Determination of the Switching Times of the Bang-bang Control for a Linear Stepping Motor by Lyapunov Functions
Kamel Ben SAAD, Ahlem MBAREK, Mohamed BENREJEB
Unité de recherche LARA automatique, Ecole Nationale d’Ingénieurs de Tunis
BP. 37 le Belvédère, 1002, Tunis, Tunisia
Abstract: The linear tubular switched reluctance stepping motor is widely used in applications requiring high-precision linear positioning operations. But, the position evolution of this kind of stepping motors is oscillatory in the case of an open-loop control. In order to damp the position response of such motor, the bang-bang control is adopted. This classical control is simple to implement in practice. However, its successful application is dependent on the suitable determination of the switching instants for which there are no systematic methods allowing their determination. In this paper, a study allowing the determination of the switching instants of the bang-bang control is presented. This study is based on a stability analysis performed by Lyapunov’s direct method which is suitable for the studied nonlinear system. The validity of the proposed method is proven by numerical simulations and in practice by the use of a test bench.
Keywords: Linear Tubular Switched Reluctance Stepping Motor, overshoot, Lyapunov fuction, Bang-Bang control
Kamel Ben Saad was born in 1976 in Tunis, Tunisia. He graduated in 2000 from “Ecole Nationale d’Ingénieurs de Tunis” (ENIT). He also received his master degree from ENIT in 2001 and his PhD degree in Electrical Engineering from “Ecole Centrale de Lille” and ENIT in 2005. He is currently Assistant Professor at the ENIT. His research interests are in the area of machines, classical and intelligent control.
Mbarek Ahlem was born in 1980 in Nabeul, Tunisia. She graduated in 2005 from Ecole Nationale d’Ingénieurs de Monastir. She received her master degree in Electrical Engineering from “Ecole Nationale d’Ingénieurs de Tunis” (ENIT) in 2007. She is currently a PhD student in ENIT. Her research interests are in the area of stepping motors control.
Mohamed Benrejeb was born in Tunisia in 1950. He obtained the Diploma of “Ingénieur IDN” in 1973, The Master degree of Automatic Control in 1974, the PhD in Automatic Control of the University of Lille in 1976 and the DSc of the same University in 1980. Full Professor at “Ecole Nationale d’Ingénieurs de Tunis” since 1985 and at “Ecole Centrale de Lille” since 2003. His research interests are in the area of analysis and synthesis of complex systems based on classical and non conventional approaches.
CITE THIS PAPER AS:
Kamel Ben SAAD, Ahlem MBAREK, Mohamed BENREJEB, Determination of the Switching Times of the Bang-bang Control for a Linear Stepping Motor by Lyapunov Functions, Studies in Informatics and Control, ISSN 1220-1766, vol. 17 (4), pp. 393-406, 2008.
The Linear Stepping Motor divides linear distances into discrete incremental movements called steps. The major advantage of this kind of motor, in comparison with a rotary stepping motor, is that the huge part of the supplied mechanical power is converted directly to linear motion. Indeed, the rotary stepping motor needs the use of rotary to linear conversion devices such as leadscrews which introduce additive mechanical loses [1-3]. The predominant inertia is the leadscrew rather than the load to be moved. Thus, most of the motor torque goes to accelerate the leadscrew. So, the linear stepping motor is considered to be the most economical linear motor positioning solution [1-3].
The studied actuator is a Linear Tubular Switched Reluctance Stepping Motor (LTSRSM) characterized by four phases. Such motor represents the linear counterpart of the rotary variable-reluctance stepper motor. Its cylindrical mechanical geometry allows simplicity of construction. The LTSRSM is simple to use and it is characterized by its precision of positioning in open-loop operation. Moreover, it offers solutions to a variety of applications requiring high linear positioning precision. Nowadays, they are widely used in automobile industry, machine tools, robotics domains and electronic industry .
As all stepping motors, the step response of the LTSRSM is generally very oscillatory. In application requiring frequent accurate positioning this poorly damped response can be a great disadvantage [5-6]. For example, if the studied actuator is used to insert pins in plastic packages of some electronic components, the oscillations can induce the destruction of the pins or a bad insertion.
There are several solutions allowing the elimination of such oscillations. We focused in this work on the bang-bang control solution which is an open-loop control technique. Such control is easy to implement. However, the determination of the switching instants constitutes the major difficulty in its application. Generally, the switching instants are determined experimentally. Such an approach is very difficult to perform.
In this paper, a method allowing a theoretical determination of such switching instants is presented. It is based on a stability study of the LTSRSM by application of Lyapunov’s direct method. The verification of the proposed approach is carried by numerical simulations and experimentally.
This paper is organized as follows: in section 2 the studied stepping motor structure and mathematical model are presented. The problem statement is introduced in section 3. Section 4 presents the proposed method for the motor oscillations suppression. Finally, the simulation results and the experimental results are given in section 5.
The LTSRSM position evolutions present some oscillations which harm some application requiring high positioning quality and resolution. In this paper a method allowing the determination of the switching instants of the bang-bang control is presented. The bang-bang control is classical open-loop control solution allowing the attenuation of the stepping motors oscillations. It is difficult to determine analytically the two bang-bang switching instants because of the complexity of the nonlinear dynamical model of the studied LTSRSM. Also, the experimental determination of these two parameters is a tedious approach. In this work the Lyapunov’s direct method is introduced to study stability of the LTSRSM. So, a Lyapunov candidate function is proposed to estimate the stability. The proposed method exploits the energy interpretation of Lyapunov’s direct method to approximate the switching instants. The simulations and the experimental results prove the efficiency of the proposed approach. The experimental tests are performed on a test bench. Thanks to the good approximation of the switching instants, the experimental implementation of the bang-bang control is very easy.
Fm: electromagnetic force
F0: dry friction force
Fc: resistant force
v : linear speed
x : plunger position
m : plunger weight
: dynamic viscosity coefficient
R : statoric resistance
Un: nominal voltage
Ui: voltage applied to phases i=A,B,C and D
ij: CURRENT OF THE PHASES j=A,B,C AND D
In: nominal current
L0: average phase inductance
L1: amplitude of the inductance variation
: fluxes density
: four step length
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