Synchronization and secure communication of chaotic systems via robust adaptive high-gain fuzzy observer

This paper proposes an alternative robust adaptive high-gain fuzzy observer design scheme and its application to synchronization and secure communication of chaotic systems. It is assumed that their states are immeasurable and their parameters are unknown. The structure of the proposed observer is represented by Takagi–Sugeno fuzzy model and has the integrator of the estimation error. It improves the performance of high-gain observer and makes the proposed observer robust against noisy measurements, uncertainties and parameter perturbations as well. Using Lyapunov stability theory, an adaptive law is derived to estimate the unknown parameters and the stability of the proposed observer is analyzed. Some simulation result of synchronization and secure communication of chaotic systems is given to present the validity of theoretical derivations and the performance of the proposed observer as an application.     

Observer-based synchronization of uncertain chaotic system and its application to secure communications

Within the drive-response configuration, this paper considers the synchronization of uncertain chaotic systems based on observers and chaos-based secure communication. Even if there are unknown disturbances and parameters in the drive system, a robust adaptive observer can be used as response system to realize chaotic synchronization. The proposed method is then applied to secure communication. The transmitter is constructed by injecting the information into the drive system with proper manner and one of the transmitting signal is the sum of one of the output and the information signal. The Lur’e chaotic system is considered as an illustrative example to demonstrate the effectiveness of the proposed approaches.     

Combining a feedback linearization controller with a disturbance observer to control a chaotic system under external excitation

This paper proposes a robust controller which combines a feedback linearization controller with a disturbance observer. This controller can suppress the chaotic motion of an unknown nonlinear system even though it receives an unknown external force. Two numerical simulations are performed to demonstrate the feasibility of the proposed method.     

Robust state estimation and fault diagnosis for uncertain hybrid nonlinear systems

Robust state estimation and fault diagnosis are challenging problems in the research of hybrid systems. In this paper, a novel robust hybrid observer is proposed for a class of uncertain hybrid nonlinear systems with unknown mode transition functions, model uncertainties and unknown disturbances. The observer consists of a mode observer for discrete mode estimation and a continuous observer for continuous state estimation. It is shown that the mode can be identified correctly and the continuous state estimation error is exponentially uniformly bounded. Robustness to unknown transition functions, model uncertainties and disturbances can be guaranteed by disturbance decoupling and selecting proper thresholds. The transition detectability and mode identifiability conditions are rigorously analyzed. Based on the robust hybrid observer, a robust fault diagnosis scheme is presented for faults modeled as discrete modes with unknown transition functions, and the analytical properties are investigated. Simulations of a hybrid three-tank system demonstrate that the proposed approach is effective.     

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.     

Robust state estimation and fault diagnosis for uncertain hybrid systems

Robust state estimation and fault diagnosis are challenging problems in the research into hybrid systems. In this paper a novel robust hybrid observer is proposed for a class of hybrid systems with unknown inputs and faults. Model uncertainties, disturbances and faults are represented as structured unknown inputs. The robust hybrid observer consists of a mode observer for mode identification and a continuous observer for continuous state estimation and mode transition detection. It is shown that the mode can be identified correctly and the continuous state estimation error is exponentially uniformly bounded. Robustness to model uncertainties and disturbances can be guaranteed for the hybrid observer by disturbance decoupling. Furthermore, the detectability and mode identifiability conditions are rigorously analyzed. On the basis of the robust hybrid observer, a robust fault detection and isolation scheme is presented also in the paper. Simulations of a hybrid four-tank system show the proposed approach is effective.     

Robust adaptive neural network synchronization controller design for a class of time delay uncertain chaotic systems

In this paper, a robust adaptive neural network synchronization controller is proposed for two chaotic systems with input time delay and uncertainty. The studied chaotic system may possess a wide class of nonlinear time-delayed input uncertainty. The radial basis function (RBF) neural network is used to approximate the unknown continuous bounded function item of the time delay uncertainty via appropriate weight value updated law. With the output of RBF neural network, a robust adaptive synchronization control scheme is presented for the time delay uncertain chaotic system. Finally, a simulation example is used to illustrate the effectiveness of the proposed synchronization control scheme.     

Robust intelligent sliding model control using recurrent cerebellar model articulation controller for uncertain nonlinear chaotic systems

In this paper, a robust intelligent sliding model control (RISMC) scheme using an adaptive recurrent cerebellar model articulation controller (RCMAC) is developed for a class of uncertain nonlinear chaotic systems. This RISMC system offers a design approach to drive the state trajectory to track a desired trajectory, and it is comprised of an adaptive RCMAC and a robust controller. The adaptive RCMAC is used to mimic an ideal sliding mode control (SMC) due to unknown system dynamics, and a robust controller is designed to recover the residual approximation error for guaranteeing the stable characteristic. Moreover, the Taylor linearization technique is employed to derive the linearized model of the RCMAC. The all adaptation laws of the RISMC system are derived based on the Lyapunov stability analysis and projection algorithm, so that the stability of the system can be guaranteed. Finally, the proposed RISMC system is applied to control a Van der Pol oscillator, a Genesio chaotic system and a Chua’s chaotic circuit. The effectiveness of the proposed control scheme is verified by some simulation results with unknown system dynamics and existence of external disturbance. In addition, the advantages of the proposed RISMC are indicated in comparison with a SMC system.     

Adaptive fuzzy sliding-mode control for multi-input multi-output chaotic systems

This paper describes an adaptive fuzzy sliding-mode control algorithm for controlling unknown or uncertain, multi-input multi-output (MIMO), possibly chaotic, dynamical systems. The control approach encompasses a fuzzy system and a robust controller. The fuzzy system is designed to mimic an ideal sliding-mode controller, and the robust controller compensates the difference between the fuzzy controller and the ideal one. The parameters of the fuzzy system, as well as the uncertainty bound of the robust controller, are tuned adaptively. The adaptive laws are derived in the Lyapunov sense to guarantee the asymptotic stability and tracking of the controlled system. The effectiveness of the proposed method is shown by applying it to some well-known chaotic systems.     

Adaptive feedback control for a class of chaotic systems

In this paper, the problem of control for a class of chaotic systems is considered. The nonlinear functions of chaotic systems are not necessarily to satisfy the Lipsichtz conditions, but bounded by a polynomial with the gains unknown. Employing adaptive method, the corresponding controller which renders the closed-loop system asymptotically stable is constructed. The designed controller is robust with respect to certain class of disturbances in the chaotic systems. Simulations on unified chaotic systems and Arneodo chaotic system are performed and the results verify the validity of the proposed techniques.     

An exponential polynomial observer for synchronization of chaotic systems

In this paper, we consider the synchronization problem via nonlinear observer design. A new exponential polynomial observer for a class of nonlinear oscillators is proposed, which is robust against output noises. A sufficient condition for synchronization is derived analytically with the help of Lyapunov stability theory. The proposed technique has been applied to synchronize chaotic systems (Rikitake and Rössler systems) by means of numerical simulation.     

Singular observer approach for chaotic synchronization and private communication

Based on the singular system observer, this paper proposes an effective approach for chaotic synchronization and private communication. When the useful information is modulated in a chaotic system and its dynamic equation is not available, we can consider the transmitted signal as an external system state. Then we can design a singular observer which has higher dimension. The advantage of such a design is that we can avoid using the derivation information of the transmitted signal. By adopting the singular system observer approach, the transmitted signal can be recovered successfully by the observer. Numerical simulations show the effectiveness of the proposed method.     

Chaos suppression of a class of unknown uncertain chaotic systems via single input

This paper deals with the design of a robust adaptive control scheme for chaos suppression of a class of chaotic systems. We assume that model uncertainties and external disturbances disturb the system’s dynamics. The bounds of both model uncertainties and external disturbances are assumed to be unknown in advance. Moreover, it is assumed that the nonlinear terms of the chaotic system dynamics are unknown bounded. Based on the global boundedness feature of the chaotic systems’ trajectories, a simple one input adaptive sliding mode control approach is proposed to suppress the chaos of the uncertain chaotic system. Furthermore, using a dynamical sliding manifold the discontinuous sign function in the control input is diverted to the first derivative of the control input to eliminate the chattering. Finally, the robustness of the proposed approach is mathematically proved and numerically illustrated.     

A recursive delayed output-feedback control to stabilize chaotic systems using linear-in-parameter neural networks

In this paper, a recursive delayed output-feedback control strategy is considered for stabilizing unstable periodic orbit of unknown nonlinear chaotic systems. An unknown nonlinearity is directly estimated by a linear-in-parameter neural network which is then used in an observer structure. An on-line modified back propagation algorithm with e-modification is used to update the weights of the network. The globally uniformly ultimately boundedness of overall closed-loop system response is analytically ensured using Razumikhin lemma. To verify the effectiveness of the proposed observer-based controller, a set of simulations is performed on a Rossler system in comparison with several previous methods.     

Synchronization of mechanical horizontal platform systems in finite time

The horizontal platform system (HPS) is a mechanical device that exhibits rich and chaotic dynamics. In this paper, the problem of finite-time synchronization of two non-autonomous chaotic HPSs is investigated. It is assumed that both drive and response systems are disturbed by model uncertainties, external disturbances and fully unknown parameters. Appropriate update laws are proposed to undertake the unknown parameters. Using the update laws and finite-time control theory, a robust adaptive controller is derived to synchronize the two uncertain HPSs in a given finite time. Subsequently, the effects of input nonlinearities are taken into account and a robust adaptive controller is introduced to synchronize the two uncertain HPSs within a finite time. The finite-time stability and convergence of the proposed schemes are analytically proved. Two illustrative examples are presented to show the robustness and applicability of the proposed adaptive finite-time control techniques.     

Unknown nonlinear chaotic gyros synchronization using adaptive fuzzy sliding mode control with unknown dead-zone input

In this paper, the problem of synchronizing two chaotic gyros in the presence of uncertainties, external disturbances and dead-zone nonlinearity in the control input is studied while the structure of the gyros, parameters of the dead-zone and the bounds of uncertainties and external disturbances are unknown. The dead-zone nonlinearity in the control input might cause the perturbed chaotic system to show unpredictable behavior. This is due to the high sensitivity of these systems to small changes in their parameters. Thereby, the effect of these issues should not be ignored in the control design for these systems. In order to eliminate the effects from the dead-zone nonlinearity, in this paper, a robust adaptive fuzzy sliding mode control scheme is proposed to overcome the synchronization problem for a class of unknown nonlinear chaotic gyros. The main contribution of our paper in comparison with other works that attempt to solve the problem of dead-zone in the synchronization of chaotic gyros is that we assume that the structure of the system, uncertainties, external disturbances, and dead-zone are fully unknown. Simulation results are provided to illustrate the effectiveness of the proposed method.     

Observer-based design of set-point tracking adaptive controllers for nonlinear chaotic systems

A gradient based approach for the design of set-point tracking adaptive controllers for nonlinear chaotic systems is presented. In this approach, Lyapunov exponents are used to select the controller gain. In the case of unknown or time varying chaotic plants, the Lyapunov exponents may vary during the plant operation. In this paper, an effective adaptive strategy is used for online identification of Lyapunov exponents and adaptive control of nonlinear chaotic plants. Also, a nonlinear observer for estimation of the states is proposed. Simulation results are provided to show the effectiveness of the proposed methodology.     

Synchronization of integer order and fractional order Chua’s systems using robust observer

In this paper we present a fractional order Chua’s circuit that behaves chaotically based on the use of a fractional order low pass filter. Next, an integer order robust observer will be designed to synchronize the fractional order Chua’s circuit as well as integer order Chua’s circuit with unknown nonlinearity. This method consists in designing a Luenberger like observer appended with an estimator of the unknown nonlinear function. The estimator assumes that the nonlinear function is slowly varying and that the observer converges quickly and uses the backward difference formula to approximate the state derivative. The efficiency of the proposed method is confirmed using numerical simulations.     

Adaptive fuzzy control for a class of chaotic systems with nonaffine inputs

A robust adaptive fuzzy control scheme is presented for a class of chaotic systems with nonaffine inputs, modeling uncertainties and external disturbances by using backstepping approach. Fuzzy logic systems (FLS) are employed to approximate the unknown parts of the virtual control and practical controls. The main characteristics of the scheme are that the number of the online adaptive parameters is no more than two times of the order of chaotic system and the tracking errors are guaranteed to be uniformly asymptotically stable with the aid of additional adaptive compensation terms. Lorenz system, Chen system, Lü system and Liu system are presented to illustrate the feasibility and effectiveness of the proposed control technique.     

Equivalent linear observer-based tracker for stochastic chaotic system with delays and disturbances

Based on the state-space self-tuning control methodology, aunified method for determining the equivalent linear observer-basedtracker for a stochastic chaotic system with input and statedelays and deterministic disturbances is developed. In thisapproach, an equivalent linear observer is obtained for theconcerned complex system, which may have unknown system parameters,system and measurement noises, and inaccessible system states.More precisely, an equivalent delay-free system is obtainedin the estimating process of the self-tuning control loop, andthen a linear controller and an observer are designed. The proposedmethod significantly simplifies the design and the implementationprocedures of the tracker.