Direct Boolean Integer and Fractional Order SMC of Switching Systems: Application to a DC-DC Buck Converter ?

In this paper, a strategy based on the Sliding Mode Control(SMC) is used to achieve the Boolean input for DC-DC buck converter. Three di¤erent surfaces are proposed in order to design integer order and fractional order controller. Unlike conventional methods, sliding mode controllers designed through this methodology directly produce Boolean control actions, avoiding the usability of pulse-width modulation (PWM) generally used to control power converters. Simulations are carried out using Matlab/SIMULINK and the results show the e¢ ciency of the proposed method to control DC-DC buck converter. Keywords: Fractional order controller; switched system; DC-DC buck converter; Sliding Mode Control; Puls width modulation. 1. INTRODUCTION Switched systems have gained an increasing place in industrial applications since the middle of the 20th century, especially in the …eld of power electronics where static converters are used extensively. For that reason, they have been acknowledged as an object of research for several decades. But their study has recently received growing attention from control researchers. Indeed, beyond their industrial interest, such systems are basically characterized by a discontinuous dynamic behavior, which makes them particularly attractive to the emergent hybrid community. From a physical point of view, switched systems are de…ned as continuous time systems including some components that evolve much faster than the time scale at which their global behavior needs to be analyzed (Richard et al. (2006)). Dynamic Models of DC-DC converters are in the class of switching systems. Applying digital methods to the control of power converters, in particular board-mounted DC-DC converters, o¤ers a rich set of possibilities from which to create new features, improved performance, and much greater product ‡exibility, and all at lower cost. DC-DC power converters are employed in a variety of applications, including power supplies for personal computers, o¢ ce equipment, spacecraft power systems, laptop computers, and telecommunications equipment, as well as DC motor drives. The input to a DC-DC converter is an unregulated DC voltage Vg. The converter produces a regulated output voltage vc, having a magnitude (and possibly polarity) that di¤ers from Vg. ? This work has been partially supported by the Research Grant TRA2008-06602-C03-02 from the Spanish Ministry of Science and Innovation. In recent years, numerous studies and applications of fractional-order systems in many areas of science and engineering have been presented (Podlubny (1999); Hilfer (2001)). Fractional calculus as old as the ordinary di¤erential calculus goes back to times when Leibniz and Newton invented di¤erential calculus. The problem raised by Leibniz in a letter dated September 30, 1695 for a fractional derivative has become an ongoing topic for more than hundreds of years. Emerging of e¤ective methods in differentiation and integration of non-integer order equations makes fractional-order systems more and more attractive for the control systems. Fractional order controllers have been investigated in many papers e.g. an analytical robust stability checking method of fractional-order linear time invariant interval uncertain system (Ahn et al. (2007)), two sets of tuning rules for fractional PIDs (Valério and Costa (2006)), a method for controlling main irrigation canals with variable dynamical parameters based on robust fractional order controllers (Feliu et al. (2007)), and several alternative methods for the control of power electronic buck converters applying fractional order control (FOC) (Calderón et al. (2006)). Recently, a work has been presented in (Richard et al. (2006)) where the authors proposed a switching control action using the SMC and Boolean law which avoid the usability of pulse-width modulation (PWM). In present paper, we propose a controller based on SMC as, u = 1 2 (1 sgn(S)) (1) where S is the surface. The stability will be proved using three surfaces, i.e. integer order PD, PID and fractional order PID. The rest of paper is organized as follows. Basic de…nitions and preliminaries of fractional order are brie‡y discussed in section 2. Mathematical model of DC-DC buck converter Proceedings of FDA’10. The 4th IFAC Workshop Fractional Differentiation and its Applications. Badajoz, Spain, October 18-20, 2010 (Eds: I. Podlubny, B. M. Vinagre Jara, YQ. Chen, V. Feliu Batlle, I. Tejado Balsera). ISBN 9788055304878. Page 1 of 6 Article no. FDA10-132