Synchronization of the unified chaotic systems using a sliding mode controller

The unified chaotic system incorporates the behaviors of the Lorenz, the Chen and the Lü chaotic systems. This paper deals with the synchronization of two identical unified chaotic systems where the slave system is assumed to have a single input. A sliding mode controller is proposed to synchronize the two systems. The asymptotic convergence to zero of the errors between the states of the master and the slave systems is shown. Simulations results are presented to illustrate the proposed controller; they indicate that the designed controller is able to synchronize the unified chaotic systems. Also, simulation results show that the proposed control scheme is robust to random bounded disturbances acting on the master system. Moreover, the proposed scheme is applied to the secure communications field, where simulation results indicate that the proposed scheme is effective.     

Synchronization of chaotic systems with parametric uncertainty using active sliding mode control

This paper presents an active sliding mode control method for synchronizing two chaotic systems with parametric uncertainty. And a sufficient condition is drawn for the robust stability of the error dynamics, and is applied to guiding the design of the controllers. Finally, numerical results are used to show the robustness and effectiveness of the proposed control strategy.     

Synchronization in reduced-order of chaotic systems via control approaches based on high-order sliding-mode observer

The reduced-order synchronization problem of two chaotic systems (master–slave) with different dimension and relative degree is considered. A control scheme based on a high-order sliding-mode observer-identifier and a feedback state controller is proposed, where the trajectories of slave can be synchronized with a canonical projection of the master. Thus, the reduced-order synchronization is achieved in spite of master/slave mismatches. Simulation results are provided in order to illustrate the performance of the proposed synchronization scheme.     

Exact differentiation and sliding mode observers for switched Lagrangian systems

The main topic of this paper is the problem of constructing observers for switched mechanical systems, which includes, as a specific case, the design of observers based on the high order sliding mode technique. The high order sliding mode is used to overcome the chattering phenomena occurring, which induce some irrelevant and undesirable phenomena for mechanical systems. The proposed approach, based on the Fliess canonical form, also allows observers to give an estimate of the discrete location of the system, which indicates the dynamic evolution. The convergence of the observers is proved and a stick–mass–friction system is used to illustrate the efficiency of the proposed hybrid observers.     

Fast synchronization of non-identical chaotic modulation-based secure systems using a modified sliding mode controller

In this paper, a secure communication scheme based on chaotic modulation is proposed using a reversible process and a robust controller with efficient cost and complexity to synchronize two different chaotic systems. In the controller design, a sliding mode control with an adaptive rule is used for non-linear inputs. The adaptive rule is applied to ensure the synchronization when uncertainties, non-modeled dynamics or external distortions are at work. The message signal is recovered at the receiver using a recursive process at the end. The effectiveness of the proposed algorithm is confirmed via the simulation results for the synchronization of the transmitted signal modulated by Chen chaotic system at the transmitter and Genesio chaotic system at the receiver, and those for the information recovery process.     

Finite-time synchronization of two different chaotic systems with unknown parameters via sliding mode technique

In this paper, the problem of finite-time chaos synchronization between two different chaotic systems with fully unknown parameters is investigated. First, a new nonsingular terminal sliding surface is introduced and its finite-time convergence to the zero equilibrium is proved. Then, appropriate adaptive laws are derived to tackle the unknown parameters of the systems. Afterwards, based on the adaptive laws and finite-time control idea, an adaptive sliding mode controller is proposed to ensure the occurrence of the sliding motion in a given finite time. It is mathematically proved that the introduced sliding mode technique has finite-time convergence and stability in both reaching and sliding mode phases. Finally, some numerical simulations are presented to demonstrate the applicability and effectiveness of the proposed technique.     

Robust synchronization of drive–response chaotic systems via adaptive sliding mode control

A robust adaptive sliding control scheme is developed in this study to achieve synchronization for two identical chaotic systems in the presence of uncertain system parameters, external disturbances and nonlinear control inputs. An adaptation algorithm is given based on the Lyapunov stability theory. Using this adaptation technique to estimate the upper-bounds of parameter variation and external disturbance uncertainties, an adaptive sliding mode controller is then constructed without requiring the bounds of parameter and disturbance uncertainties to be known in advance. It is proven that the proposed adaptive sliding mode controller can maintain the existence of sliding mode in finite time in uncertain chaotic systems. Finally, numerical simulations are presented to show the effectiveness of the proposed control scheme.     

Synchronizing different chaotic systems using active sliding mode control

An active sliding mode controller is designed to synchronize three pairs of different chaotic systems (Lorenz–Chen, Chen–Lü, and Lü–Lorenz) in drive–response structure. It is assumed that the system parameters are known. The closed loop error dynamics depend on the linear part of the response systems and parameters of the controller. Therefore, the synchronization rate can be adjusted through these parameters. Analysis of the stability for the proposed method is derived based on the Lyapunov stability theorem. Finally, numerical results are presented to show the effectiveness of the proposed control technique.     

Synchronization of two different uncertain chaotic systems with unknown parameters using a robust adaptive sliding mode controller

In this paper, a robust adaptive sliding mode controller (RASMC) is proposed to realize chaos synchronization between two different chaotic systems with uncertainties, external disturbances and fully unknown parameters. It is assumed that both master and slave chaotic systems are perturbed by uncertainties, external disturbances and unknown parameters. The bounds of the uncertainties and external disturbances are assumed to be unknown in advance. Suitable update laws are designed to tackle the uncertainties, external disturbances and unknown parameters. For constructing the RASMC a simple sliding surface is first designed. Then, the RASMC is derived to guarantee the occurrence of the sliding motion. The robustness and stability of the proposed RASMC is proved using Lyapunov stability theory. Finally, the introduced RASMC is applied to achieve chaos synchronization between three different pairs of the chaotic systems (Lorenz–Chen, Chen–Lorenz, and Liu–Lorenz) in the presence of the uncertainties, external disturbances and unknown parameters. Some numerical simulations are given to demonstrate the robustness and efficiency of the proposed RASMC.     

A chaotic system in synchronization and secure communications

In this paper we deal with the synchronization and parameter estimations of an uncertain Rikitake system and its application in secure communications employing chaotic parameter modulation. The strategy consists of proposing a receiver system which tends to follow asymptotically the unknown Rikitake system, refereed as transmitter system. The gains of the receiver system are adjusted continually according to a convenient high order sliding-mode adaptative controller (HOSMAC), until the measurable output errors converge to zero. By using HOSMAC, synchronization between transmitter and receiver is achieved and message signals are recovered. The convergence analysis is carried out by using Barbalat’s Lemma.     

Control of a novel class of fractional-order chaotic systems via adaptive sliding mode control approach

In this paper, an adaptive sliding mode controller for a novel class of fractional-order chaotic systems with uncertainty and external disturbance is proposed to realize chaos control. The bounds of the uncertainty and external disturbance are assumed to be unknown. Appropriate adaptive laws are designed to tackle the uncertainty and external disturbance. In the adaptive sliding mode control (ASMC) strategy, fractional-order derivative is introduced to obtain a novel sliding surface. The adaptive sliding mode controller is shown to guarantee asymptotical stability of the considered fractional-order chaotic systems in the presence of uncertainty and external disturbance. Some numerical simulations demonstrate the effectiveness of the proposed ASMC scheme.     

Design of sliding mode controller for a class of fractional-order chaotic systems

In this paper, a sliding mode control law is designed to control chaos in a class of fractional-order chaotic systems. A class of unknown fractional-order systems is introduced. Based on the sliding mode control method, the states of the fractional-order system have been stabled, even if the system with uncertainty is in the presence of external disturbance. In addition, chaos control is implemented in the fractional-order Chen system, the fractional-order Lorenz system, and the same to the fractional-order financial system by utilizing this method. Effectiveness of the proposed control scheme is illustrated through numerical simulations.     

Full-order and reduced-order observers for one-sided Lipschitz nonlinear systems using Riccati equations

This paper aims to design full-order and reduced-order observers for one-sided Lipschitz nonlinear systems. The system under consideration is an extension of its known Lipschitz counterpart and possesses inherent advantages with respect to conservativeness. For such system, we first develop a novel Riccati equation approach to design a full-order observer, for which rigorous mathematical analysis is performed. Consequently, we show that the conditions under which a full-order observer exists also guarantee the existence of a reduced-order observer. A design method for the reduced-order observer that is dependent on the solution of the Riccati equation is then presented. The proposed conditions are easily and numerically tractable via standard numerical software. Furthermore, it is theoretically proven that the obtained conditions are less conservative than some existing ones in recent literature. The effectiveness of the proposed observers is illustrated via a simulative example.     

Synchronization of non-identical chaotic delayed fuzzy cellular neural networks based on sliding mode control

-In this paper, we investigate the synchronization problems of chaotic fuzzy cellular neural networks with time-varying delays. To overcome the difficulty that complete synchronization between non-identical chaotic neural networks cannot be achieved only by utilizing output feedback control, we use a sliding mode control approach to study the synchronization of non-identical chaotic fuzzy cellular neural networks with time-varying delays, where the parameters and activation functions are mismatched. This research demonstrates the effectiveness of application in secure communication. Numerical simulations are carried out to illustrate the main results.     

Adaptive fuzzy sliding mode control for synchronization of uncertain fractional order chaotic systems

This paper deals with chaos synchronization between two different uncertain fractional order chaotic systems based on adaptive fuzzy sliding mode control (AFSMC). With the definition of fractional derivatives and integrals, a fuzzy Lyapunov synthesis approach is proposed to tune free parameters of the adaptive fuzzy controller on line by output feedback control law and adaptive law. Moreover, chattering phenomena in the control efforts can be reduced. The sliding mode design procedure not only guarantees the stability and robustness of the proposed AFSMC, but also the external disturbance on the synchronization error can be attenuated. The simulation example is included to confirm validity and synchronization performance of the advocated design methodology.     

Robust stabilization and synchronization of a class of fractional-order chaotic systems via a novel fractional sliding mode controller

This paper proposes a novel fractional-order sliding mode approach for stabilization and synchronization of a class of fractional-order chaotic systems. Based on the fractional calculus a stable integral type fractional-order sliding surface is introduced. Using the fractional Lyapunov stability theorem, a single sliding mode control law is proposed to ensure the existence of the sliding motion in finite time. The proposed control scheme is applied to stabilize/synchronize a class of fractional-order chaotic systems in the presence of model uncertainties and external disturbances. Some numerical simulations are performed to confirm the theoretical results of the paper. It is worth noticing that the proposed fractional-order sliding mode controller can be applied to control a broad range of fractional-order dynamical systems.     

Synchronization of the unified chaotic system and application in secure communication

This paper studies the synchronization of the unified chaotic system via optimal linear feedback control and the potential use of chaos in cryptography, through the presentation of a chaos-based algorithm for encryption.     

Synchronization of non-autonomous chaotic systems with time-varying delay via delayed feedback control

In this paper, we investigate the synchronization of non-autonomous chaotic systems with time-varying delay via delayed feedback control. Using a combination of Riccati differential equation approach, Lyapunov-Krasovskii functional, inequality techniques, some sufficient conditions for exponentially stability of the error system are formulated in form of a solution to the standard Riccati differential equation. The designed controller ensures that the synchronization of non-autonomous chaotic systems are proposed via delayed feedback control and intermittent linear state delayed feedback control. Numerical simulations are presented to illustrate the effectiveness of these synchronization criteria.     

Synchronization of two coupled systems of J-J type using active sliding mode control

This paper first presents a rigorous mathematical analysis for the active sliding mode control method proposed recently by Haeri and Emadzadeh. Second, the technique is applied to achieve synchronization between two coupled systems of J-J. Numerical simulations are used to verify the above analytical results.     

Observer-based synchronization of uncertain chaotic system and its application to secure communications

Within the drive-response configuration, this paper considers the synchronization of uncertain chaotic systems based on observers and chaos-based secure communication. Even if there are unknown disturbances and parameters in the drive system, a robust adaptive observer can be used as response system to realize chaotic synchronization. The proposed method is then applied to secure communication. The transmitter is constructed by injecting the information into the drive system with proper manner and one of the transmitting signal is the sum of one of the output and the information signal. The Lur’e chaotic system is considered as an illustrative example to demonstrate the effectiveness of the proposed approaches.