Suppression of chaotic behavior in horizontal platform systems based on an adaptive sliding mode control scheme

This work presents an adaptive sliding mode control scheme to elucidate the robust chaos suppression control of non-autonomous chaotic systems. The proposed control scheme utilizes extended systems to ensure that continuous control input is obtained in order to avoid chattering phenomenon as frequently in conventional sliding mode control systems. A switching surface is adopted to ensure the relative ease in stabilizing the extended error dynamics in the sliding mode. An adaptive sliding mode controller (ASMC) is then derived to guarantee the occurrence of the sliding motion, even when the chaotic horizontal platform system (HPS) is undergoing parametric uncertainties. Based on Lyapunov stability theorem, control laws are derived. In addition to guaranteeing that uncertain horizontal platform chaotic systems can be stabilized to a steady state, the proposed control scheme ensures asymptotically tracking of any desired trajectory. Furthermore, the numerical simulations verify the accuracy of the proposed control scheme, which is applicable to another chaotic system based on the same design scheme.     

Chaos control in AFM system using sliding mode control by backstepping design

This paper presents a robust algorithm to control the chaotic atomic force microscope system (AFMs) by backstepping design procedure. The proposed feedback controller is composed by a sliding mode control (SMC) and a backstepping feedback, so its implementation is quite simple and can be made on the basis of the measured signal. The developed control scheme allows chaos suppression despite uncertainties in the model as well as system external disturbances. The concept of extended system is used such that a continuous sliding mode control effort is generated using backstepping scheme. It is guaranteed that under the proposed control law, uncertain AFMs can asymptotically track target orbits. The converging speed of error states can be arbitrary turned by assigning the corresponding dynamics of the sliding surfaces. Numerical simulations demonstrate its advantages by stabilizing the unstable periodic orbits of the AFMs and this method can also be easily extended to elimination chaotic motion in any types of chaotic AFMs.     

An adaptive feedback control of linearizable chaotic systems

This paper proposes an adaptive feedback controller for a class of chaotic systems. This controller can be used for tracking a smooth orbit that can be a limit cycle or a chaotic orbit of another system. Based on Lyapunov approach, the adaptation law is determined to tune the controller gain vector in order to track a predetermined linearizing feedback control. To demonstrate the efficiency of the proposed scheme, two well-known chaotic systems namely Chua’s circuit and a Lur’e-like system are considered as illustrative examples.     

A new method for chaos control in communication systems

With the increasing needs of global communication, the improvement of secure communication is of vital importance. This study proposes a new scheme for establishing secure communication systems. The new scheme separates white Gaussian noises from the chaotic signals with modified Independent Component Analysis (ICA) and then controls each chaotic signal. This scheme is able to deal with white Gaussian noises in the natural world. However, the signals separated by traditional ICA shows opposite phase and unequal amplitude, making chaos control impossible. Our study proposed a modified ICA, which can calculate accurately the phase and amplitude and ensure control of the chaotic systems. The result indicates that our proposed system can successfully separate white Gaussian noise and stabilize all the chaotic signals.     

Anti-control of Hopf bifurcation in the new chaotic system with two stable node-foci

In order to further understand a complex 3D dynamical system showing strange chaotic attractors with two stable node-foci near Hopf bifurcation point, we propose nonlinear control scheme to the system and the controlled system, depending on five parameters, can exhibit codimension one, two, and three Hopf bifurcations in a much larger parameter regain. The control strategy used keeps the equilibrium structure of the chaotic system and can be applied to degenerate Hopf bifurcation at the desired location with preferred stability.     

Chaos control of Lorenz systems using adaptive controller with input saturation

This paper presents an adaptive sliding mode control scheme for Lorenz chaos subject to saturating input. The state of Lorenz system can be asymptotically driven to an equilibrium point in spite of the presence of input saturation and external disturbance using the proposed control scheme. Numerical simulations demonstrate the effectiveness of its application to chaotic system control. It also shows that the settling time will be decreased, if the saturation bound of control input is relaxed.     

Estimating parameters of chaotic systems under noise-induced synchronization

Kim et al. introduced in 2002 [Kim CM, Rim S, Kye WH. Sequential synchronization of chaotic systems with an application to communication. Phys Rev Lett 2002;88:014103] a hierarchically structured communication scheme based on sequential synchronization, a modification of noise-induced synchronization (NIS). We propose in this paper an approach that can estimate the parameters of chaotic systems under NIS. In this approach, a dimensionally-expanded parameter estimating system is first constructed according to the original chaotic system. By feeding chaotic transmitted signal and external driving signal, the parameter estimating system can be synchronized with the original chaotic system. Consequently, parameters would be estimated. Numerical simulation shows that this approach can estimate all the parameters of chaotic systems under two feeding modes, which implies the potential weakness of the chaotic communication scheme under NIS or sequential synchronization.     

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.     

Observer-based adaptive control of chaos in nonlinear discrete-time systems using time-delayed state feedback

A new approach to adaptive control of chaos in a class of nonlinear discrete-time-varying systems, using a delayed state feedback scheme, is presented. It is discussed that such systems can show chaotic behavior as their parameters change. A strategy is employed for on-line calculation of the Lyapunov exponents that will be used within an adaptive scheme that decides on the control effort to suppress the chaotic behavior once detected. The scheme is further augmented with a nonlinear observer for estimation of the states that are required by the controller but are hard to measure. Simulation results for chaotic control problem of Jin map are provided to show the effectiveness of the proposed scheme.     

Identify fully uncertain parameters and design controllers based on synchronization

For a chaotic system with specified structure, all unknown model parameters can be simultaneously identified by a simple combination of adaptive scheme and linear feedback. Furthermore, based on the Lyapunov stability theory, a sufficient condition for chaos synchronization is derived analytically, which guarantees that the system with fully uncertain parameters and the controlled system achieve chaos synchronization. Numerical simulations are presented for demonstration.     

Identify fully uncertain parameters and design controllers based on synchronization

For a chaotic system with specified structure, all unknown model parameters can be simultaneously identified by a simple combination of adaptive scheme and linear feedback. Furthermore, based on the Lyapunov stability theory, a sufficient condition for chaos synchronization is derived analytically, which guarantees that the system with fully uncertain parameters and the controlled system achieve chaos synchronization. Numerical simulations are presented for demonstration.     

Robust synchronization of drive–response chaotic systems via adaptive sliding mode control

A robust adaptive sliding control scheme is developed in this study to achieve synchronization for two identical chaotic systems in the presence of uncertain system parameters, external disturbances and nonlinear control inputs. An adaptation algorithm is given based on the Lyapunov stability theory. Using this adaptation technique to estimate the upper-bounds of parameter variation and external disturbance uncertainties, an adaptive sliding mode controller is then constructed without requiring the bounds of parameter and disturbance uncertainties to be known in advance. It is proven that the proposed adaptive sliding mode controller can maintain the existence of sliding mode in finite time in uncertain chaotic systems. Finally, numerical simulations are presented to show the effectiveness of the proposed control scheme.     

Adaptive anti control of chaos for robot manipulators with experimental evaluations

Roughly speaking, anti control of chaos consists in injecting a chaotic behavior to a system by means of a control scheme. This note introduces a new scheme to solve the anti control of chaos for robot manipulators. The proposed controller uses an adaption law to estimate the robot parameters on line. Thus, the controller does not require any knowledge of the physical parameters of the manipulator, such as masses, lengths of the links, moments of inertia, etc. The new scheme is based in the velocity field control paradigm, hence the specification of a chaotic system to define a desired velocity field is required. Experimental results in a two degrees-of-freedom direct-drive robot illustrate the practical feasibility of the introduced theory. In order to achieve anti control of chaos of our experimental system, two different chaotic attractors are used: the Genesio-Tesi system and a Jerk-type system. Results showed that the controller is able to inject the chaotic behavior to the robot while the robot parameters are estimated on line.     

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.     

Synchronization and anti-synchronization of Lu and Bhalekar–Gejji chaotic systems using nonlinear active control

This paper aims at synchronization and anti-synchronization between Lu chaotic system, a member of unified chaotic system, and recently developed Bhalekar–Gejji chaotic system, a system which cannot be derived from the member of unified chaotic system. These synchronization and anti-synchronization have been achieved by using nonlinear active control since the parameters of both the systems are known. Lyapunov stability theory is used and required condition is derived to ensure the stability of error dynamics. Controller is designed by using the sum of relevant variables in chaotic systems. Simulation results suggest that proposed scheme is working satisfactorily.     

Tracking control of nonlinear chaotic systems with dynamics uncertainties

This paper deals with the tracking control of nonlinear chaotic systems with dynamics uncertainties. A robust control strategy is developed to control a class of nonlinear chaotic systems with uncertainties. The proposed strategy is an input-output control scheme which comprises an uncertainty estimator and a linearizing-like feedback. The control time is explicitly computed. Computer simulations of the Duffing system are provided to verify the validity of the proposed control scheme.     

Chaos suppression on a class of uncertain nonlinear chaotic systems using an optimal H∞ adaptive PID controller

This paper introduces an optimal H_∞ adaptive PID (OHAPID) control scheme for a class of nonlinear chaotic system in the presence system uncertainties and external disturbances. Based on Lyapunov stability theory, it is shown that the proposed control scheme can guarantee the stability robustness of closed-loop system with H_∞ tracking performance. In the core of proposed controller, to achieve an optimal performance of OHAPID, the Particle Swarm Optimization (PSO) algorithm is utilized. To show the feasibility of proposed OHAPID controller, it is applied on the chaotic gyro system. Simulation results demonstrate that it has highly effective in providing an optimal performance.     

Control uncertain Genesio–Tesi chaotic system: Adaptive sliding mode approach

An adaptive sliding mode control (ASMC) technique is introduced in this paper for a chaotic dynamical system (Genesio–Tesi system). Using the sliding mode control technique, a sliding surface is determined and the control law is established. An adaptive sliding mode control law is derived to make the states of the Genesio–Tesi system asymptotically track and regulate the desired state. The designed control scheme can control the uncertain chaotic behaviors to a desired state without oscillating very fast and guarantee the property of asymptotical stability. An illustrative simulation result is given to demonstrate the effectiveness of the proposed adaptive sliding mode control design.     

A chaos secure communication scheme based on multiplication modulation

A secure spread spectrum communication scheme using multiplication modulation is proposed. The proposed system multiplies the message by chaotic signal. The scheme does not need to know the initial condition of the chaotic signals and the receiver is based on an extended Kalman filter (EKF). This signal encryption scheme lends itself to cheap implementation and can therefore be used effectively for ensuring security and privacy in commercial consumer electronics products. To illustrate the effectiveness of the proposed scheme, a numerical example based on Genesio-Tesi system and also Chen dynamical system is presented and the results are compared.     

Function vector synchronization based on fuzzy control for uncertain chaotic systems with dead‐zone nonlinearities

In this article, a fuzzy adaptive control scheme is designed to achieve a function vector synchronization behavior between two identical or different chaotic (or hyperchaotic) systems in the presence of unknown dynamic disturbances and input nonlinearities (dead‐zone and sector nonlinearities). This proposed synchronization scheme can be considered as a generalization of many existing projective synchronization schemes (namely the function projective synchronization, the modified projective synchronization, generalized projective synchronization, and so forth) in the sense that the master and slave outputs are assumed to be some general function vectors. To practically deal with the input nonlinearities, the adaptive fuzzy control system is designed in a variable‐structure framework. The fuzzy systems are used to appropriately approximate the uncertain nonlinear functions. A Lyapunov approach is used to prove the boundedness of all signals of the closed‐loop control system as well as the exponential convergence of the corresponding synchronization errors to an adjustable region. The synchronization between two identical systems (chaotic satellite systems) and two different systems (chaotic Chen and Lü systems) are taken as two illustrative examples to show the effectiveness of the proposed method. © 2015 Wiley Periodicals, Inc. Complexity 21: 234–249, 2016