Adaptative high-gain extended Kalman filter and applications

tel-00559107, version 1 - 24 Jan 2011
4.1 Modeling of the Series-connected DC Machine and Observability Normal
Form
is the input of the system, u(t), and the current, I(t), is the measured output. The resulting
following SISO model for the series-connected DC motor is:
⎧ � � �
⎨ LI˙ u − RI − Laf ωrI
=
J ˙ωr
Laf I
⎩
2 �
− Bωr − Tl
(4.1)
y = I
This model is used to simulate the DC motor by means of a Matlab/Simulink S-function.
4.1.2 Observability Cannonical Form
Before implementing the observer to reconstruct the state vector of this system, we test the
system’s observability property. We use the differentiation approach, i.e. we check differential
observability (which implies observability):
− if I(t) is known with time, then ˙
I =(u − R.I − Laf ωrI)/L is known and as long as u(t),
R, Laf and L are known then ωr can be computed,
− because ωr(t) is known, ωr ˙ =(Laf I2 − Bωr − Tl)/J can also be computed. From the
knowledge of I(t), Laf , B,and J, then Tl can be estimated.
We conclude that a third variable can be added to the state vector in order to reconstruct
the load torque applied to the shaft of the motor along with the state of the system. We
assume that the load torque is constant over time. Sudden changes of the load torque then,
are interpreted as non modeled perturbations. The estimation of the load torque is made
possible including the constraint ˙
Tl = 0 in equation (4.1). We now need to find the coordinate
transformation that puts this systems into the observability canonical form.
From the equation y = I, we choose x1 = I and then
which by setting x2 = Iωr becomes
x1 ˙ = 1
L (u(t) − RI − Laf Iωr),
x1 ˙ = − Laf
L x2 + 1
L (u(t) − Rx1) =α2(u)x2 + b1(x1,u). (4.2)
We now compute the time derivative of x2:
x2 ˙ = ˙ Iωr + Iωr ˙ = − 1
J TlI − B
J Iωr + Laf
J I3 − Laf
L ω2 rI + u(t)
L ωr − R
L ωrI
provided that I>0 (i.e. x1 > 0). This constraint represents a reasonable assumption since
when I, the current of the circuit, equals zero there is no power being supplied to the engine
and therefore there is nothing to observe. We have ωr = x2
x1 , and x3 = TlI. The above
equation then becomes
x2 ˙ = − 1
J x3 − B
J x2 + Laf
J x31 = α3(u)x3 + b2(x1,x2,u)
59
− Laf
L
x 2 2
x1
u(t) x2 R
+ L − x1 L x2
(4.3)