Novel Cryptography Technique via Chaos Synchronization of Fractional-Order Derivative Systems

Synchronization of fractional-order-derivative systems for cryptography purpose is still exploratory and despite an increase in cryptography research, several challenges remain in designing a powerful cryptosystem. This chapter addresses the problem of synchronization of fractional-order-derivative chaotic systems using random numbers generator for a novel technique to key distribution in cryptography. However, there is evidence that researchers have approached the problem using integer order derivative chaotic systems. Consequently, the aim of the chapter lies in coding and decoding a text via chaos synchronization of fractional-order derivative, the performance analysis and the key establishment scheme following an application on a text encryption using the chaotic Mathieu-Van Der Pol fractional system. In order to improve the level of the key security, the Fibonacci Q-matrix is used in the key generation process and the initial condition; the order of the derivative of the responder system secretly shared between the responder and the receiver are also involved. It followed from this study that compared to the existing cryptography techniques, this proposed method is found to be very efficient due to the fact that, it improves the key security.

Fuzzy Adaptive Controller for Synchronization of Uncertain Fractional-Order Chaotic Systems

In this chapter, the projective synchronization problem of different multivariable fractional-order chaotic systems with both uncertain dynamics and external disturbances is studied. More specifically, a fuzzy adaptive controller is investigated for achieving a projective synchronization of uncertain fractional-order chaotic systems. The adaptive fuzzy-logic system is used to online estimate the uncertain nonlinear functions. The latter is augmented by a robust control term to efficiently compensate for the unavoidable fuzzy approximation errors, external disturbances as well as residual error due to the use of the so-called e-modification in the adaptive laws. A Lyapunov approach is employed to derive the parameter adaptation laws and to prove the boundedness of all signals of the closed-loop system. Numerical simulations are performed to verify the effectiveness of the proposed synchronization scheme.

Robust Synchronization of Fractional-Order Arneodo Chaotic Systems Using a Fractional Sliding Mode Control Strategy

This chapter investigates a new control design methodology for the synchronization of fractional-order Arneodo chaotic systems using a fractional-order sliding mode control configuration. This class of nonlinear fractional-order systems shows a chaotic behavior for a set of model parameters. The stability analysis of the proposed fractional-order sliding mode control law is performed by means of the Lyapunov stability theory. Simulation examples on fractional-order Arneodo chaotic systems synchronization are provided in presence of disturbances and noises. These results illustrate the effectiveness and robustness of this control design approach.

Stabilization of a Class of Fractional-Order Chaotic Systems via Direct Adaptive Fuzzy Optimal Sliding Mode Control

In this chapter a new direct adaptive fuzzy optimal sliding mode control approach is proposed for the stabilization of fractional chaotic systems with different initial conditions of the state under the presence of uncertainties and external disturbances. Using Lyapunov analysis, the direct adaptive fuzzy optimal sliding mode control approach illustrates asymptotic convergence of error to zero as well as good robustness against external disturbances and uncertainties. The authors present a method for optimum tuning of sliding mode control system parameter using particle swarm optimization (PSO) algorithm. PSO is a robust stochastic optimization technique based on the movement and intelligence of swarm, applying the concept of social interaction to problem solving. Simulation examples for the control of nonlinear fractional-order systems are given to illustrate the effectiveness of the proposed fractional adaptive fuzzy control strategy.

Chaos Synchronization of Optical Systems via a Fractional-Order Sliding Mode Controller

In this chapter, one investigates the chaos synchronization of a class of uncertain optical chaotic systems. More precisely, one also presents a systematic approach for designing a fractional-order (FO) sliding mode controller to achieve a rapid, robust, and perfect chaos synchronization. By this robust controller, it is rigorously proven that the associated synchronization error is Mittag-Leffler (or asymptotically) stable. In a numerical simulation framework, this synchronization scheme is tested on many chaotic optical systems taken from the open literature. The obtained results clearly show that the proposed chaos synchronization controller is not only strongly robust with respect to the unavoidable system's uncertainties (as unmodeled dynamics, and parameters' variation and uncertainty) and eventual dynamical external disturbances, but also can significantly reduce the chattering effect.

Adaptive Neural Control for Unknown Nonlinear Time-Delay Fractional-Order Systems With Input Saturation

This chapter focuses on the adaptive neural control of a class of uncertain multi-input multi-output (MIMO) nonlinear time-delay non-integer order systems with unmeasured states, unknown control direction, and unknown asymmetric saturation actuator. The design of the controller follows a number of steps. Firstly, based on the semi-group property of fractional order derivative, the system is transformed into a normalized fractional order system by means of a state transformation in order to facilitate the control design. Then, a simple linear state observer is constructed to estimate the unmeasured states of the transformed system. A neural network is incorporated to approximate the unknown nonlinear functions while a Nussbaum function is used to deal with the unknown control direction. In addition, the strictly positive real (SPR) condition, the Razumikhin lemma, the frequency distributed model, and the Lyapunov method are utilized to derive the parameter adaptive laws and to perform the stability proof.

Dynamics and Synchronization of the Fractional-Order Hyperchaotic System

The fractional-order Lorenz hyperchaotic system is solved as a discrete map by applying Adomian decomposition method (ADM). Dynamics of this system versus parameters are analyzed by LCEs, bifurcation diagrams, and SE and C0 complexity. Results show that this system has rich dynamical behaviors. Chaos and hyperchaos can be generated by decreasing the fractional derivative order q in this system. It also shows that the system is more complex when q takes smaller values. Moreover, coupled synchronization of fractional-order chaotic system is investigated theoretically. The synchronization performances with synchronization controller parameters and derivative order varying are analyzed. Synchronization and complexity of intermediate variables which generated by ADM are investigated. It shows that intermediate variables between the driving system and response system are synchronized and higher complexity values are found. It lays a technical foundation for secure communication applications of fractional-order chaotic systems.

Neural Approximation-Based Adaptive Control for Pure-Feedback Fractional-Order Systems With Output Constraints and Actuator Nonlinearities

In this chapter, an adaptive control approach-based neural approximation is developed for a category of uncertain fractional-order systems with actuator nonlinearities and output constraints. First, to overcome the difficulties arising from the actuator nonlinearities and nonaffine structures, the mean value theorem is introduced. Second, to deal with the uncertain nonlinear dynamics, the unknown control directions and the output constraints, neural networks, smooth Nussbaum-type functions, and asymmetric barrier Lyapunov functions are employed, respectively. Moreover, for satisfactorily designing the control updating laws and to carry out the stability analysis of the overall closed-loop system, the Backstepping technique is used. The main advantage about this research is that (1) the number of parameters to be adapted is much reduced, (2) the tracking errors converge to zero, and (3) the output constraints are not transgressed. At last, simulation results demonstrate the feasibility of the newly presented design techniques.

Secure Digital Data Communication Based on Fractional-Order Chaotic Maps

The aim of the chapter is twofold. First, a literature review on synchronization methods of fractional-order discrete-time systems is exposed. Second, a secure digital data communication based on synchronization of fractional-order discrete-time chaotic systems is proposed. Two synchronization methods based on observers are proposed to synchronize two fractional-order discrete-time chaotic systems. The first method concerns the impulsive synchronization where sufficient conditions for the synchronization error of the states are given. The second method concerns the exact synchronization which is based on a step-by-step delayed observer. In the same way, conditions are provided in order to allow the reconstruction of the states and the unknown input which is the message in this case. The two synchronization methods are combined in order to design a novel robust secure digital data communication. The performance of the proposed communication system is illustrated in numerical simulations where digital image signal is considered.

Design and Optimization of Generalized PD-Based Control Scheme to Stabilize and to Synchronize Fractional-Order Hyperchaotic Systems

This chapter will establish the importance and significance of studying the fractional-order control of nonlinear dynamical systems and emphasize the link between the factional calculus and famous PID control design. It will lay the foundation related to the research scope, problem formulation, objectives and contributions. As a case study, a fractional-order PD-based feedback (Fo-PDF) control scheme with optimal knowledge base is developed in this work for achieving stabilization and synchronization of a large class of fractional-order chaotic systems (FoCS). Based and derived on Lyapunov stabilization arguments of fractional-order systems, the stability analysis of the closed-loop control system is investigated. The design and multiobjective optimization of Fo-PDF control law is theoretically rigorous and presents a powerful and simple approach to provide a reasonable tradeoff between simplicity, numerical accuracy, and stability analysis in control and synchronization of FoCS. The feasibility and validity of this developed Fo-PDF scheme have been illustrated by numerical simulations using the fractional-order Mathieu-Van Der Pol hyperchaotic system.