Generalized finite-time synchronization between coupled chaotic systems of different orders with unknown parameters

This letter investigates the adaptive finite-time synchronization of different coupled chaotic (or hyperchaotic) systems with unknown parameters. The sufficient conditions for achieving the generalized finite-time synchronization of two chaotic systems are derived based on the theory of finite-time stability of dynamical systems. By the adaptive control technique, the control laws and the corresponding parameters update laws are proposed such that the generalized finite-time synchronization of nonidentical chaotic (or hyperchaotic) systems is to be obtained. These results obtained are in good agreement with the existing one in open literature and it is shown that the technique introduced here can be further applied to various finite-time synchronizations between dynamical systems. Finally, numerical simulations are given to demonstrate the effectiveness of the proposed scheme.     

Finite-time combination-combination synchronization of four different chaotic systems with unknown parameters via sliding mode control

In this paper, we apply the nonsingular terminal sliding mode control technique to realize the novel combination-combination synchronization between combination of two chaotic systems as drive system and combination of two chaotic systems as response system with unknown parameters in a finite time. On the basic of the adaptive laws and finite-time stability theory, an adaptive combination sliding mode controller is proposed to ensure the occurrence of the sliding motion in a given finite time for four different chaotic systems. In theory, it is proved that the sliding mode technique can realize fast convergence for four different chaotic systems in the finite time. Some criteria and corollaries are derived for finite-time combination-combination synchronization of four different chaotic systems. Numerical simulation results are shown to verify the effectiveness and correctness of the combination-combination synchronization.     

Synchronization of nonlinear chaotic electromechanical gyrostat systems with uncertainties

This paper solves the problem of robust synchronization of nonlinear chaotic gyrostat systems in a given finite time. The parameters of both master and slave chaotic gyrostat systems are assumed to be unknown in advance. In addition, the gyrostat systems are disturbed by unknown model uncertainties and external disturbances. Suitable update laws are proposed to estimate the unknown parameters. Based on the finite-time control idea and update laws, appropriate control laws are designed to ensure the stabilization of the closed-loop system in finite time. The precise value of the convergence time is given. A numerical simulation demonstrates the applicability and efficiency of the proposed finite-time synchronization strategy.     

Adaptive control for electromechanical systems considering dead-zone phenomenon

The electromechanical gyrostat is a fourth-order nonautonomous system that exhibits very rich behavior such as chaos. In recent years, synchronization of nonautonomous chaotic systems has found many useful applications in nonlinear science and engineering fields. On the other hand, it is well known that the finite-time control techniques demonstrate good robustness and disturbance rejection properties. This paper studies the potential application of the finite-time control techniques for synchronization of nonautonomous chaotic electromechanical gyrostat systems in finite time. It is assumed that all the parameters of both drive and response systems are unknown parameters in advance. Moreover, the effects of dead-zone nonlinearities in the control inputs are also taken into account. Some adaptive controllers are introduced to synchronize two gyrostat systems in different scenarios within a given finite-time. Two illustrative examples are presented to demonstrate the efficiency and robustness of the proposed finite-time synchronization strategy.     

Global finite-time synchronization of a class of the non-autonomous chaotic systems

This paper investigates the global finite-time synchronization of a class of the second-order nonautonomous chaotic systems via a master?Cslave coupling. A?continuous generalized linear state-error feedback controller with simple structure is introduced into the synchronization scheme. Some easily implemented algebraic criteria for achieving the global finite-time synchronization are proven and then optimized for the purposes of improving their sharpness. The optimized criteria are applied to a practical master?Cslave synchronization scheme for the single-machine-infinite-bus (SMIB) systems, obtaining the precise corresponding synchronization conditions. Several numerical examples are provided to illustrate the effectiveness of the new synchronization criteria.     

A general nonlinear adaptive control scheme for finite-time synchronization of chaotic systems with uncertain parameters and nonlinear inputs

In this paper, the problem of finite-time chaos synchronization between two different uncertain chaotic systems with unknown parameters and input nonlinearities is investigated. It is assumed that both master and slave systems are perturbed by unknown model uncertainties, external disturbances, and fully unknown parameters. Proper update laws are proposed to estimate the systems?? unknown parameters. Based on the update laws and finite-time control technique, a robust adaptive controller is introduced to guarantee the convergence of the slave system trajectories to the trajectories of the master system in a given finite time. Two illustrative examples are presented to illustrate the effectiveness and applicability of the proposed finite-time controller and to validate the theoretical results of the paper.     

Finite-time real combination synchronization of three complex-variable chaotic systems with unknown parameters via sliding mode control

The problem of real combination synchronization between three complex-variable chaotic systems with unknown parameters is investigated by nonsingular terminal sliding mode control in a finite time. Based on the adaptive laws and finite-time stability theory, a nonsingular terminal sliding mode control is designed to ensure the real combination synchronization of three complex-variable chaotic systems in a given finite time. It is theoretically gained that the introduced sliding mode technique has finite-time convergence and stability in both arriving and sliding mode phases. Numerical simulation results are given to show the effectiveness and reliability of the finite-time real combination synchronization.     

Adaptive <Emphasis Type="Italic">Q</Emphasis>–<Emphasis Type="Italic">S</Emphasis> synchronization of non-identical chaotic systems with unknown parameters

This work investigates the adaptive Q–S synchronization of non-identical chaotic systems with unknown parameters. The sufficient conditions for achieving Q–S synchronization of two different chaotic systems (including different dimensional systems) are derived, based on Lyapunov stability theory. By the adaptive control technique, the control laws and the corresponding parameter update laws are proposed such that the non-identical chaotic systems are to have Q–S synchronization. Finally, four illustrative numerical simulations are also given to demonstrate the effectiveness of the proposed scheme.     

Robust finite-time tracking for a square fully actuated class of nonlinear systems

In this paper, the robust finite-time tracking problem is addressed for a square fully actuated class of nonlinear systems subjected to disturbances and uncertainties. Firstly, two applicable lemmas are derived and novel nonlinear sliding surfaces (manifolds) are defined by applying these lemmas. Secondly, by developing the nonsingular terminal sliding mode control, two different types of robust nonlinear control inputs are designed to meet and accomplish the aforementioned finite-time tracking objective. The global finite-time stability of the closed-loop nonlinear system is evaluated analytically and mathematically. The proposed control inputs are utilized to tackle and solve two interesting issues containing (a): the finite-time tracking problem of the unified chaotic system and (b): the finite-time synchronization of two non-identical hyperchaotic systems. Finally, based on MATLAB software, two numerical simulations are carried out to illustrate and demonstrate the effectiveness and performance of the proposed robust finite-time nonlinear control schemes.

     

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.     

Practical finite-time synchronization of jerk systems: Theory and experiment

This paper addresses the problem of finite-time synchronization of jerk chaotic systems through a simple linear feedback control. The controller is designed such that practical finite-time synchronization could be achieved. As example, we use a new jerk system obtained thanks to the chaotification of the Duffing system using jerk architecture and simplification via a single silicon p-n junction diode. Mathematic proof, numerical and PSpice simulations, and practical results are presented to show the feasibility of the proposed scheme. The proposed method could be applied to all jerk-like systems.     

Adaptive generalized function projective synchronization of uncertain chaotic complex systems

The complex nonlinear systems appear in many important fields of physics and engineering, which are very useful for cryptography and secure communication. This paper investigates adaptive generalized function projective synchronization (AGFPS) between two different dimensional chaotic complex systems with fully or partially unknown parameters via both reduced order and increased order. Based on the Lyapunov stability theorem and adaptive control technique, a general adaptive controller with corresponding parameter update rule is constructed to achieve AGFPS between two nonidentical chaotic complex systems with distinct orders, and identify the unknown parameters simultaneously. This scheme is then applied to obtain AGFPS between the hyperchaotic complex Lü system and the chaotic complex Lorenz system with fully unknown parameters, and between the uncertain chaotic complex Chen system and the uncertain hyperchaotic complex Lorenz system, respectively. Corresponding simulations results are performed to show the feasibility and effectiveness of the proposed synchronization method.     

Optimal nonlinear observers for chaotic synchronization with message embedded

This paper presents an optimal nonlinear observer for synchronizing the transmitter-receiver pair with guaranteed optimal performance. In the proposed scheme, a generalized nonlinear state-space observer via uniform matrix transformations is constructed to estimate the transmitter state and the information signal, simultaneously. A nonlinear optimal design approach is used to synchronize chaotic systems. Solving the Hamilton–Jacobi–Bellman (H–J–B) equations we can obtain a linear optimal feedback scheme for piecewise-linear chaotic systems. Moreover, a robust scheme derived from the H _∞ optimization theory improves the synchronization performance of general nonlinear chaotic systems by suppressing the influence of their high order residual terms. Finally, two numerical simulation examples are illustrated by the chaotic Chua’s circuit system and the Lorenz chaotic system to demonstrate the effectiveness of our scheme.     

Synchronization of two different chaotic systems with discontinuous coupling

This paper is concerned with the problem of synchronization between two different chaotic systems with discontinuous coupling. Based on the stability theory and the comparison theorem of differential equations, we derive less restrictive synchronization conditions than those resulting from the Lyapunov theory. The theoretical results show that generalized synchronization between two different chaotic systems can be achieved if the time-average coupling strength is large enough. Finally, the corresponding numerical simulations are presented to demonstrate the effectiveness of proposed schemes.     

Robust synchronization of a chaotic mechanical system with nonlinearities in control inputs

Centrifugal flywheel governors are known as chaotic non-autonomous mechanical devices used for automatic control of the speed of engines. The main characteristic of them is avoiding the damage caused by sudden change of the load torques. In this paper, the problem of robust finite-time synchronization of centrifugal flywheel governor systems is studied. The effects of unknown parameters, model uncertainties, external noises, and input nonlinearities are fully taken into account. We propose some adaptive laws to overcome the side effects of the unknown parameters of the system on the synchronization performance. Then, a robust adaptive switching controller is introduced to synchronize centrifugal flywheel governors with nonlinear control inputs in a given finite time. The finite-time fast convergence property of the proposed scheme is analytically proved and numerically illustrated.     

Secure communication in wireless sensor networks based on chaos synchronization using adaptive sliding mode control

Due to resource constraints in wireless sensor networks and the presence of unwanted conditions in communication systems and transmission channels, the suggestion of a robust method which provides battery lifetime increment and relative security is of vital importance. This paper considers the secure communication in wireless sensor networks based on new robust adaptive finite time chaos synchronization approach in the presence of noise and uncertainty. For this purpose, the modified Chua oscillators are added to the base station and sensor nodes to generate the chaotic signals. Chaotic signals are impregnated with the noise and uncertainty. At first, we apply the modified independent component analysis to separate the noise from the chaotic signals. Then, using the adaptive finite-time sliding mode controller, a control law and an adaptive parameter-tuning method is proposed to achieve the finite-time chaos synchronization under the noisy conditions and parametric uncertainties. Synchronization between the base station and each of the sensor nodes is realized by multiplying a selection matrix by the specified chaotic signal which is broadcasted by the base station to the sensor nodes. Simulation results are presented to show the effectiveness and applicability of the proposed technique.     

Adaptive feedback control and synchronization of non-identical chaotic fractional order systems

This paper addresses the reliable synchronization problem between two non-identical chaotic fractional order systems. In this work, we present an adaptive feedback control scheme for the synchronization of two coupled chaotic fractional order systems with different fractional orders. Based on the stability results of linear fractional order systems and Laplace transform theory, using the master-slave synchronization scheme, sufficient conditions for chaos synchronization are derived. The designed controller ensures that fractional order chaotic oscillators that have non-identical fractional orders can be synchronized with suitable feedback controller applied to the response system. Numerical simulations are performed to assess the performance of the proposed adaptive controller in synchronizing chaotic systems.     

Nonlinear active observer-based generalized synchronization in time-delayed systems

When two different chaotic oscillators are coupled, generalized synchronization can occur. It may imply a very complicated relation between the states of drive and response systems. We propose a method that can be used to detect and characterize the generalized synchronization in modulated time-delayed systems. Using Krasovskii–Lyapunov theory, sufficient condition for generalized synchronization is derived. The proposed technique has been applied to synchronize prototype and Ikeda models by numerical simulation.     

Generalized synchronization of the fractional-order chaos in weighted complex dynamical networks with nonidentical nodes

A fractional-order weighted complex network consists of a number of nodes, which are the fractional-order chaotic systems, and weighted connections between the nodes. In this paper, we investigate generalized chaotic synchronization of the general fractional-order weighted complex dynamical networks with nonidentical nodes. The well-studied integer-order complex networks are the special cases of the fractional-order ones. Based on the stability theory of linear fraction-order systems, the nonlinear controllers are designed to make the fractional-order complex dynamical networks with distinct nodes asymptotically synchronize onto any smooth goal dynamics. Numerical simulations are provided to verify the theoretical results. It is worth noting that the synchronization effect sensitively depends on both the fractional order ?? and the feedback gain k _i . Moreover, generalized synchronization of the fractional-order weighted networks can still be achieved effectively with the existence of noise perturbation.     

Synchronization between integer-order chaotic systems and a class of fractional-order chaotic system based on fuzzy sliding mode control

In this paper, we focus on the synchronization between integer-order chaotic systems and a class of fractional-order chaotic system using the stability theory of fractional-order systems. A new fuzzy sliding mode method is proposed to accomplish this end for different initial conditions and number of dimensions. Furthermore, three examples are presented to illustrate the effectiveness of the proposed scheme, which are the synchronization between a fractional-order Lü chaotic system and an integer-order Liu chaotic system, the synchronization between a fractional-order hyperchaotic system based on Chen??s system and an integer-order hyperchaotic system based upon the Lorenz system, and the synchronization between a fractional-order hyperchaotic system based on Chen??s system, and an integer-order Liu chaotic system. Finally, numerical results are presented and are in agreement with theoretical analysis.