Synchronization Of Uncertain Chaotic Systems Research Articles

Synchronization of uncertain chaotic systems with minimal parametric information

The chaotic synchronization is mainly hampered by uncertain system dynamics in terms of underlying parameters. To obtain accurate parameter estimates for the proper synchronization of uncertain chaotic systems (UCSs) through adaptive control, it is necessary to satisfy the persistence of excitation (PE) condition. Furthermore, the challenges imposed by the explosion of complexity in sequential stabilization procedures, slow performances, and lack of information in UCSs hinder the synchronization process through existing control techniques. Therefore, the contribution of this research is two-fold: First, a systematic stabilization approach with minimal calculations is proposed to achieve proper synchronization between chaotic drive and response systems. The key concept of the proposed theory is to obtain an invariant manifold by immersing the error dynamics (resulting from the mismatch between the drive and response systems) into lower-order target dynamics. From there, the control law for synchronizing these chaotic systems can be derived by defining passive outputs and associated storage functions. The proposed framework, which combines the concepts of immersion and passivity, is referred to as the Passivity and Immersion (P&I) approach. Second, unlike adaptive control, the synchronization of UCSs is made possible by selecting the appropriate target dynamics without any parameter estimation (or with minimal parametric information in certain cases). The simplicity and superiority of the proposed approach over existing techniques are preserved and demonstrated by application to a number of well-known chaotic systems.

Observer-based adaptive neural network control design for projective synchronization of uncertain chaotic systems

This paper addresses the design of an observer-based adaptive neural network chaos synchronization scheme for a general class of uncertain chaotic systems. The controller consists of an adaptive neural network control law and an extended state observer. The parameterization of the designed extended observer and the sufficient stability conditions are derived in the light of the singular perturbation theory. The extended observer is incorporated into the controller to reconstruct the synchronization error vector as well as to estimate the error between an ideal control law and the actual control. These estimated error signals are utilized in the adaptation mechanism of the neural network weight vector. In the presented chaos synchronization method, the knowledge of the models of the master–slave systems is not required and the controller only needs the projective synchronization error for its implementation. Numerical simulations are performed along with a comparative study to demonstrate the efficiency and effectiveness of the suggested chaos synchronization approach.

Synchronization of a Class of Chaotic Systems with Different Dimensions

In this paper, two scaling matrices are used to research the synchronization of different dimensional chaotic systems with unknown parameters. Firstly, the definition of synchronization of chaotic systems with different dimensions is introduced. Secondly, based on Lyapunov stability theorem and adaptive control method, an adaptive feedback hybrid controller and parameter adaptive laws are designed to realize synchronization of uncertain chaotic systems with different dimensions. Finally, three numerical experiments are carried out to verify the effectiveness of the proposed method.

Robust prescribed-time stabilization for fuzzy sliding mode synchronization for uncertain chaotic systems

This paper focuses on the problem of robust fuzzy sliding mode synchronization of uncertain nonlinear systems. A relevant fuzzy sliding mode control(FSMC) law is designed to force the state errors onto the specified surface. The prescribed-time boundedness (PTB) of the dynamical error is guaranteed by introducing the partitioning strategy. The chattering phenomenon, which is inherent to a conventional sliding mode control, is avoided by proper fuzzy tuning of the reaching law. The control law is deduced from the stability analysis of th closed-loop system in the sense of lyapunov. Furthermore, robustness issue is investigated and a prescribed-time property is obtained with sufficient conditions, established in terms of linear matrix inequalities (LMIs) using Lyapunov and cost functions. The results obtained show the effectiveness of the proposed control scheme and the reachability of the synchronization before the fixed time. The outcomes of this research are illustrated using computer simulations for the control problem of chaotic Chua’s system, in the presence of uncertainties, which clearly reveals that FSMC is better than SMC in terms of robustness. The present proposal provides a consistent procedure to design FSMC for a general class of nonlinear systems.

An application of adaptive synchronization of uncertain chaotic system in secure communication systems

ABSTRACT Chaotic secure communication via synchronization of chaotic systems has attracted increasing attention in recent years. This note aims to introduce the function projective synchronization (FPS) of coupled fractional-order (FO) complex chaotic Lü system in the master-slave system formalism up to the desired scaling function matrix in the presence of bounded model uncertainties. An efficient chaotic secure communication scheme is proposed using the chaotic masking approach. In order to increase the unpredictability of the information signal, the transmitted signals to the channel are considered as the FO derivative of the product of the information signal and the states of the chaotic system. At the receiver side, by employing the Lyapunov stability theorem, an appropriate adaptive FO control law is derived to synchronize the master and slave systems. After the robust synchronization is achieved, the information signal can be recovered accurately with high speed in the transient response. Finally, to verify the effectiveness of the theoretical results, two numerical simulations are performed and some comparisons based on the adaptation parameter of the control law are provided.

Adaptive fuzzy control design for synchronization of chaotic time-delay system

This paper focuses on an adaptive synchronization for a class of uncertain chaotic systems (CSs) with time delay. The chaotic systems are represented by Takagi–Sugeno (T-S) fuzzy model. In order to estimate the unknown parameters of the uncertain chaotic systems, we use advanced adaptive laws with smooth projection operator to solve it. In addition, we take the adaptive law as a part of Lyapunov function to make the chaotic systems stable. Different from the previous study of adaptive synchronization of uncertain chaotic systems, this paper fully considers the effect of time delay on uncertain chaotic systems. Finally, by constructing a suitable Lyapunov function (LKF) and using linear matrix inequality (LMI) theory, we can obtain the stability condition of closed-loop system. Furthermore, the simulation cases in this paper illustrate the robust performance of the error system is guaranteed under the uncertain dynamic equation modeling.

Robust finite-time composite nonlinear feedback control for synchronization of uncertain chaotic systems with nonlinearity and time-delay

This paper proposes a combination of finite-time robust-tracking theory and composite nonlinear feedback approach for the finite-time and high performance synchronization of the chaotic systems in the presence of the external disturbances, parametric uncertainties, Lipschitz nonlinearities and time delays. The composite nonlinear feedback control technique develops an accurate and high-performance response for the following of the master chaotic system and the finite time concept provides the convergence of the error signals to zero in the finite time. Therefore, in this work, we will develop a new finite time robust tracking and model following control approach based on the composite nonlinear feedback scheme. Using the Lyapunov stability approach, we have proved that the tracking errors of the uncertain chaotic system converge to the origin in the finite time. Moreover, a sufficient criterion is derived to guarantee the robust asymptotic stability of the synchronization error dynamics. Simulation results on Chua's chaotic system are presented to prove the performance of the suggested controller compared to the other technique.

Chaos synchronization of uncertain chaotic systems using composite nonlinear feedback based integral sliding mode control

This paper proposes a combination of composite nonlinear feedback and integral sliding mode techniques for fast and accurate chaos synchronization of uncertain chaotic systems with Lipschitz nonlinear functions, time-varying delays and disturbances. The composite nonlinear feedback method allows accurate following of the master chaotic system and the integral sliding mode control provides invariance property which rejects the perturbations and preserves the stability of the closed-loop system. Based on the Lyapunov- Krasovskii stability theory and linear matrix inequalities, a novel sufficient condition is offered for the chaos synchronization of uncertain chaotic systems. This method not only guarantees the robustness against perturbations and time-delays, but also eliminates reaching phase and avoids chattering problem. Simulation results demonstrate that the suggested procedure leads to a great control performance.

<inline-formula> <tex-math notation="LaTeX">$H_\infty$ </tex-math> </inline-formula>Synchronization for Uncertain Time-Delay Chaotic Systems With One-Sided Lipschitz Nonlinearity

This paper addresses $H_\infty $ synchronization for uncertain chaotic systems with one-sided Lipschitz nonlinearity under the output and intrinsic state delays. By utilizing the one-sided Lipschitz condition and quadratic inner boundedness, constructing an appropriate Lyapunov–Krasovskii (LKF), robust controller design conditions based on Lyapunov stability theory are derived for synchronization of chaotic systems under disturbances or perturbations bounded by $L_{2}$ norm. By introducing the delay-derivative limits and delay-interval bounds into LKF, the intrinsic state time-varying delay can be tackled by the delay-range-dependent strategy. Less conservative stability condition can be obtained by the further improved inequality of Jensen inequality and reciprocally convex approach, which can lead to the tighter upper bound for integral inequality. Numerical simulations are provided to verify the validity of the proposed methodology for synchronization of chaotic systems.

Robust finite-time synchronization of uncertain chaotic systems: application on Duffing-Holmes system and chaos gyros

ABSTRACTThis paper considers the finite-time synchronization of chaotic systems in the presence of model uncertainties and/or external disturbances. The synchronization happens between the two nonlinear master and slave systems. Control law is designed in such a way that the state variables of the slave system follow the state variables of the master system in the presence of uncertainties and external disturbances. In order to design a robust finite-time controller, first, a novel terminal sliding surface is proposed and its finite-time convergence to zero is analytically proved. Then a terminal sliding mode controller is designed which can conquer the uncertainties and guarantees the finite-time stability of the sliding motion equations. In this regard, a theorem is proposed and according to the Lyapunov approach it is proved that the synchronization happenes in finite-time. Additionally, in order to show the applicability of the proposed controller, it is applied on two practical systems, the Duffing–Holmes system and chaotic gyroscope system. Computer simulations verify the theoretical results and also display the effective performance of the proposed controller.

Bounded Scaling Function Projective Synchronization of Chaotic Systems with Adaptive Finite-Time Control

This paper investigates the bounded scaling function projective synchronization of uncertain chaotic systems using adaptive finite-time control. Based on finite-time control and inequality principle, the new adaptive finite-time controller is designed to achieve two chaotic systems scaling function projective synchronized, and uncertain parameters of chaotic systems are also identified. Moreover, in comparison with those of the existing scaling function synchronization, the given scaling function can be more complex bounded functions. Some numerical are also given to show the effectiveness of the proposed method.

Synchronization of uncertain chaotic systems via complete-adaptive-impulsive controls

The paper presents a novel synchronization scheme for uncertain chaotic systems via complete-adaptive-impulsive controls. The controllers are designed in the form of linear-error feedback coupling, but the control gains are completely adaptive. More details on minimizing interaction terms and accelerating synchronization process are revealed. The interaction terms can be selected on the largest invariant set minimally, but would be optimized corroboratively to promote the stabilization. The analytic expressions of parameter update laws for identifying uncertain parameters are derived from a reasonable truncation directly. A representative chaotic system is employed to show that the present scheme is not only a tactful way of synchronizing chaotic systems with uncertainties imposed on nonlinear terms, but a more radical approach on achieving synchronization with relatively moderate control gains than existed methods.

Prescribed performance synchronization for uncertain chaotic systems with input saturation based on neural networks

In this paper, a prescribed performance adaptive neural network synchronization is investigated for a class of unknown chaotic systems in the presence of input saturation and external unknown disturbances. A prescribed performance function is employed to transform the constraint problem of chaotic synchronization control error into the problem of guaranteeing the boundedness of the transformed error. By introducing the Gaussian error function, the input saturation is handled. A neural network-based synchronization control scheme is then developed. Under the developed synchronization control scheme, the synchronization of uncertain chaotic systems is achieved with different initial conditions. Numerical simulation results further demonstrate the effectiveness of the proposed synchronization control scheme for unknown chaotic systems subject to external unknown disturbances and input saturation.

Impulsive Control and Practical Generalized Synchronization of a Class of Uncertain Chaotic System with a Given Manifold

In this paper, we study practical generalized synchronization of uncertain chaotic system with a given manifold Y = H(X). We construct a class of the bi-directionally coupled chaotic systems with impulsive control, and demonstrate theoretically that the bi-coupled systems could realize practical generalized synchronization on the basis of stability theory of impulsive differential equations. Numerical simulations with super-chaotic system are provided to further demonstrate the effectiveness and generality of our approach.

Adaptive synchronization of uncertain chaotic systems with definite integration scaling function

The problem of adaptive synchronization for the uncertain chaotic systems with definite integration scaling function is investigated in this paper. A kind of the new scaling function is given, in comparison to those of the existing scaling function synchronization, the scaling function is definite integration in this paper, and the upper and lower limit of integration are adaptive. Finally, to illustrate the implementation of the proposed method, some numerical simulations are given.

Neural network-based synchronization of uncertain chaotic systems with unknown states

In this paper, synchronization of chaotic systems with unknown parameters and unmeasured states is investigated. Two nonidentical chaotic systems in the framework of a master and a slave are considered for synchronization. It is assumed that both systems have uncertain dynamics, and states of the slave system are not measured. To tackle this challenging synchronization problem, a novel neural network-based adaptive observer and an adaptive controller have been designed. Moreover, a neural network is utilized to approximate the unknown dynamics of the slave system. The proposed method imposes neither restrictive assumption nor constraint on the dynamics of the systems. Furthermore, the stability of the entire closed-loop system in the presence of the observer dynamics has been established. Finally, effectiveness of the proposed scheme is demonstrated via computer simulation.

Adaptive Fuzzy Synchronization for Uncertain Chaotic Systems with Different Dimensions and Disturbances

This paper focuses on the design of a novel adaptive fuzzy controller to synchronize the uncertain master–slave chaotic systems with different dimensions. In this paper, completely unknown nonlinear structures and disturbances exist in both the master and slave systems. For such systems, the definition of the generalized linear master–slave synchronization (GLMSS) is proposed. And different from the existing literatures, it is the unknown norm bound function of the uncertain structure to be approximated by the fuzzy logic system. Thus, regardless of the number of the nonlinear functions in the master and slave systems’ unknown structures, only two fuzzy logic systems are needed to synthesize the adaptive controller which guarantees the two systems achieving GLMSS. Three numerical examples are given to verify the effectiveness and feasibility of the fuzzy adaptive control scheme in this paper.

Two-mode Indirect Adaptive Control Approach for the Synchronization of Uncertain Chaotic Systems by the Use of a Hierarchical Interval Type-2 Fuzzy Neural Network

Two-mode adaptive controllers have two phases of operation: a learning phase and an operation phase. This paper presents a two-mode indirect adaptive control approach for the synchronization of chaotic systems, using a hierarchical interval type-2 fuzzy neural network (HT2FNN). Its contribution to the existing literature is the adaptation laws derived for the parameters of the membership functions, based on Lyapunov stability analysis. Since, in hierarchical case, each T2FNN has only two inputs, the computing of the derivatives is much simpler than the case in classical interval type-2 FNN. Moreover, a novel approach is presented for the compensation of the approximation error. The tuning of the parameters of the membership functions (MF) and the use of an interval type-2 FNN ensures that the estimation error is very small so that it can be negligible. Furthermore, the number of MF required is seen to be less than that needed with type-1 fuzzy sets. The simulation results confirm the efficacy of the proposed scheme in the synchronization of a uncertain nonidentical chaotic systems.

Adaptive synchronization on uncertain dynamics of high-order nonlinear multi-agent systems with partition of unity approach

An adaptive synchronization of uncertain chaotic system is presented using partition of unity method. First, the uncertainties of chaotic systems are approximate by the linear combination of partition of unity. Subsequently, the sliding mode adaptive controllers are proposed for synchronization of the uncertain chaotic systems. The proposed approach offers a systematic design procedure for adaptive synchronization of a class of uncertain chaotic system in the chaos research literature. We also illustrate examples of chaotic systems such as Chen and Lorenz chaotic systems to show the effectiveness of the approach.

A kind of binary scaling function projective lag synchronization of chaotic systems with stochastic perturbation

This paper investigates a kind of binary function projective lag synchronization of uncertain chaotic systems with stochastic perturbation. In comparison with those of the existing scaling function synchronization, we assume that the given scaling function can be the binary boundary function, even an n-ary boundary function. Based on the LaSalle-type invariance principle for stochastic differential equation, the adaptive control law is derived to make the state of two chaotic systems function projective lag synchronized. Some numerical is also given to show the effectiveness of the proposed method.