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Active Sliding Mode for Synchronization of a Wide Class of Four-Dimensional Fractional-Order Chaotic Systems  [PDF]
Bin Wang,Yuangui Zhou,Jianyi Xue,Delan Zhu
ISRN Applied Mathematics , 2014, DOI: 10.1155/2014/472371
Abstract: We focus on the synchronization of a wide class of four-dimensional (4-D) chaotic systems. Firstly, based on the stability theory in fractional-order calculus and sliding mode control, a new method is derived to make the synchronization of a wide class of fractional-order chaotic systems. Furthermore, the method guarantees the synchronization between an integer-order system and a fraction-order system and the synchronization between two fractional-order chaotic systems with different orders. Finally, three examples are presented to illustrate the effectiveness of the proposed scheme and simulation results are given to demonstrate the effectiveness of the proposed method. 1. Introduction Chaos synchronization is the concept of closeness of the frequencies between different periodic oscillations generated by two chaotic systems, one of which is the master and the other is the slave. Since the pioneering work of Pecora and Carroll [1] who proposed a method to synchronize two identical chaotic systems, chaos synchronization has attracted a lot of attention in a variety of research fields over the last two decades. This is because chaos synchronization can be used in many areas such as physics, engineering, and particularly in secure communication [2–5]. Many methods have been proposed to synchronize chaotic systems including active control [6], back-stepping control [7], linear feedback control [8], adaptive control theory [9], sliding mode control [10, 11], and fuzzy control [12]. For example, Bhalekar and Daftardar-Gejji [13] used active control for the problem of synchronization of fractional-order Liu system with fractional-order Lorenz system. Based on the idea of tracking control and stability theory of fractional-order systems, Zhou and Ding [14] designed a controller to synchronize the fractional-order Lorenz chaotic system via fractional-order derivative. Zhang and Yang [15] dealt with the lag synchronization of fractional-order chaotic systems with uncertain parameters. Projective synchronization of a class of fractional-order hyperchaotic system with uncertain parameters was studied by Bai et al. [16] as well, but the derivative orders of the state in response system was the same with drive system. Chen et al. [17] designed a sliding mode controller for a class of fractional-order chaotic systems. However, most of the above-mentioned work on chaos synchronization has focused on fractional-order chaos and integer-order systems, respectively. To the best of our knowledge, there has been little information available about the synchronization between
Adaptive synchronization of fractional-order chaotic system via sliding mode control
基于滑模控制的分数阶混沌系统的自适应同步

Cao He-Fei,Zhang Ruo-Xun,
曹鹤飞
,张若洵

物理学报 , 2011,
Abstract: Based on sliding mode control theory and adaptive control theory, this paper investigates the synchronization of three-dimensional chaotic systems, designs a fractional order proportional integral switching surface, and proposes a single adaptive-feedback controller for fractional-order chaos synchronization. Simulation results for fractional-order unified chaotic system and Arneodo chaotic systems are provided to illustrate the effectiveness of the proposed scheme.
Sliding mode observers and observability singularity in chaotic synchronization  [PDF]
L. Boutat-Baddas,J. P. Barbot,D. Boutat,R. Tauleigne
Mathematical Problems in Engineering , 2004, DOI: 10.1155/s1024123x04309038
Abstract: We present a new secured data transmission based on a chaotic synchronization and observability singularity. For this, we adopt an approach based on an inclusion of the message in the system structure and we use a sliding mode observer for system with unknown input in order to recover the information. We end the paper with an example of chaotic system with an observability bifurcation. Moreover, this example highlights some benefits of the so-called step-by-step sliding mode observer.
Sliding mode observers and observability singularity in chaotic synchronization
Boutat-Baddas L.,Barbot J. P.,Boutat D.,Tauleigne R.
Mathematical Problems in Engineering , 2004,
Abstract: We present a new secured data transmission based on a chaotic synchronization and observability singularity. For this, we adopt an approach based on an inclusion of the message in the system structure and we use a sliding mode observer for system with unknown input in order to recover the information. We end the paper with an example of chaotic system with an observability bifurcation. Moreover, this example highlights some benefits of the so-called step-by-step sliding mode observer.
Sliding Controller Design of Hybrid Synchronization of Four-Wing Chaotic Systems
V. Sundarapandian,S. Sivaperumal
International Journal of Soft Computing , 2012, DOI: 10.3923/ijscomp.2011.224.231
Abstract: This study investigates the sliding controller design of hybrid synchronization of Four-Wing Chaotic Systems. In this study, researchers derive new results based on the Sliding Mode Control (SMC) for the hybrid synchronization of identical Qi 3D Four-Wing Chaotic Systems (2008) and identical Liu 3D Four-Wing Chaotic Systems (2009). The stability results for the hybrid synchronization schemes derived in this paper using SMC are established using the Lyapunov Stability theory. Since, the Lyapunov exponents are not required for these calculations, the sliding controller design is very effective and convenient to achieve global hybrid synchronization of the identical Qi Four-Wing Chaotic Systems and the identical Liu Four-Wing Chaotic Systems. Numerical simulations are presented to demonstrate the effectiveness of the synchronization results derived in this study.
ANTI-SYNCHRONIZATION OF ARNEODO CHAOTIC SYSTEMS BY SLIDING MODE CONTROL  [PDF]
Dr. V. SUNDARAPANDIAN,S. SIVAPERUMAL
International Journal of Engineering Science and Technology , 2011,
Abstract: In this paper, we derive new results based on the sliding mode control for the anti-synchronization of identical Arneodo chaotic systems (1980). The stability results for the sliding mode control based antisynchronization schemes derived in this paper are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is effective and convenient to achieve global chaos anti-synchronization of the identical Arneodo systems. Numericalsimulations are shown to illustrate the effectiveness of the sliding mode controller results derived in this paper for the identical Arneodo systems.
Synchronization of Unified Chaotic Systems Using Sliding Mode Controller  [PDF]
Yi-You Hou,Ben-Yi Liau,Hsin-Chieh Chen
Mathematical Problems in Engineering , 2012, DOI: 10.1155/2012/632712
Abstract: This paper presents a method for synchronizing the unified chaotic systems via a sliding mode controller (SMC). The unified chaotic system and problem formulation are described. Two identical unified chaotic systems can be synchronized using the SMC technique. The switching surface and its controller design are developed in detail. Simulation results show the feasibility of a chaotic secure communication system based on the synchronization of the Lorenz circuits via the proposed SMC. 1. Introduction Chaos theory is an extensively studied branch of the theory of nonlinear systems. Lorenz presented the first well-known chaotic system, which was a third-order autonomous system with only two multiplication-type quadratic terms but displayed very complex dynamical behaviors [1]. A chaotic system is a very complex, dynamical nonlinear system whose response has intrinsic characteristics such as broadband noise-like waveforms, difficult predictability, and sensitivity to initial condition variations [2, 3]. These properties are advantageous in secure communication systems. Therefore, the synchronization of chaotic circuits for secure communication has received a lot of research attention [4–7]. Studies have shown that it is possible to set up a chaotic communication system to obtain secure communication [8]. The synchronization between master (transmitter) and slave (receiver) chaotic systems has potential applications for secure communication [9–13]. Several control schemes have been developed for the synchronization of chaotic systems. Sliding mode control is a popular nonlinear control strategy [14–19]. For sliding mode controller (SMC) design, the Lyaponov stability method is applied to keep the nonlinear system under control. The sliding mode approach transforms a higher-order system into a lower-order system, allowing a simple control algorithm to be applied, making the system very straightforward and robust. Sliding mode control has been applied to the synchronization of chaotic systems [5, 6, 11]. The present study designs an SMC-based chaotic secure communication system. To achieve this goal, a proportional-integral (PI) switching surface is first designed for the considered error dynamics system in sliding motion, and then, based on it, a sliding mode controller is derived. This controller is effective and guarantees both the occurrence of sliding motion and synchronization of the master-slave unified chaotic systems. Finally, an example is given to illustrate the usefulness of the proposed SMC. 2. Problem Formulation and Main Results A sliding mode
具有纠缠项的分数阶五维混沌系统滑模同步的两种方法
Two methods for sliding mode synchronization of five-dimensional fractional-order chaotic systems with entanglement iterms
 [PDF]

王东晓
- , 2018, DOI: 10.6040/j.issn.1672-3961.0.2018.139
Abstract: 根据滑模和积分滑模两种方法研究具有3个纠缠项的分数阶五维混沌系统的滑模同步,给出滑模面和控制器的两种设计方法,得到纠缠混沌系统取得滑模同步的2个充分条件。研究表明:一定条件下,分数阶五维纠缠混沌系统取得滑模同步。通过数值仿真,验证了控制器的正确定性和有效性。
Sliding mode synchronization of five-dimensional fractional-order chaotic systems with three entanglement iterms was studied based on classical and integral sliding mode methods. Sliding mode surfaces and controllers were designed in two methods and two sufficient conditions were arrived for entanglement chaotic systems to acquire sliding mode synchronization. The research conclusion illustrated that five-dimensional fractional-order entanglement chaotic systems were sliding mode synchronization under certain conditions. Numerical simulation showed the correctness and the effectiveness of the designed controller.
Sliding Mode Control in Finite Time Stabilization for Synchronization of Chaotic Systems  [PDF]
Zhan-Shan Zhao,Jing Zhang,Lian-Kun Sun
ISRN Applied Mathematics , 2013, DOI: 10.1155/2013/320180
Abstract: An adaptive sliding mode control for chaotic systems synchronization is considered. The design of robust finite time convergent controller is based on geometric homogeneity and integral sliding mode manifold. The knowledge of the upper bound of the system uncertainties is not prior required. The chaos synchronization is presented to system stability based on the Lyapunov stability theory. The simulation results show the effectiveness of the proposed method. 1. Introduction Nowadays, chaos has been seen to have a lot of useful applications in many engineering systems such as secure communications, optics, power converters, chemical and biological systems, and neural networks [1–5]. Chaotic systems are dynamical systems and their response exhibits a lot of specific characteristics, including an excessive sensitivity to the initial conditions, fractal properties of the motion in phase space, and broad spectrums of Fourier transform. The main feature of chaotic systems is that a very small change in initial conditions leads to very large differences in the system states. Several other control methods have been successfully applied to chaotic motion control. For example, adaptive control [6, 7] presents chaos control of chaotic dynamical systems by using backstepping design method and so forth. Sliding mode control (SMC) is a popular robust control approach for its robustness against parameter variations and external disturbances under matching conditions of nonlinear systems operating under uncertainty conditions, as the controllers can be designed to compensate for the uncertainties or disturbances [8–15]. References [8–11] are concerned with sliding mode control of continuous-time switched stochastic systems. In practice, classic SMC suffers from high frequency chattering, as the infinite switching frequency required by ideal sliding mode is not achievable. Keeping the main advantages of the standard sliding mode control, the chattering effect is reduced and finite time convergence is provided. Interesting high-order sliding modes (HOSM) are proposed in [14–18] with the robustness of the system during the entire response. In order to reduce the chattering, the approach in [19] is modified and applied Synchronization in chaotic dynamic systems, so that a continuous feedback is produced combining the robustness of HOSM and finite-time stabilization by continuous control. The aim of the modified method is to deal with unknown but bounded system uncertainties. The upper bounds of uncertainties are not required to be known in advance. System stability is
Adaptive Sliding Mode Control of a Novel Class of Fractional Chaotic Systems  [PDF]
Jian Yuan,Bao Shi,Wenqiang Ji
Advances in Mathematical Physics , 2013, DOI: 10.1155/2013/576709
Abstract: Recently, control and synchronization of fractional chaotic systems have increasingly attracted much attention in the fractional control community. In this paper we introduce a novel class of fractional chaotic systems in the pseudo state space and propose an adaptive sliding mode control scheme to stabilize the chaotic systems in the presence of uncertainties and external disturbances whose bounds are unknown. To verify the effectiveness of the proposed adaptive sliding mode control technique, numerical simulations of control design of fractional Lorenz's system and Chen's system are presented. 1. Introduction Fractional calculus is an old and yet novel topic whose infancy dates back to the end of the 17th century, the time when Newton and Leibniz established the foundations of classical calculus. For three centuries, fractional calculus developed mainly as a pure theoretical mathematical field without applications. However, in the last two decades it has attracted the interest of researchers in several areas including mathematics, physics, chemistry, material, engineering, finance, and even social science. The stability of fractional differential equations (FDEs) and fractional control have both gained rapid development very recently [1–3]. One of the most important areas of application is the chaos theory. In recent years, fractional chaotic systems have intensively attracted a great deal of attention due to the ease of their electronics implementations and the rapid development of the stability of FDEs. More and more fractional dynamics described in the pseudo state space exhibiting chaos have been found, such as the fractional Chua circuit [4], the fractional Van der Pol oscillator [5–7], the fractional Lorenz system [8, 9], the fractional Chen system [10–12], the fractional Lü system [13], the fractional Liu system [14], the fractional R?ssler system [15, 16], the fractional Arneodo system [17], the fractional Newton-Leipnik system [18–20], the fractional Lotka-Volterra system [21, 22], the fractional finance system [20, 23], and the fractional Rucklidge system [24]. Most of the above papers have used numerical methods to present chaotic behaviors. In particular, control and synchronization of fractional chaotic systems have increasingly attracted much attention in the fractional control community. Moreover, several control and synchronization methods have been proposed based on the stability of fractional differential equations in the pseudo state space [22]. The linear state feedback control algorithm based upon the stability criterion of linear
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