Controlling Chaotic Behavior of the Stepper Motor Using Genetic Algorithms
In this paper we suggest a method to control the chaotic behavior of the stepper motor into a periodic one. Using the supply frequency as a bifurcation parameter we show that as the frequency is increased beyond a critical value the motor steps become irr
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Controlling Chaotic Behavior of the Stepper Motor Using Genetic Algorithms Yosra Miladi, Hanene Medhaffar, Moez Feki, and Nabil Derbel
17.1 Introduction The hybrid two-phased stepper motor is a common electromechanical converter widely used in robotic field and small devices positioning systems such as disk drives and X-ray scanning equipments. Originally, stepper motors were designed to provide precise discrete positioning in an open-loop control mode. However, it has been shown in (Robert et al. 2000) that using the stepper motor in an open loop configuration gives poor performance if it is driven using higher stepping rates than advised by the constructor. Indeed, authors have shown that quasiperiodic as well as chaotic behaviors appear as the power supply frequency is increased and this is due to incompatibility between the motor inertia and the driving speed. Therefore, controlling the chaotic behavior of the stepper motor becomes a worthwhile endeavor. The interest in controlling chaotic systems has revived after the pioneering work of (Ott et al. 1990). Since then, several strategies to control chaos have been developed (Pyragas 1992). Several controlling methods are considered worthless since they simply consider chaotic systems as nonlinear systems and their aim is to stabilize the equilibrium points. Some other methods are worthier since they concern stabilizing the unstable periodic orbits of chaotic systems such as in power converters (El Aroudi et al. 2009; Kaoubaâ et al. 2010; Robert and Feki 2011). A widely considered controlling strategy is Pyragas’ method which consists in adding a time delayed input signal to the chaotic system to attempt to stabilize an unstable periodic orbit. This strategy has been approached using linear control as well as nonlinear control methods (Fourati et al. 2010; Postlethwaite and Mary
Y. Miladi • H. Medhaffar • M. Feki () • N. Derbel Research group CEMLab, National Engineering School of Sfax, University of Sfax, BP 1173, Sfax 3038, Tunisia e-mail: [email protected] S. Banerjee and S. ¸ S. ¸ Erçetin (eds.), Chaos, Complexity and Leadership 2012, Springer Proceedings in Complexity, DOI 10.1007/978-94-007-7362-2__17, © Springer ScienceCBusiness Media Dordrecht 2014
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2007). In (Robert et al. 2006) the time delayed control method was used in conjunction with the absolute stability theory to extend the stability of the stepper periodic behavior for higher values of the power supply frequency. In this work, we design a new controlling method to extend the operating domain of the stepper motor to frequencies larger than advised by the constructor. Our approach consists in using a genetic algorithm to calculate the optimal switching instances to excite the stepper to advance one step forward in a periodic manner while respecting the predetermined angular velocity. This paper will be organized as follows, in Sect. 17.2 we present the dynamical model of the stepper motor. In Sect. 17.3, we introduce basics of the genetic algorithms and we expla
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