bounded intervals. The problem arises from the exclusion of
0 as a possible value for x1. This is solved by picking any small ɛ > 0 and considering that for I = x1 < ɛ the
motor is running too slowly to be of any practical use. Those trajectories are now in a compact subset of the
state space.
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