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
In this chapter, we focus now on the implementation of the adaptive high-gain extended
Kalman filter that was introduced in Chapter 3. We provide a full definition of the observer,
in which an adaptation function is explicitly given. A methodology is advanced for tuning
the several parameters. Our goal is to demonstrate that the adaptive high-gain extended
Kalman filter can be used in practice even in the case of a relatively fast process, e.g. 100
Hz.
The process we consider is a series-connected DC motor, modeled via a nonlinear SISO 1
system when current and voltage are the only observables. This process has been used in
previous studies, allowing us to compare our results here with those from the earlier works (see
[87, 93]). Moreover, the machine itself is readily available and experiments can be considered
quite realistic. Although the process is quite simple, the implementation of the observer in a
real-time environment raises interesting questions that a simulation does not.
The modeling itself of the process and the observability study are investigated in Section
4.1. The implementation of the process in a simulation is the subject of Section 4.2. The
methodology for the tuning the parameters is also developed in this section. Finally, a set of
real experiments performed using an actual machine using a hard real-time operating system
is detailed in Section 4.3.
4.1 Modeling of the Series-connected DC Machine and Observability
Normal Form
Basically, an electric motor converts electrical energy into mechanical energy. In a DC
motor, the stator (also denoted field) is composed of an electromagnet, or a permanent
magnet, that immerses the rotor in a magnetic field. The rotor (also denoted armature) is
made of an electromagnet that once supplied with current creates a second magnetic field. The
stator is kept fixed while the rotor is allowed to move — i.e., rotate. The attraction/repelling
behavior of magnets generates the rotative motion.
In order to make the rotative motion permanent, one of the two magnetic fields has
to be switched at appropriate moments. The magnetic field created by the stator remains
fixed. The rotor windings are connected to a commutator causing the direction of the current
flowing through the armature coils to switch during the rotation. This reverses the polarity
of the armature magnetic field. Successive commutations then maintain the rotating motion
of the machine.
A DC motor whose field circuit and armature circuit are connected in series, and therefore
fed by the same power supply, is referred to as a series-connected DC motor [77].
4.1.1 Mathematical Model
The model of the series-connected DC motor is obtained from the equivalent circuit representation
shown in Figure 4.1. We denote If as the current flowing through the field part
of the circuit (between points A and C), and Ia as the current flowing through the armature
circuit (between points C and B). When the shaft of the motor is turned by an external force,
the motor acts as a generator and produces an electromotive force. In the case of the DC
1 Single input single output.
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