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
motor, this force will act against the current applied to the circuit and is then denoted back
or counter electromotive force (BEMF or CEMF). The electrical balance leads to
for the field circuit, and to
for the armature circuit. The notations are:
Lf ˙
If + Rf If = VAC
La ˙
Ia + RaIa = VCB − E
− Lf and Rf for the inductance and the resistance of the field circuit,
− La and Ra for the inductance and the resistance of the armature circuit,
− E for the Back EMF.
Kirchoff’s laws give us the relations:
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The total electrical balance is
I = Ia = If
V = VAC + VCB.
L ˙
I + RI = V − E,
where L = Lf + La and R = Rf + Ra. The field flux is denoted byΦ . We have Φ= f(If )=
f(I), and E = KmΦωr where Km is a constant and ωr is the rotational speed of the shaft.
The second equation of the model is given by the mechanical balance of the shaft of the
motor using Newton’s second law of motion. We consider that the only forces applied to the
shaft are the electromechanical torque Te, the viscous friction torque and the load torque Tl
leading to
J ˙
ωr = Te − Bωr − Tl
where J denotes the rotor inertia, and B the viscous friction coefficient. The electromechanical
torque is given by Te = KeΦI with Ke denoting a constant parameter. We consider that
the motor is operated below saturation [93]. In this case, the field flux can be expressed
by the linear expression Φ= Laf I, where Laf denotes the mutual inductance between the
field and the rotating armature coils. To conclude with the modeling of the DC, motor we
impose the ideal hypothesis of 100% efficiency of conservation of energy, which is expressed
as K = Km = Ke. For simplicity in the notation, we write Laf instead of KLaf . The voltage
Rf
Lf
La
Figure 4.1: Series-connected DC Motor: equivalent circuit representation.
58
Ra
C
A
+
- B