Chaos control in AFM systems using nonlinear delayed feedback via sliding mode control

In this paper a nonlinear delayed feedback control is proposed to control chaos in an Atomic Force Microscope (AFM) system. The chaotic behavior of the system is suppressed by stabilizing one of its first-order Unstable Periodic Orbits (UPOs). At first, it is assumed that the system parameters are known, and a nonlinear delayed feedback control is designed to stabilize the UPO of the system. Then, in the presence of model parameter uncertainties, the proposed delayed feedback control law is modified via sliding mode scheme. The effectiveness of the presented methods is numerically investigated by stabilizing the unstable first-order periodic orbit of the AFM system. Simulation results show the high performance of the methods for chaos elimination in AFM systems.     

Chaos control of chaotic dynamical systems using backstepping design

This work presents chaos control of chaotic dynamical systems by using backstepping design method. This technique is applied to achieve chaos control for each of the dynamical systems Lorenz, Chen and Lü systems. Based on Lyapunov stability theory, control laws are derived. We used the same technique to enable stabilization of chaotic motion to a steady state as well as tracking of any desired trajectory to be achieved in a systematic way. Numerical simulations are shown to verify the results.     

Chaos synchronization in noisy environment using nonlinear filtering and sliding mode control

This paper presents an algorithm for synchronizing two different chaotic systems, using a combination of the extended Kalman filter and the sliding mode controller. It is assumed that the drive chaotic system has a random excitation with a stochastically chaotic behavior. Two different cases are considered in this study. At first it is assumed that all state variables of the drive system are available, i.e. complete state measurement, and a sliding mode controller is designed for synchronization. For the second case, it is assumed that the output of the drive system does not contain the whole state variables of the drive system, and it is also affected by some random noise. By combination of extended Kalman filter and the sliding mode control, a synchronizing control law is proposed. As a case study, the presented algorithm is applied to the Lur’e-Genesio chaotic systems as the drive-response dynamic systems. Simulation results show the good performance of the algorithm in synchronizing the chaotic systems in presence of noisy environment.     

High-order sliding mode controller with backstepping design for aeroelastic systems

A nonlinear system for controlling flutter in an aeroelastic system is proposed. The dynamic model describes the plunge and pitch motion of a wing. Interacting nonlinear forces such as structural and aerodynamic forces cause destabilizing phenomena such as flutter and limit cycle oscillation on the wing. Aeroelastic models have a wing section with only a single trailing-edge control surface for suppressing limit cycle oscillation. When modeling a single control surface, the controller design can achieve trajectory control of either plunge displacement or pitch angle, but not both, and internal dynamics describe the residual motion in closed-loop systems. Internal dynamics of aeroelasticity depend on model parameters such as freestream velocity and spring constant. Since single control surfaces have limited effectiveness, this study used leading- and trailing-edge control surfaces to improve control of limit-cycle oscillation. Moreover, two control surfaces were used to provide sufficient flexibility to shape both the plunge and the pitch responses. In this study, high order sliding mode control (HOSMC) with backstepping design achieved system stability and eliminated limit cycle phenomenon. Compared to the conventional sliding mode control design, the proposed control law not only preserves system robustness, but also avoids chatter phenomenon. Simulation results show that the proposed controller effectively regulate the response to origin in state space even under saturated controller input.     

Chaos synchronization of coupled neurons via adaptive sliding mode control

In this paper, an adaptive neural network (NN) sliding mode controller (SMC) is proposed to realize the chaos synchronization of two gap junction coupled FitzHugh–Nagumo (FHN) neurons under external electrical stimulation. The controller consists of a radial basis function (RBF) NN and an SMC. After the RBFNN approximating the uncertain nonlinear part of the error dynamical system, the SMC realizes the desired control property regardless of the existence of the approximation errors and external disturbances. The weights of the NN are tuned online based on the sliding mode reaching law. According to the Lyapunov stability theory, the stability of the closed error system is guaranteed. The control scheme is robust to the uncertainties such as approximate error, ionic channel noise and external disturbances. Chaos synchronization is obtained by the proper choice of the control parameters. The simulation results demonstrate the effectiveness of the proposed control method.     

An adaptive sliding mode backstepping control for the mobile manipulator with nonholonomic constraints

To solve disturbances, nonlinearity, nonholonomic constraints and dynamic coupling between the platform and its mounted robot manipulator, an adaptive sliding mode controller based on the backstepping method applied to the robust trajectory tracking of the wheeled mobile manipulator is described in this article. The control algorithm rests on adopting the backstepping method to improve the global ultimate asymptotic stability and applying the sliding mode control to obtain high response and invariability to uncertainties. According to the Lyapunov stability criterion, the wheeled mobile manipulator is divided into several stabilizing subsystems, and an adaptive law is designed to estimate the general nondeterminacy, which make the controller be capable to drive the trajectory tracking error of the mobile manipulator to converge to zero even in the presence of perturbations and mathematical model errors. We compare our controller with the robust neural network based algorithm in nonholonomic constraints and uncertainties, and simulation results prove the effectivity and feasibility of the proposed method in the trajectory tracking of the wheeled mobile manipulator.     

Nonlinear and chaos control of a micro-electro-mechanical system by using second-order fast terminal sliding mode control

In this paper, a novel second-order fast terminal sliding mode control (SFTSMC) scheme is proposed to suppress the chaotic motion of a micro-mechanical resonator with system uncertainty and external disturbance. To obtain a better disturbance rejection property, a fuzzy logic system is introduced to estimate the upper boundary of the sum of system uncertainty and external disturbance. Moreover, we employ the finite-time technique to obtain the properties of fast response and high precision. Finally, numerical simulations demonstrate the effectiveness of the proposed control scheme.     

Synchronizing the noise-perturbed Genesio chaotic system by sliding mode control

This paper investigates the chaos synchronization between Genesio chaotic systems with noise perturbation. It is proved theoretically that the synchronization between such noise-perturbed systems can be implemented by choosing a suitable sliding mode surface and designing a sliding mode controller. Numerical simulations show the effectiveness of the theoretical analysis. This proposed method is important because it can be applied to many other chaotic systems.     

Chaos control of 4D chaotic systems using recursive backstepping nonlinear controller

This paper examines chaos control of two four-dimensional chaotic systems, namely: the Lorenz–Stenflo (LS) system that models low-frequency short-wavelength gravity waves and a new four-dimensional chaotic system (Qi systems), containing three cross products. The control analysis is based on recursive backstepping design technique and it is shown to be effective for the 4D systems considered. Numerical simulations are also presented.     

Controlling uncertain Lü system using backstepping design

In this paper, we discussed how to control Lü system with unknown parameters. Firstly we designed an observer to identify the unknown parameter of Lü system, then we used backstepping design method to control the system, and track any desired trajectory by the same way. At the same time we gave the numerical simulation for the results we had gained.     

Adaptive backstepping sliding mode control for chaos synchronization of two coupled neurons in the external electrical stimulation

In this paper, a robust control system combining backstepping and sliding mode control techniques is used to realize the synchronization of two gap junction coupled chaotic FitzHugh-Nagumo (FHN) neurons in the external electrical stimulation. A backstepping sliding mode approach is applied firstly to compensate the uncertainty which occur in the control system. However, the bound of uncertainty is necessary in the design of the backstepping sliding mode controller. To relax the requirement for the bound of uncertainty, an adaptive backstepping sliding mode controller with a simple adaptive law to adapt the uncertainty in real time is designed. The adaptive backstepping sliding mode control system is robust for time-varying external disturbances. The simulation results demonstrate the effectiveness of the control scheme.     

A nonlinear programming approach for the sliding mode control design

We treat the sliding mode control problem by formulating it as a two phase problem consisting of reaching and sliding phases. We show that such a problem can be formulated as bicriteria nonlinear programming problem by associating each of these phases with an appropriate objective function and constraints. We then scalarize this problem by taking weighted sum of these objective functions. We show that by solving a sequence of such formulated nonlinear programming problems it is possible to obtain sliding mode controller feedback coefficients which yield a competitive performance throughout the control. We solve the nonlinear programming problems so constructed by using the modified subgradient method which does not require any convexity and differentiability assumptions. We illustrate validity of our approach by generating a sliding mode control input function for stabilization of an inverted pendulum.     

Iterative sliding mode control

Iterative Learning Control (ILC) methods are described and applied ever-increasingly as powerful tools to control dynamics nowadays.

ILC’s methods in most studies are described as based on repetitive process from the beginning to the end of process or as a kind of repetitive control.

Our newly designed controllers based on a particular case of iterative learning control radically differ from conventional methods in attempting to stabilize a class of non linear systems.

In this paper two kinds of ILC method are introduced in two separate sections. In the first, our newly designed method satisfies the condition of a Lyapunov stability theorem in a class of non linear systems in which their structures have the Lipschitz property. In the second, by freezing the time and moving to a new virtual axis, called the index axis, this newly designed method tries to find the best value for control at this time step and can be used in two modes, on-line and off-line.

In both methods, by satisfying the convergence condition of our designed ILC, closed loop stability is obtained automatically.     

Synchronizing different chaotic systems using active sliding mode control

An active sliding mode controller is designed to synchronize three pairs of different chaotic systems (Lorenz–Chen, Chen–Lü, and Lü–Lorenz) in drive–response structure. It is assumed that the system parameters are known. The closed loop error dynamics depend on the linear part of the response systems and parameters of the controller. Therefore, the synchronization rate can be adjusted through these parameters. Analysis of the stability for the proposed method is derived based on the Lyapunov stability theorem. Finally, numerical results are presented to show the effectiveness of the proposed control technique.     

Synchronization of nuclear spin generator system using active backstepping design

In this letter, an active backstepping design control method is presented for synchronizing two identical nuclear spin generator system with each other. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A generic design methodology for sliding mode control of switched systems

The aim of this paper is to present a generic methodology to design sliding mode controllers for multivariable switched systems affine in control such as dc–dc power converters. An original formulation of the so-called reachability condition, suitable for this class of systems, is established. Based on the choice of a Lyapunov-like function and parameterized by a single weighting matrix, it allows several kinds of control strategies to be derived, namely conventional piecewise continuous strategies as well as discrete (Boolean) strategies. Its application to the important subclass of linear time invariant systems is investigated more specifically. In the Boolean case, the present approach is also compared to another hybrid one called the stabilizing approach. Eventually, its efficiency as a design methodology, as well as the performance of the resulting control, are shown by simulating it on non-trivial examples of power converters.     

Optimal sliding mode control of single degree-of-freedom hysteretic structural system

This paper addresses the controller design problem of a nonlinear single degree-of-freedom structural system excited by the earthquake. Bouc–Wen model, as an efficient hysteresis modeling method, is used to model the system nonlinearity. Sliding mode control (SMC), due to its robustness in dealing with uncertainty, is utilized as the main control strategy. An optimal sliding surface is presented which minimizes the displacement and control force in terms of a quadratic cost function. Two numerical examples are given to illustrate the effectiveness of the proposed strategy subject to three earthquakes of El-Centro, Rinaldi and Kobe. Simulation results show a significant and considerable reduction in structural response and indicate that the performance of suggested optimal SMC strategy is remarkable.     

Adaptive fuzzy sliding mode control of the cart-pole underactuated system

In this work, an intelligent control scheme is proposed for the stabilization of the cart-pole underactuated system. The adopted approach is primarily based on a smooth sliding mode controller, but an adaptive fuzzy inference system is embedded within the boundary layer in order to improve the control performance. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Chaos synchronization using single variable feedback based on backstepping method

In recent years, backstepping method has been developed in the field of nonlinear control, such as controller, observer and output regulation. In this paper, an effective backstepping design is applied to chaos synchronization. There are some advantages in this method for synchronizing chaotic systems, such as (a) the synchronization error is exponential convergent; (b) only one variable information of the master system is needed; (c) it presents a systematic procedure for selecting a proper controller. Numerical simulations for the Chua's circuit and the Rössler system demonstrate that this method is very effective.     

Self-organizing adaptive wavelet CMAC backstepping control system design for nonlinear chaotic systems

A novel self-organizing wavelet cerebellar model articulation controller (CMAC) is proposed. This self-organizing wavelet CMAC (SOWC) can be viewed as a generalization of a self-organizing neural network and of a conventional CMAC, and it has better generalizing, faster learning and faster recall than a self-organizing neural network and a conventional CMAC. The proposed SOWC has the advantages of structure learning and parameter learning simultaneously. The structure learning possesses the ability of on-line generation and elimination of layers to achieve optimal wavelet CMAC structure, and the parameter learning can adjust the interconnection weights of wavelet CMAC to achieve favorable approximation performance. Then a SOWC backstepping (SOWCB) control system is proposed for the nonlinear chaotic systems. This SOWCB control system is composed of a SOWC and a fuzzy compensator. The SOWC is used to mimic an ideal backstepping controller and the fuzzy compensator is designed to dispel the residual of approximation errors between the ideal backstepping controller and the SOWC. Moreover, the parameters of the SAWCB control system are on-line tuned by the derived adaptive laws in the Lyapunov sense, so that the stability of the feedback control system can be guaranteed. Finally, two application examples, a Duffing–Holmes chaotic system and a gyro chaotic system, are used to demonstrate the effectiveness of the proposed control method. The simulation results show that the proposed SAWCB control system can achieve favorable control performance and has better tracking performance than a fuzzy neural network control system and a conventional adaptive CMAC.