Reduced-order synchronization of uncertain chaotic systems via adaptive control

We consider the coupling of two uncertain dynamical systems with different orders using an adaptive feedback linearization controller to achieve reduced-order synchronization between the two systems. Reduced-order synchronization is the problem of synchronization of a slave system with projection of a master system. The synchronization scheme is an exponential linearizing-like controller and a state/uncertainty estimator. As an illustrative example, we show that the dynamical evolution of a second-order driven oscillator can be synchronized with the canonical projection of a fourth-order chaotic system. Simulation results indicated that the proposed control scheme can significantly improve the synchronousness performance. These promising results justify the usefulness of the proposed output feedback controller in the application of secure communication.     

Robust adaptive neural-fuzzy-network control for the synchronization of uncertain chaotic systems

This paper proposes a robust adaptive neural-fuzzy-network control (RANFC) to address the problem of controlled synchronization of a class of uncertain chaotic systems. The proposed RANFC system is comprised of a four-layer neural-fuzzy-network (NFN) identifier and a supervisory controller. The NFN identifier is the principal controller utilized for online estimation of the compound uncertainties. The supervisory controller is used to attenuate the effects of the approximation error so that the perfect tracking and synchronization of chaotic systems are achieved. All the parameter learning algorithms are derived based on Lyapunov stability theorem to ensure network convergence as well as stable synchronization performance. Finally, simulation results are provided to verify the effectiveness and robustness of the proposed RANFC methodology.     

Exponential generalized synchronization of uncertain coupled chaotic systems by adaptive control

In this paper, the exponential generalized synchronization for a class of coupled systems with uncertainties is defined. A novel and powerful method is proposed to investigate the generalized synchronization based on the adaptive control technique. According to the Lyapunov stability theory, rigorous proof is given for the exponential stability of error system. In comparison with previous schemes, the presented method shortens the synchronization time and is more applicable in practice. Besides, it is shown that the synchronization effect is robust against the uncertain factors. Some typical chaotic and hyper-chaotic systems are taken as examples to illustrate above approach. The corresponding numerical simulations are demonstrated to verify the effectiveness of proposed method.     

Image encryption based on synchronization of fractional chaotic systems

This paper deals with a synchronization scheme for two fractional chaotic systems which is applied in image encryption. Based on Pecora and Carroll (PC) synchronization, fractional-order Lorenz-like system forms a master–slave configuration, and the sufficient conditions are derived to realize synchronization between these two systems via the Laplace transformation theory. An image encryption algorithm is introduced where the original image is encoded by a nonlinear function of a fractional chaotic state. Simulation results show that the original image is well masked in the cipher texts and recovered successfully through chaotic signals. Further, the cryptanalysis is conducted in detail through histogram, information entropy, key space and sensitivity to verify the high security.     

Output-feedback synchronization of chaotic systems based on sum-of-squares approach

This paper investigates the synchronization of chaotic systems using an output feedback polynomial controller. As only output system states are considered, it makes the controller design and system analysis more challenging compared to the full-state feedback control schemes. To study the system stability and synthesize the output feedback polynomial controller, Lyapunov stability theory is employed. Sufficient stability conditions are derived in terms of sum of squares (SOS) conditions to guarantee the system stability and aid the controller synthesis. A genetic algorithm-based SOS technique is proposed to find the solution to the SOS conditions and the parameter values of the output feedback polynomial controller. A simulation example is employed to illustrate the effectiveness of the proposed approach.     

Robust adaptive synchronization of different uncertain chaotic systems subject to input nonlinearity

In this paper, an adaptive controller is designed to ensure robust synchronization of two different chaotic systems with input nonlinearities. For this purpose, a stable sliding surface is defined and an adaptive sliding mode controller is designed to achieve robust synchronization of the systems when the control input is influenced through nonlinearities produced by actuator or external uncertainty recourses. The adaptation law guarantees the synchronization assuming of unknown model uncertainty. Furthermore by adding an integrator and incorporating a saturation function in the control law, the chattering phenomenon caused by the sign function is avoided. The simulation results for synchronization of Chua’s circuit and Genesio systems show the efficiency of the proposed technique.     

Sector-condition-based results for adaptive control and synchronization of chaotic systems under input saturation

This paper addresses the design of adaptive feedback controllers for two problems (namely, stabilization and synchronization) of chaotic systems with unknown parameters by considering input saturation constraints. A novel generalized sector condition is developed to deal with the saturation nonlinearities for synthesizing the nonlinear and the adaptive controllers for the stabilization and synchronization control objectives. By application of the proposed sector condition and rigorous regional stability analysis, control and adaptation laws are formulated to guarantee local stabilization of a nonlinear system under actuator saturation. Further, simple control and adaptation laws are developed to synchronize two chaotic systems under uncertain parameters and input saturation nonlinearity. Numerical simulation results for Rössler and FitzHugh–Nagumo models are provided to demonstrate the effectiveness of the proposed adaptive stabilization and synchronization control methodologies.     

Finite-time synchronization of uncertain unified chaotic systems based on CLF

This paper studies the problem of finite-time synchronization for the unified chaotic systems. We prove that global finite-time synchronization can be achieved for unified chaotic systems which have uncertain parameters. Simulation results for Lorenz, Lü and Chen chaotic systems are provided to illustrate the effectiveness of the proposed scheme.     

Global synchronization criterion and adaptive synchronization for new chaotic system

This paper proposes two schemes of synchronization of two four-scorll chaotic attractor, a simple global synchronization and adaptive synchronization in the presence of unknown system parameters. Based on Lyapunov stability theory and matrix measure, a simple generic criterion is derived for global synchronization of four-scorll chaotic attractor system with a unidirectional linear error feedback coupling. This methods are applicable to a large class of chaotic systems where only a few algebraic inequalities are involved. Numerical simulations are presented to show the effectiveness of the proposed chaos synchronization method.     

On synchronization of three chaotic systems

In this paper, a simple but efficient method is applied to the synchronization of three chaotic systems, i.e., the chaotic Lorenz, Chua, and Chen systems. Numerical simulations show this method works very well.     

Symplectic synchronization of different chaotic systems

In this paper, a new symplectic synchronization of chaotic systems is studied. Traditional generalized synchronizations are special cases of the symplectic synchronization. A sufficient condition is given for the asymptotical stability of the null solution of an error dynamics. The symplectic synchronization may be applied to the design of secure communication. Finally, numerical results are studied for a Quantum-CNN oscillators synchronized with a Rössler system in three different cases.     

Finite time synchronization of chaotic systems

Using finite time control techniques, continuous state feedback control laws are developed to solve the synchronization problem of two chaotic systems. We demonstrate that these two chaotic systems can be synchronized in finite time. Examples of Duffing systems, Lorenz systems are presented to verify our method.     

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.     

A simple adaptive control for full and reduced-order synchronization of uncertain time-varying chaotic systems

In this paper, a simple adaptive feedback control is proposed for full and reduced-order synchronization of time-varying and strictly uncertain chaotic systems. Our method uses only one feedback gain with parameter adaptation law and converges very fast even in the presence of noise. For full synchronization, a drive-response system consisting of two second-order identical parametrically excited oscillators achieve global synchronization; while for reduced-order synchronization, the dynamical evolution of a second-order parametrically driven oscillator is synchronized with the projection of a third-order time-varying chaotic system. The effectiveness of our approach is demonstrated using numerical simulations.     

Projective lag synchronization in chaotic systems

In this paper, a new projective lag synchronization is proposed, where a driven chaotic system synchronizes the past state of the driver up to a scaling factor α. An active control method is employed to design a controller to achieve the global synchronization of two identical chaotic systems. Based on Lyapunov stability theorem, a sufficient condition is then given for the asymptotical stability of the null solution of an error dynamics. The effectiveness of the proposed schemes is verified via numerical simulations.     

On synchronization of chaotic systems based on the Thau observer design

In this paper we revisit the Thau observer design and concern its application to the synchronization problem of two Lorenz name related systems in the master-slave formalism. The first one is the Lorenz-Stenflo system possessing a positively invariant ellipsoid while another one is the hyperchaotic Lorenz system possessing a positively invariant cylinder. Information about loci of these invariant domains is applied for the observer design. Further, we present one assertion related to one spectral inequality arisen in the process of assigning stable spectrum to the observer matrix and show its use in the observer design. We demonstrate the efficiency of synchronization schemes for the both of systems with help of numerical simulation.     

Fast synchronization of directionally coupled chaotic systems

This paper studies the fast synchronization of directionally coupled chaotic systems under a chained interaction topology. Firstly, by applying finite-time stability theory, it is shown that all chaotic systems can achieve synchronization in finite time as long as the coupling strength is strong enough. Secondly, it is proved that the settling times are determined by the interaction strength, system parameters and initial conditions of the chaotic systems. Furthermore, it is found that the settling times are mainly dependent on the bounded value and dimension of the coupled chaotic systems when the individual chaotic sub-system is bounded. Finally, illustrative examples and numerical simulations are given to show the correctness of theoretical results.     

Linear generalized synchronization of continuous-time chaotic systems

This paper develops a general approach for constructing a response system to implement linear generalized synchronization (GS) with the drive continuous-time chaotic system. Some sufficient conditions of global asymptotic linear GS between the drive and response continuous-time chaotic systems are attained from rigorously modern control theory. Finally, we take Chua’s circuit as an example for illustration and verification.     

Quadratic optimal synchronization of uncertain chaotic systems

This paper investigates the quadratic optimal synchronization of uncertain chaotic systems with parameter mismatch, parametric perturbations and external disturbances on both master and slave systems. A robust control scheme based on Lyapunov stability theory and quadratic optimal control approach is derived to realize chaotic synchronization. The sufficient criterion for stability condition is formulated in a linear matrix inequality (LMI) form. The effect of uncertain parameters and external disturbance is suppressed to an H_∞ norm constraint. An adaptive algorithm is proposed to adjust the uncertain bound in the robust controller avoiding the chattering phenomena. The simulation results for synchronization of the Chua’s circuit system and the Lorenz system demonstrate the effectiveness of the proposed scheme.     

Adaptive fuzzy observer based synchronization design and secure communications of chaotic systems

This paper proposes a synchronization design scheme based on an alternative indirect adaptive fuzzy observer and its application to secure communication of chaotic systems. It is assumed that their states are unmeasurable and their parameters are unknown. Chaotic systems and the structure of the fuzzy observer are represented by the Takagi–Sugeno fuzzy model. Using Lyapunov stability theory, an adaptive law is derived to estimate the unknown parameters and the stability of the proposed system is guaranteed. Through this process, the asymptotic synchronization of chaotic systems is achieved. The proposed observer is applied to secure communications of chaotic systems and some numerical simulation results show the validity of theoretical derivations and the performance of the proposed observer.