4094 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 11, NOVEMBER 2008 Some Applications of Fractional Calculus in Suppression of Chaotic Oscillations Mohammad Saleh Tavazoei, Mohammad Haeri, Member, IEEE, Saeid Jafari, Sadegh Bolouki, and Milad Siami Abstract—This paper presents two different stabilization meth- ods based on the fractional-calculus theory. The first method is proposed via using the fractional differentiator, and the other is constructed based on using the fractional integrator. It has been shown that the proposed techniques can be used to suppress chaotic oscillations in 3-D chaotic systems. To show the practical capability of the methods, some experimental results on the control of chaos in chaotic circuits are presented. Index Terms—Chaos, chaotic circuit, fractional-order con- troller, stabilization. I. I NTRODUCTION F RACTIONAL calculus is a powerful mathematical tool with a long history, but its application to engineering has attracted much attention only in recent years [1]. Nowadays, some fractional-order controllers have been introduced and studied in practical applications. The tilt-integral derivative controller [2], the fractional PI λ D μ controller [3], [4], the CRONE controllers [5], [6], the fractional lead–lag compen- sator [7], optimal fractional controllers [8], and fractional adap- tive controllers [9] are some of the well-known fractional-order controllers introduced up to now. In addition, the efficiency of fractional-order controllers has been reported in some practical applications such as backlash vibration suppression control of torsional systems [10], reducing engine vibrations in automo- biles [11], position control of manipulators [12], control of main irrigation canals [13], speed control of electromechanical systems [14], lateral and longitudinal control of autonomous vehicles [15], control of power electronic buck converters [16], temperature control [17], control of robotic systems [18], flexible spacecraft attitude control [19], and other industrial applications [20]. Fractional-order controllers can be used to control both integer-order and fractional-order systems. However, since most of the available models for the controlled plants are in integer-order form, it is more common to design these controllers to control the integer-order systems in practical investigation [20]. Manuscript received February 4, 2008; revised May 5, 2008. First published May 20, 2008; current version published October 31, 2008. M. S. Tavazoei, S. Jafari, S. Bolouki, and M. Siami are with the Depart- ment of Electrical Engineering, Sharif University of Technology, Tehran, Iran (e-mail: m_tavazoei@ee.sharif.edu). M. Haeri is with the Advanced Control System Laboratory, Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran (e-mail: haeri@sina.sharif.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2008.925774 On the other hand, due to the importance of suppressing the undesirable behaviors such as chaotic oscillations in industrial applications [21], chaos control has gained increasing attention in the past few decades. Consequently, many control techniques for suppression of chaotic oscillation have been proposed by the control community (for more details, see [22] and references therein). The chaotic dynamics of fractional-order models have been studied in the literature (see [23, Ch. 7] and [24]). In addition, chaos control in chaotic fractional-order systems has been the subject of some previously published papers [25]–[27]. In this paper, some applications of fractional calculus in chaos control are proposed. Theoretical arguments and experimental results are given to show the efficiency of some simple fractional- calculus-based methods for suppressing of chaotic oscillations. This paper is organized as follows. Section II outlines some basic concepts in the fractional-calculus theory and the fractional-order systems. Section III introduces a fractional- differentiator-based stabilization method which can be used to control chaos in 3-D chaotic systems. Experimental results are also given in this section to show the efficiency of the proposed method. In Section IV, another fractional-calculus- based technique is introduced. This technique uses a fractional integrator and can be employed to control chaos. This section also contains some experimental results to verify the practical capability of the introduced method. A conclusion in Section V closes this paper. II. PRELIMINARY By extending the concept of integer-order integral and deriv- ative, the fractional integral and derivative have been defined. The definition of the fractional integral is an outgrowth of Cauchy formula for evaluating the integration. The αth or- der fractional integral of function f (t) with respect to t is defined by J α f (t)= 1 Γ(α) t  0 (t − τ ) α−1 f (τ )dτ. (1) In addition, there are some definitions for fractional derivatives such as Riemann–Liouville definition, Grunwald– Letnikov definition, and Caputo definition [28]. For example, the Riemann–Liouville fractional derivative of order α for function f (t) is defined by D α f (t)= d r dt r  J r−α f (t)  (2) 0278-0046/$25.00 © 2008 IEEE