IEEE/CAA JOURNAL OF AUTOMATICA SINICA 1 Fractional-Order Control for a Novel Chaotic System without Equilibrium Shu-Yi Shao and Mou Chen Member, IEEE, Abstract—The control problem is discussed for a chaotic sys- portant results have been reported. In the early 1990s, the tem without equilibrium in this paper. On the basis of the linear synchronization of chaotic systems was achieved by Pecora mathematical model of the two-wheeled self-balancing robot, a and Carroll[14;15], which was a trailblazing result, and the novel chaotic system which has no equilibrium is proposed. The basic dynamical properties of this new system are studied via result promoted the development of chaos control and chaos [16;17] Lyapunov exponents and Poincare´ map. To further demonstrate synchronization . In recent years, different chaos control the physical realizability of the presented novel chaotic system, a and chaos synchronization strategies have been developed chaotic circuit is designed. By using fractional-order operators, for chaotic systems. The sliding mode control method was a controller is designed based on the state-feedback method. applied to chaos control[18;19] and chaos synchronization[20]. According to the Gronwall inequality, Laplace transform and Mittag-Leffler function, a new control scheme is explored for the In [21], the feedback control method and the adaptive control whole closed-loop system. Under the developed control scheme, method were used to realize chaos control for the energy the state variables of the closed-loop system are controlled to resource chaotic system. The chaos control problems were stabilize them to zero. Finally, the numerical simulation results investigated for Lorenz system, Chen system and Lu¨ system of the chaotic system with equilibrium and without equilibrium based on backstepping design method in [22]. By using illustrate the effectiveness of the proposed control scheme. adaptive control method, the problems of chaos control[23;24] Index Terms—Chaotic system, Circuit implementation, and chaotic synchronization[25] were studied for chaotic sys- Fractional-order, Stabilization. tems. The neural adaptive control method was developed for a class of chaotic systems with uncertain dynamics, input I. INTRODUCTION and output constraints in [26]. In [27], on the basis of ROM From the simplified equation of convection roll- impulsive control method, the problems of the stabilization F s in the equations of the atmosphere, the first three- and synchronization were explored for Lorenz systems. The dimensional chaotic system was derived by Lorenz in 1963[1]. synchronization problem was resolved for a class of chaotic With the development and applying of chaos theory, a number systems by using a fractional-order observer-based method and of chaotic systems, hyperchaotic systems, fractional-order the synchronization was applied to secure communication in chaotic systems and fractional-order hyperchaotic systems [28]. In [29], the synchronization was studied for fractional- have been proposed, such as Rossler¨ chaotic system[2], Liu order systems based on the output feedback sliding mode chaotic system[3], hyperchaotic Chen system[4], hyperchaotic control method. A new synchronization strategy was presented Lu¨ system[5], fractional-order financial system[6], fractional- for two fractional-order systems and the synchronization was order Lotka-Volterra equation[7], fractional-order hyperchaos applied in image encryption in [30]. The above literature works Lorenz system[8], a modified four-dimensional fractional order focused on chaos control and chaos synchronization in prac- hyperchaotic system[9] and so on. The above mentioned chaot- tical chaotic systems with equilibrium points. However, the ic systems have equilibrium. In addition, there are a number of control of chaotic systems without equilibrium has rarely been [13] chaotic systems without equilibrium which have been studied investigated . Meanwhile, for most of the above mentioned by [10-13]. As a result, chaos control became one of the works, fractional-order controllers have rarely been used to important issues for chaotic systems. Due to great potential realize the chaos control of integer-order chaotic systems, application in electrical engineering, information processing although some important results on the fractional-order con- − and secure communication, it is important to investigate new trollers have been proposed for various systems[31 33]. In [31], control methods for chaotic systems. a well-known fractional-order controller was presented. In Over the past few decades, chaos control and chaos syn- [32], the concept of a fractional-order PIλDµ controller was chronization have received much attention and many im- proposed and the fractional-order controller included fractional order integrator and fractional-order differentiator. In [33], on This article has been accepted for publication in a future issue of this the basis of the Lyapunov stability theory, a novel fractional- journal, but has not been fully edited. Content may change prior to final publication. order controller was given, and fractional-order chaotic and hy- This work is supported by National Natural Science Foundation of China perchaotic systems were controlled by the proposed fractional- (61573184), Jiangsu Natural Science Foundation of China (SBK20130033), order controller. The fractional-order controllers are effective Six Talents Peak Project of Jiangsu Province (2012-XXRJ-010) and Funda- mental Research Funds for the Central Universities (NE2016101). Recom- to control systems, which have been proved in the mentioned mended by Associate Editor Antonio Visioli. works. Therefore, it is valuable to further explore the chaos Shu-Yi Shao and Mou Chen are with the College of Automation Engineer- control of integer-order chaotic systems without equilibrium ing, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, China (e-mail: [email protected], [email protected]). by using fractional-order controllers. Digital Object Identifier 10.1109/JAS.2016.7510124 Inspired by the above discussions, the objective of this paper 2 IEEE/CAA JOURNAL OF AUTOMATICA SINICA is to design an efficient fractional-order controller and the control input u, (1) can be described as stability is realized for the closed-loop system. A novel chaotic 2 3 2 3 2 3 x_ 0 1 0 0 x system without equilibrium is proposed based on the model of r r 6 x¨ 7 6 0 0 −23:6701 0 7 6 x_ 7 two-wheeled self-balancing robot. Meanwhile, the presented 6 r 7 = 6 7 6 r 7 4 θ_ 5 4 0 0 0 1 5 4 θ 5 new system is used to verify the effectiveness of the proposed p p θ¨ 0 0 124:5128 0 θ_ control scheme. p 2 3 p 0 The organization of the paper is as follows. Section II 6 7 details the problem formulation. A novel chaotic system is 6 4:5974 7 _ ¨ + 4 5 Φ(xr; x_ r; θp; θp) + u (2) proposed and the chaotic system circuit is designed in Section 0 III. Section IV presents the fractional-order controller based on −19:0414 the state-feedback method. The numerical simulation studies T where the control input u = [u1; u2; u3; u4] . are presented to demonstrate the effectiveness of the developed This paper aims at constructing a novel chaotic system control method in Section V, followed by some concluding without equilibrium and developing a fractional-order control remarks in Section VI. scheme, so that the stabilization of the whole closed-loop sys- tem is realized based on the designed control strategy. Under designed fractional-order controller, the state variables of the II. PROBLEM STATEMENT AND PRELIMINARIES closed-loop system will be asymptotically stable. To develop In this paper, a novel chaotic system will be proposed by the fractional-order control scheme, we firstly introduce the following definitions and lemmas: only considering the straight line position xr and the pitch Definition 1[35]. The Caputo fractional derivative operator, angle θp of the two-wheeled self-balancing robot of Googol Technology as shown in Fig. 1. A mathematical model related which is one of the most widely used fractional derivative to the two-wheeled self-balancing robot of Googol Technology operators, is defined for the function f(t) as follows : Z was established in [34]. The linear mathematical model for t (m) α 1 f (τ) x and θ of the two-wheeled self-balancing robot of Googol D f(t) = − dτ; (3) r p Γ(m − α) (t − τ)α m+1 Technology is described in the form t0 2 3 2 3 2 3 where α is the fractional order and m − 1 < α < m, m = x_ r 0 1 0 0 xr [α] + 1, [α] denotes the integer part of α, and Γ(·) is gamma R 1 6 x¨ 7 6 − 7 6 7 − m−α−1 −t 6 r 7 6 0 0 23:6701 0 7 6 x_ r 7 function, which is defined as Γ(m α) = 0 t e dt. 4 _ 5 = 4 5 4 5 θp 0 0 0 1 θp The main advantage of (3) is that Caputo derivative of a ¨ _ ≤ θp 0 0 124:5128 0 θp constant is equal to zero. Particularly, when 0 < α 1, 2 3 L f α g α − α−1 0 we have D f(t) = s F (s) s f(0): The Laplace 6 7 transform of fractional integral at t = 0 has the following 6 4:5974 7 0 + 4 5 Cθ (1) form: 0 { } −19:0414 L D−αf(t) = s−αL ff(t)g = s−αF (s); (α > 0); (4) where Cθ denotes the pitch torque. where t and s are the variables in the time domain and Laplace domain, respectively. F (s) = L(f(t)) and L(·) stands for the Laplace transform. In this paper, the fractional-order controller will be de- scribed by using Caputo definition with lower limit of integral t0 = 0 and the order 1 < α < 2.