Stabilization of fractional-order chaotic system via a single state adaptive-feedback controller

This letter investigates the stabilization of three-dimensional fractional-order chaotic systems, and proposes a single state adaptive-feedback controller for fractional-order chaos control based on Lyapunov stability theory, fractional order differential inequality, and adaptive control theory. The present controller which only contains a single state variable is simple both in design and implementation. Simulation results for several fractional-order chaotic systems are provided to illustrate the effectiveness of the proposed scheme.     

Lyapunov-based fractional-order controller design to synchronize a class of fractional-order chaotic systems

In this paper, a novel adaptive fractional-order feedback controller is first developed by extending an adaptive integer-order feedback controller. Then a simple but practical method to synchronize almost all familiar fractional-order chaotic systems has been put forward. Through rigorous theoretical proof by means of the Lyapunov stability theorem and Barbalat lemma, sufficient conditions are derived to guarantee chaos synchronization. A wide range of fractional-order chaotic systems, including the commensurate system and incommensurate case, autonomous system, and nonautonomous case, is just the novelty of this technique. The feasibility and validity of presented scheme have been illustrated by numerical simulations of the fractional-order Chen system, fractional-order hyperchaotic Lü system, and fractional-order Duffing system.     

Adaptive synchronization of fractional-order chaotic systems via a single driving variable

This letter investigates the synchronization of a class of three-dimensional fractional-order chaotic systems. Based on sliding mode variable structure control theory and adaptive control technique, a single-state adaptive-feedback controller containing a novel fractional integral sliding surface is developed to synchronize a class of fractional-order chaotic systems. The present controller, which only contains a single driving variable, is simple both in design and implementation. Simulation results for three fractional-order chaotic systems are provided to illustrate the effectiveness of the proposed scheme.     

Observer-based adaptive stabilization of the fractional-order chaotic MEMS resonator

Compared to the integer-order chaotic MEMS resonator, the fractional-order system can better model its hereditary properties and exhibit complex dynamical behavior. Following the increasing attention to adaptive stabilization in controller design, this paper deals with the observer-based adaptive stabilization issue of the fractional-order chaotic MEMS resonator with uncertain function, parameter perturbation, and unmeasurable states under electrostatic excitation. To compensate the uncertainty, a Chebyshev neural network is applied to approximate the uncertain function while its weight is tuned by a parametric update law. A fractional-order state observer is then constructed to gain unmeasured feedback information and a tracking differentiator based on a super-twisting algorithm is employed to avoid repeated derivative in the framework of backstepping. Based on the Lyapunov stability criterion and the frequency-distributed model of the fractional integrator, it is proved that the adaptive stabilization scheme not only guarantees the boundedness of all signals, but also suppresses chaotic motion of the system. The effectiveness of the proposed scheme for the fractional-order chaotic MEMS resonator is illustrated through simulation studies.     

No-chatter variable structure control for fractional nonlinear complex systems

This paper concerns the problem of robust control of uncertain fractional-order nonlinear complex systems. After establishing a simple linear sliding surface, the sliding mode theory is used to derive a novel robust fractional control law for ensuring the existence of the sliding motion in finite time. We use a nonsmooth positive definitive function to prove the stability of the controlled system based on the fractional version of the Lyapunov stability theorem. In order to avoid the chattering, which is inherent in conventional sliding mode controllers, we transfer the sign function of the control input into the first derivative of the control signal. The proposed sliding mode approach is also applied for control of a class of nonlinear fractional-order systems via a single control input. Simulation results indicate that the proposed fractional variable structure controller works well for stabilization of hyperchaotic and chaotic complex fractional-order nonlinear systems. Moreover, it is revealed that the control inputs are free of chattering and practical.     

A Lyapunov-based control scheme for robust stabilization of fractional chaotic systems

This paper concerns the problem of robust stabilization of autonomous and non-autonomous fractional-order chaotic systems with uncertain parameters and external noises. We propose a simple efficient fractional integral-type sliding surface with some desired stability properties. We use the fractional version of the Lyapunov theory to derive a robust sliding mode control law. The obtained control law is single input and guarantees the occurrence of the sliding motion in a given finite time. Furthermore, the proposed nonlinear control strategy is able to deal with a large class of uncertain autonomous and non-autonomous fractional-order complex systems. Also, Rigorous mathematical and analytical analyses are provided to prove the correctness and robustness of the introduced approach. At last, two illustrative examples are given to show the applicability and usefulness of the proposed fractional-order variable structure controller.     

Stability analysis for nonlinear fractional-order systems based on comparison principle

This work constructs a theoretical framework for the stability analysis of nonlinear fractional-order systems. A new definition, the generalized Caputo fractional derivative, is proposed for the first time. Based on that, the comparison principles for scalar and vector fractional-order systems are constructed, respectively. Furthermore, a sufficient theorem for stability analysis is proved, and how to use this theorem in stabilization is also discussed. Three examples have been presented to illustrate how to use the developed theory to analyze the stability and to design stabilization controllers. With the proposed method, the problems of stabilization and synchronization of the fractional-order chaotic fractional-order systems can be easily solved with linear feedback control.     

Controller design for fractional-order interconnected systems with unmodeled dynamics

This paper focuses on the output feedback tracking control for fractional-order interconnected systems with unmodeled dynamics. The reduced order high gain K-filters are designed to construct the estimation of the unavailable system state. Unmodeled dynamics is extended to the general fractional-order dynamical systems for the first time which is characterized by introducing a dynamical signal r(t). An adaptive output feedback controller is established using the fractional-order Lyapunov methods and proposed by novel dynamic surface control strategy. Then, it is confirmed that the considered system is semi-globally bounded stable and the errors between outputs and the desired trajectories can concentrate to a small neighborhood of the origin. Finally, a simulation example is introduced to demonstrate the correctness of the supplied controller.

     

Adaptive synchronization control based on QPSO algorithm with interval estimation for fractional-order chaotic systems and its application in secret communication

In this paper, the synchronization problem and its application in secret communication are investigated for two fractional-order chaotic systems with unequal orders, different structures, parameter uncertainty and bounded external disturbance. On the basis of matrix theory, properties of fractional calculus and adaptive control theory, we design a feedback controller for realizing the synchronization. In addition, in order to make it better apply to secret communication, we design an optimal controller based on optimal control theory. In the meantime, we propose an improved quantum particle swarm optimization (QPSO) algorithm by introducing an interval estimation mechanism into QPSO algorithm. Further, we make use of QPSO algorithm with interval estimation to optimize the proposed controller according to some performance indicator. Finally, by comparison, numerical simulations show that the controller not only can achieve the synchronization and secret communization well, but also can estimate the unknown parameters of the systems and bounds of external disturbance, which verify the effectiveness and applicability of the proposed control scheme.     

Adaptive sliding mode control of uncertain chaotic systems with input nonlinearity

This paper is concerned with the stabilization problem of uncertain chaotic systems with input nonlinearity. The slope parameters of this nonlinearity are unmeasured. A new sliding function is designed, then an adaptive sliding mode controller is established such that the trajectory of the system converges to the sliding surface in a finite time and finite-time reachability is theoretically proved. Using a virtual state feedback control technique, sufficient condition for the asymptotic stability of sliding mode dynamics is derived via linear matrix inequality (LMI). Then the results can be extended to uncertain chaotic systems with disturbances and adaptive sliding mode H _∞ controllers are designed. Finally, a simulation example is presented to show the validity and advantage of the proposed method.     

Application of Takagi–Sugeno fuzzy model to a class of chaotic synchronization and anti-synchronization

In this study, we investigate a class of chaotic synchronization and anti-synchronization with stochastic parameters. A controller is composed of a compensation controller and a fuzzy controller which is designed based on fractional stability theory. Three typical examples, including the synchronization between an integer-order Chen system and a fractional-order Lü system, the anti-synchronization of different 4D fractional-order hyperchaotic systems with non-identical orders, and the synchronization between a 3D integer-order chaotic system and a 4D fractional-order hyperchaos system, are presented to illustrate the effectiveness of the controller. The numerical simulation results and theoretical analysis both demonstrate the effectiveness of the proposed approach. Overall, this study presents new insights concerning the concepts of synchronization and anti-synchronization, synchronization and control, the relationship of fractional and integer order nonlinear systems.     

Stabilizing the unstable periodic orbits of a chaotic system using model independent adaptive time-delayed controller

In this paper we deal with the control of chaotic systems. Knowing that a chaotic attractor contains a myriad of unstable periodic orbits (UPO’s), the aim of our work is to stabilize some of the UPO’s embedded in the chaotic attractor and which have interesting characteristics. First, using the input-to-state linearization method in conjunction with a time-delayed state feedback, we design a control signal that can achieve stabilization. Next, an adaptive time-delayed state feedback is proposed which shows at once efficiency and simplicity and circumvents the construction complexity of the first controller. Finally, we propose a reduced order sliding mode observer to estimate the necessary states for the design of an adaptive time delayed state feedback controller. This last controller has one main advantage, it in fact achieves UPO stabilization without using the system model. The efficacy of the proposed methods is illustrated by numerical simulations onto Chua’s system.     

Robust chaos synchronization of fractional-order chaotic systems with unknown parameters and uncertain perturbations

Chaotic systems in practice are always influenced by some uncertainties and external disturbances. This paper investigates the problem of practical synchronization of fractional-order chaotic systems. Based on Lyapunov stability theory and a fractional-order differential inequality, a modified adaptive control scheme and adaptive laws of parameters are developed to robustly synchronize coupled fractional-order chaotic systems with unknown parameters and uncertain perturbations. This synchronization approach is simple, global and theoretically rigorous. Simulation results for two fractional-order chaotic systems are provided to illustrate the effectiveness of the proposed scheme.     

Using fractional-order integrator to control chaos in single-input chaotic systems

This paper deals with a fractional calculus based control strategy for chaos suppression in the 3D chaotic systems. It is assumed that the structure of the controlled chaotic system has only one control input. In the proposed strategy, the controller has three tuneable parameters and the control input is constructed as fractional-order integration of a linear combination of linearized model states. The tuning procedure is based on the stability theorems in the incommensurate fractional-order systems. To evaluate the performance of the proposed controller, the design method is applied to suppress chaotic oscillations in a 3D chaotic oscillator and in the Chen chaotic system.     

Design of fractional robust adaptive intelligent controller for uncertain fractional-order chaotic systems based on active control technique

In this paper, a robust fractional-order adaptive intelligent controller is proposed for stabilization of uncertain fractional-order chaotic systems. The intelligent neuro-fuzzy network is used to estimate unknown dynamics of system, while the neuro-fuzzy network parameters as well as the upper bounds of the model uncertainties, disturbances and approximation errors are adaptively estimated via separate adaptive rules. An SMC scheme, with a fractional-order sliding surface, is employed, as the controller to improve the velocity and performance of the proposed control system and to eliminate the unknown but bounded uncertainties, external disturbances and approximation errors. The Lyapunov stability theorem has been also employed to show the stability of the closed-loop system, robustness against uncertainties, external disturbances and approximation errors, while the control signal remains bounded. Explanatory examples and simulation results are given to confirm the effectiveness of the proposed procedure, which consent well with the analytical results.     

Modified projective synchronization of fractional-order chaotic systems via active sliding mode control

This paper is devoted to study the problem of modified projective synchronization of fractional-order chaotic system. Base on the stability theorems of fractional-order linear system, active sliding mode controller is proposed to synchronize two different fractional-order systems. Moreover, the controller is robust to the bounded noise. Numerical simulations are provided to show the effectiveness of the analytical results.     

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.     

Generalized passivity-based chaos synchronization

In this paper, a new passivity-based synchronization method for a general class of chaotic systems is proposed. Based on the Lyapunov theory and the linear matrix inequality （LMI） approach, the passivity-based controller is presented to make the synchronization error system not only passive but also asymptotically stable. The proposed controller can be obtained by solving a convex optimization problem represented by the LMI. Simulation studies for the Genesio-Tesi chaotic system and the Qi chaotic system are presented to demonstrate the effectiveness of the proposed scheme.     

Bursting generation mechanism in a three-dimensional autonomous system,chaos control,and synchronization in its fractional-order form

The bifurcation mechanism of bursting oscillations in a three-dimensional autonomous slow-fast Kingni et al. system (Nonlinear Dyn. 73, 1111–1123, 2013) and its fractional-order form are investigated in this paper. The stability analysis of the system is carried out assuming that the slow subsystem evolves on quasi-static state. It is reveaved that the bursting oscillations found in the system result from the system switching between the unstable and the stable states of the only equilibrium point of the fast subsystem. We refer this class of bursting to “source/bursting.” The coexistence of symmetrical bursting limit cycles and chaotic bursting attractors is observed. In addition, the fractional-order chaotic slow-fast system is studied. The lowest order of the commensurate form of this system to exhibit chaotic behavior is found to be 2.199. By tuning the commensurate fractional-order, the chaotic slow-fast system displays Chen- and Lorenz-like chaotic attractors, respectively. The stability analysis of the controlled fractional-order-form of the system to its equilibria is undertaken using Routh–Hurwitz conditions for fractional-order systems. Moreover, the synchronization of chaotic bursting oscillations in two identical fractional-order systems is numerically studied using the unidirectional linear error feedback coupling scheme. It is shown that the system can achieve synchronization for appropriate coupling strength. Furthermore, the effect of fractional derivatives orders on chaos control and synchronization is analyzed.     

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.