Exponential synchronization of chaotic Lur’e systems with delayed feedback control

The exponential synchronization problem is studied in this paper for a class of chaotic Lur’e systems by using delayed feedback control. An augmented Lyapunov functional based approach is proposed to deal with this issue. A delay-dependent condition is established such that the controlled slave system can exponentially synchronize with the master system. It is shown that the delayed feedback gain matrix and the exponential decay rate can be obtained by solving a set of linear matrix inequalities. The decay coefficient can be also easily calculated. Finally, as an example, the Chua’s circuit is used to illustrate the effectiveness of the developed approach and the improvement over some existing results.     

A new chaotic system with fractional order and its projective synchronization

Based on Rikitake system, a new chaotic system is discussed. Some basic dynamical properties, such as equilibrium points, Lyapunov exponents, fractal dimension, Poincaré map, bifurcation diagrams and chaotic dynamical behaviors of the new chaotic system are studied, either numerically or analytically. The obtained results show clearly that the system discussed is a new chaotic system. By utilizing the fractional calculus theory and computer simulations, it is found that chaos exists in the new fractional-order three-dimensional system with order less than 3. The lowest order to yield chaos in this system is 2.733. The results are validated by the existence of one positive Lyapunov exponent and some phase diagrams. Further, based on the stability theory of the fractional-order system, projective synchronization of the new fractional-order chaotic system through designing the suitable nonlinear controller is investigated. The proposed method is rather simple and need not compute the conditional Lyapunov exponents. Numerical results are performed to verify the effectiveness of the presented synchronization scheme.     

Adaptive control for synchronization of a four-dimensional chaotic system via a single variable

This paper studies the control for synchronization of a four-dimensional system via a single variable, and a linear feedback controller and an adaptive controller are proposed. Based on the Lyapunov stability theory, the correctness of the proposed methods is strictly demonstrated. The numerical simulations further show their effectiveness.     

A revisit to synchronization of Lurie systems with time-delay feedback control

This paper revisits the problem of synchronization for general Lurie systems with time-delay feedback control. Differently from most of existing results, the more restrictively slope restrictions on the nonlinearities of Lurie systems are considered in view of the fact that the slope restrictions may improve synchronization conditions compared with the sector ones. The Kalman–Yakubovich–Popov (KYP) lemma and the Schur complement formula are applied to get novel and less conservative synchronization criteria, which have the forms of linear matrix inequalities (LMIs). Numerical examples are presented to illustrate the efficiency of the proposed results.     

A novel active pinning control for synchronization and anti-synchronization of new uncertain unified chaotic systems

This paper discusses the synchronization and anti-synchronization of new uncertain unified chaotic systems (UUCS). Based on the idea of active control, a novel active Pinning control strategy is presented, which only needs a state of new UUCS. The proposed controller can achieve synchronization between a response system and a drive system, and ensure the synchronized robust stability of new UUCS. Numerical simulations of new UUCS show that the controller can make chaotic systems achieve synchronization or anti-synchronization in a quite short period and both are of good robust stability.     

Robust adaptive synchronization of uncertain unified chaotic systems

The problem of synchronizing a unified chaotic system in the presence of parameter variations, unstructured uncertainties, and external disturbances is addressed. To tackle such perturbations whose bounds may be unknown, two robust adaptive algorithms are proposed. The stability analysis is presented based on the Lyapunov stability theorem. Simulation results demonstrate the performance of the developed synchronization schemes.     

Observer-based impulsive chaotic synchronization of discrete-time switched systems

This paper investigates impulsive chaotic synchronization of discrete-time switched systems with state-dependent switching strategy. The parameter-dependent Lyapunov function (PDLF) technique is used to establish stability criteria for a class of switched systems consisting of both stable and unstable subsystems. With these criteria, sufficient conditions are given to achieve observer-based impulsive chaotic synchronization. Examples are presented to illustrate the criteria.     

Constrained predictive synchronization of discrete-time chaotic Lur’e systems with time-varying delayed feedback control

This paper presents a predictive synchronization method for discrete-time chaotic Lur’e systems with input constraints by using time-varying delayed feedback control. Based on the model predictive control scheme, a delay-dependent stabilization criterion is derived for the synchronization of chaotic systems that is represented by Lur’e systems with input constraints. By constructing a suitable Lyapunov–Krasovskii functional and combining with a reciprocally convex combination technique, a delay-dependent stabilization condition for synchronization is obtained via linear matrix inequality (LMI) formulation. The control inputs are obtained by solving a min-max problem subject to cost monotonicity, which is expressed in terms of LMIs. The effectiveness of the proposed method will be verified throughout a numerical example.     

New results on synchronization of chaotic systems with time-varying delays via intermittent control

This paper is concerned with the problem of exponential synchronization for chaotic systems with time-varying delays by using periodically intermittent control. Some new and useful synchronization criteria are obtained based on the differential inequality method and the analysis technique. It is noteworthy that the methods used in this paper are different from the techniques employed in the existing works, and the derived conditions are less conservative. Especially, a strong constraint on the control width that the control width should be large than the time delay imposed by the current references is released in this paper. Moreover, the new synchronization criteria do not impose any restriction on the size of time delay. Numerical examples are finally presented to illustrate the effectiveness of the theoretical results.     

Adaptive bidirectionally coupled synchronization of?chaotic?systems with unknown parameters

This paper investigates the chaos synchronization of two bidirectionally coupled chaotic systems. In comparison with previous methods (identical bidirectionally coupled synchronization), the present control scheme is different bidirectionally coupled synchronization, which includes different complete bidirectionally coupled synchronization and different partial bidirectionally coupled synchronization. Based on the Lasalle invariance principle, adaptive schemes are designed to make two different bidirectionally coupled chaotic systems asymptotically synchronized, and unknown parameters are identified simultaneously in the process of synchronization. Theoretical analysis and numerical simulations are shown to verify the results.     

Lag synchronization of a class of chaotic systems with unknown parameters

Research on chaos synchronization of dynamical systems has been largely reported in literature. However, synchronization of different structure—uncertain dynamical systems—has received less attention. This paper addresses synchronization of a class of time-delay chaotic systems containing uncertain parameters. A unified scheme is established for synchronization between two strictly different time-delay uncertain chaotic systems. The synchronization is successfully achieved by designing an adaptive controller with the estimates of the unknown parameters and the nonlinear feedback gain. The result is rigorously proved by the Lyapunov stability theorem. Moreover, we illustrate the application of the proposed scheme by numerical simulation, which demonstrates the effectiveness and feasibility of the proposed synchronization method.     

Complete (anti-)synchronization of chaotic systems with fully uncertain parameters by adaptive control

This paper addresses a unified mathematical expression describing a class of chaotic systems, for which the problem of synchronization and anti-synchronization between different chaotic systems with fully uncertain parameters and different structure are studied. Based on the Lyapunov stability theory, a novel, simple, and systemic adaptive synchronization controller is designated, the analytic expression of the controller and the adaptive laws of parameters are developed. Moreover, the proposed scheme can be extended to anti-synchronize a class of chaotic systems. Two chaotic systems with different structure and fully uncertain parameters are employed as the examples to show the effectiveness of the proposed adaptive synchronization and anti-synchronization schemes. Additionally, the robustness and noise immunity of the adaptive synchronization scheme is investigated by measuring the mean squared error of the systems.     

Achievement of chaotic synchronization trajectories of master–slave manipulators with feedback control strategy

This paper addresses a master-slave synchro- nization strategy for complex dynamic systems based on feedback control. This strategy is applied to 3-DOF pla- nar manipulators in order to obtain synchronization in such complicated as chaotic motions of end-effectors. A chaotic curve is selected from Duffing equation as the trajectory of master end-effector and a piecewise approximation method is proposed to accurately represent this chaotic trajectory of end-effectors. The dynamical equations of master-slave manipulators with synchronization controller are derived, and the Lyapunov stability theory is used to determine the stability of this controlled synchronization system. In numer- ical experiments, the synchronous motions of end-effectors as well as three joint angles and torques of master-slave manipulators are studied under the control of the proposed synchronization strategy. It is found that the positive gain matrix affects the implementation of synchronization con- trol strategy. This synchronization control strategy proves the synchronization＇s feasibility and controllability for com- plicated motions generated by master-slave manipulators.     

A new Lyapunov approach for global synchronization of non-autonomous chaotic systems

This paper proposes a new approach for finding the Lyapunov function to study the sufficient global synchronization criterion of master-slave non-autonomous chaotic systems via linear state error feedback control. The approach is first demonstrated in a synchronization scheme for the second-order non-autonomous chaotic systems and then generalized to the schemes for the nth-order non-autonomous chaotic systems. Some algebraic synchronization criteria for the second-order chaotic systems are obtained. The sharpness of the new criteria is compared with that of the existing criteria of the same type by numerical examples.     

Reliable impulsive lag synchronization for a class of nonlinear discrete chaotic systems

The problem of reliable impulsive lag synchronization for a class of nonlinear discrete chaotic systems is investigated in this paper. Firstly a reliable impulsive controller is designed by the impulsive control theory. Then, some sufficient conditions for reliable impulsive lag synchronization between the drive system and the response system are obtained. Numerical simulations are given to show the effectiveness of the proposed method.     

Finite-time chaos control of unified chaotic systems with uncertain parameters

This paper is concerned with finite-time chaos control of unified chaotic systems with uncertain parameters. Based on the finite-time stability theory in the cascade-connected system, a nonlinear control law is presented to achieve finite-time chaos control. The controller is simple and easy to be constructed. Simulation results for Lorenz, Lü, and Chen chaotic systems are provided to illustrate the effectiveness of the proposed scheme. Supported by the National Natural Science Foundation of China (Grant No. 60674024).     

Observer-based synchronization scheme for a class of chaotic systems using contraction theory

In this paper, an adaptive synchronization scheme is proposed for a class of nonlinear systems. The design utilizes an adaptive observer, which is quite useful in establishing a transmitter–receiver kind of synchronization scheme. The proposed approach is based on contraction theory and provides a very simple way of establishing exponential convergence of observer states to actual system states. The class of systems addressed here has uncertain parameters, associated with the part of system dynamics that is a function of measurable output only. The explicit conditions for the stability of the observer are derived in terms of gain selection of the observer. Initially, the case without uncertainty is considered and then the results are extended to the case with uncertainty in parameters of the system. An application of the proposed approach is presented to synchronize the family of N chaotic systems which are coupled through the output variable only. The numerical results are presented for designing an adaptive observer for the chaotic Chua system to verify the efficacy of the proposed approach. Explicit bounds on observer gains are derived by exploiting the properties of the chaotic attractor exhibited by Chua’s system. Convergence of uncertain parameters is also analyzed for this case and numerical simulations depict the convergence of parameter estimates to their true value.     

Synchronization for chaotic Lur’e systems with sector-restricted nonlinearities via delayed feedback control

In this paper, the effects of time delay on chaotic master–slave synchronization scheme are considered. Using delayed feedback control scheme, a delay-dependent stability criterion is derived for the synchronization of chaotic systems that are represented by Lur’e system with sector-restricted nonlinearities. The derived criterion is a sufficient condition for absolute stability of error dynamics between the master and the slave system. Using a convex representation of the nonlinearity, the stability condition based on the Lyapunov–Krasovskii functional is obtained via LMI formulation. The proposed delay-dependent synchronization criterion is less conservative than the existing ones. The effectiveness of our work is verified through numerical examples.     

Adaptive fuzzy H ∞ tracking design of SISO uncertain nonlinear fractional order time-delay systems

In this paper, a fuzzy logic controller equipped with training algorithms is developed such that the H ^?? tracking performance should be satisfied for a model-free nonlinear fractional order time delay system which is infinite dimensional in nature and time delay is a source of instability. In order to deal with the linguistic uncertainties caused from delay terms, the adaptive time delay fuzzy logic system is constructed to approximate the unknown time delay system functions. By incorporating Lyapunov stability criterion with H ^?? tracking design technique, the free parameters of the adaptive fuzzy controller can be tuned on line by output feedback control law and adaptive law. Moreover, the tracking error and external disturbance can be attenuated to arbitrary desired level. The numerical results show the effectiveness of the proposed adaptive H ^?? tracking scheme.     

Nonlinear observer-based impulsive synchronization in chaotic systems with multiple attractors

The issue of impulsive synchronization of the coupled Newton–Leipnik system is investigated. Based on the impulsive stability theory, nonlinear observer-based impulsive synchronization scheme is derived. A new and less conservative criteria for impulsive synchronization via nonlinear observer is proposed. The boundary of the stable regions is also estimated. One important advantage of the proposed method is that it is also applicable for the systems with more than one attractor. Numerical simulations on Newton–Leipnik system are illustrated to verify the theoretical results.