Control of chaos in atomic force microscopes using delayed feedback based on entropy minimization

Active chaos control of a tapping mode atomic force microscope (AFM) model via delayed feedback method is presented. The feedback gain is obtained and adapted according to a minimum entropy (ME) algorithm. In this method, stabilizing an unstable fixed point of the system Poincare map is achieved by minimizing the entropy of points distribution on the Poincare section. Simulation results show the feasibility of the proposed method in applying the delayed feedback technique for chaos control of an AFM system.     

Delayed feedback control of chaotic spinning disk via minimum entropy approach

Chaos control of a spinning disk model via delayed feedback method is presented. The feedback gain is obtained and adapted according to a minimum entropy (ME) algorithm. In this method, stabilizing an unstable fixed point of the system Poincare map is achieved by minimizing the entropy of point distribution on the Poincare section. Simulation results show the feasibility of the proposed method in applying the delayed feedback technique for chaos control of spinning disks.     

Multi-variable control of chaos using PSO-based minimum entropy control

The minimum entropy (ME) control is a chaos control technique which causes chaotic behavior to vanish by stabilizing unstable periodic orbits of the system without using mathematical model of the system. In this technique some controller type, normally delayed feedback controller, with an adjustable parameter such as feedback gain is used. The adjustable parameter is determined such that the entropy of the system is minimized. Proposed in this paper is the PSO-based multi-variable ME control. In this technique two or more control parameters are adjusted concurrently either in a single or in multiple control inputs. Thus it is possible to use two or more feedback terms in the delayed feedback controller and adjust their gains. Also the multi-variable ME control can be used in multi-input systems. The minimizing engine in this technique is the particle swarm optimizer. Using online PSO, the PSO-based multi-variable ME control technique is applied to stabilize the 1-cycle fixed points of the Logistic map, the Hénon map, and the chaotic Duffing system. The results exhibit good effectiveness and performance of this controller.     

Controlling chaos in tapping mode atomic force microscopes using improved minimum entropy control

Minimum entropy control technique, an approach for controlling chaos without using the dynamical model of the system, can be improved by being combined with a nature-based optimization technique. In this paper, an ACO-based optimization algorithm is employed to minimize the entropy function of the chaotic system. The feedback gain of a delayed feedback controller is adjusted in the ACO algorithm. The effectiveness of the idea is investigated on suppressing chaos in the tapping-mode atomic force microscope equations. Results show a good performance. The PSO-based version of the minimum entropy control technique is also used to control the chaotic behavior of the AFM, and corresponding results are compared showing almost a same functionality for the two optimization algorithms of PSO and ACO as the minimizing engines of the minimum entropy strategy.     

Delayed state feedback and chaos control for time-periodic systems via a symbolic approach

This paper presents a symbolic method for a delayed state feedback controller (DSFC) design for linear time-periodic delay (LTPD) systems that are open loop unstable and its extension to incorporate regulation and tracking of nonlinear time-periodic delay (NTPD) systems exhibiting chaos. By using shifted Chebyshev polynomials, the closed loop monodromy matrix of the LTPD system (or the linearized error dynamics of the NTPD system) is obtained symbolically in terms of controller parameters. The symbolic closed loop monodromy matrix, which is a finite dimensional approximation of an infinite dimensional operator, is used in conjunction with the Routh–Hurwitz criterion to design a DSFC to asymptotically stabilize the unstable dynamic system. Two controllers designs are presented. The first design is a constant gain DSFC and the second one is a periodic gain DSFC. The periodic gain DSFC has a larger region of stability in the parameter space than the constant gain DSFC. The asymptotic stability of the LTPD system obtained by the proposed method is illustrated by asymptotically stabilizing an open loop unstable delayed Mathieu equation. Control of a chaotic nonlinear system to any desired periodic orbit is achieved by rendering asymptotic stability to the error dynamics system. To accommodate large initial conditions, an open loop controller is also designed. This open loop controller is used first to control the error trajectories close to zero states and then the DSFC is switched on to achieve asymptotic stability of error states and consequently tracking of the original system states. The methodology is illustrated by two examples.     

A simple method of chaos control for a class of chaotic discrete-time systems

In this paper, a simple method is proposed for chaos control for a class of discrete-time chaotic systems. The proposed method is built upon the state feedback control and the characteristic of ergodicity of chaos. The feedback gain matrix of the controller is designed using a simple criterion, so that control parameters can be selected via the pole placement technique of linear control theory. The new controller has a feature that it only uses the state variable for control and does not require the target equilibrium point in the feedback path. Moreover, the proposed control method cannot only overcome the so-called “odd eigenvalues number limitation” of delayed feedback control, but also control the chaotic systems to the specified equilibrium points. The effectiveness of the proposed method is demonstrated by a two-dimensional discrete-time chaotic system.     

Optimal feedback control of the forced van der Pol system

A simple feedback control strategy for chaotic systems is investigated using the forced van der Pol system as an example. The strategy regards chaos control as an optimization problem, where the maximum magnitude Floquet multiplier of a target unstable periodic orbit (UPO) is used as a cost function that needs to be minimized. Thus, the method obtains the optimal control gain in terms of the stability of the target UPO. This strategy was recently proposed for the proportional feedback control (PFC) method. Here, it is extended to the highly popular delayed feedback control (DFC) method. Since the DFC method treats the system as a delay-differential equation whose phase space is infinite-dimensional, the characteristic multipliers are found through a truncation in the number of delayed states. Control of a target UPO is achieved for several values of the forcing amplitude. We compare the DFC and PFC methods in terms of stability of the controlled orbit, steady state error and control effort.     

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.     

Controlling chaotic behaviour for spin generator and Rossler dynamical systems with feedback control

Different methods are proposed to control chaotic behaviour of the Nuclear Spin Generator (NSG) and Rossler continuous dynamical systems. Linear and nonlinear feedback control techniques are used to suppress chaos. The stabilization of unstable fixed point or unstable periodic solution of chaotic behaviour is achieved. The controlled system is stable under some conditions on the parameters of the system. Stability of the controlled system is determined by the Routh–Hurwitz criterion and Lyapunov direct method. Numerical simulation results are included to show the control process.     

Chaos control in lateral oscillations of spinning disks via nonlinear feedback

In this paper the problem of chaos control in single mode lateral oscillations of spinning disks is studied. At first, using the harmonic balance method, one of the periodic orbits of system is evaluated. Then proposing a nonlinear feedback strategy a control law is presented for chaos elimination by tracking the mentioned periodic solution. It is shown that although the system is not input-state feedback linearizable, by defining an output signal and using the input–output linearization method, the objective of complete periodic orbit tracking is achieved. The sufficient condition for this purpose is presented, and the performance of proposed method is examined by numerical simulation.     

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.     

Controlling chaos in a nonlinear pendulum using an extended time-delayed feedback control method

Chaos control is employed for the stabilization of unstable periodic orbits (UPOs) embedded in chaotic attractors. The extended time-delayed feedback control uses a continuous feedback loop incorporating information from previous states of the system in order to stabilize unstable orbits. This article deals with the chaos control of a nonlinear pendulum employing the extended time-delayed feedback control method. The control law leads to delay-differential equations (DDEs) that contain derivatives that depend on the solution of previous time instants. A fourth-order Runge–Kutta method with linear interpolation on the delayed variables is employed for numerical simulations of the DDEs and its initial function is estimated by a Taylor series expansion. During the learning stage, the UPOs are identified by the close-return method and control parameters are chosen for each desired UPO by defining situations where the largest Lyapunov exponent becomes negative. Analyses of a nonlinear pendulum are carried out by considering signals that are generated by numerical integration of the mathematical model using experimentally identified parameters. Results show the capability of the control procedure to stabilize UPOs of the dynamical system, highlighting some difficulties to achieve the stabilization of the desired orbit.     

一种双寡头垄断Cournot-Puu模型的混沌控制研究

基于非线性动力学的基本原理,研究了经济系统中的双寡头垄断Cournot-Puu模型及其混沌控制方法.Cournot-Puu模型具有双曲线形需求函数和彼此不同的不变边际成本,离散化的差分系统显示出其复杂的非线性、分岔和混沌行为.在此基础上,结合Cournot-Puu模型的基本特征,应用延迟反馈控制方法以及自适应控制方法对该系统的混沌行为进行了研究.在结合实际经济意义的条件下,对该模型的输出进行调整并实现混沌控制.     

On control and synchronization in chaotic and hyperchaotic systems via linear feedback control

This paper presents the control and synchronization of chaos by designing linear feedback controllers. The linear feedback control problem for nonlinear systems has been formulated under optimal control theory viewpoint. Asymptotic stability of the closed-loop nonlinear system is guaranteed by means of a Lyapunov function which can clearly be seen to be the solution of the Hamilton–Jacobi–Bellman equation thus guaranteeing both stability and optimality. The formulated theorem expresses explicitly the form of minimized functional and gives the sufficient conditions that allow using the linear feedback control for nonlinear system. The numerical simulations were provided in order to show the effectiveness of this method for the control of the chaotic Rössler system and synchronization of the hyperchaotic Rössler system.     

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.     

Stabilization of the second-order linear time-invariant control systems by a delayed feedback

We consider a static stabilization problem for a two-dimensional linear time-invariant control system with a delayed feedback. We obtain the necessary and sufficient conditions for the stabilizability of the system under consideration. The theorems proved in this paper show that such a delayed feedback approach is efficient in stabilizing the second-order linear systems.     

Minimum entropy control of chaos via online particle swarm optimization method

One of the recently developed approaches for control of chaos is the minimum entropy (ME) control technique. In this method an entropy function based on the Shannon definition, is defined for a chaotic system. The control action is designed such that the entropy as a cost function is minimized which results in more regular pattern of motion for the system trajectories. In this paper an online optimization technique using particle swarm optimization (PSO) method is developed to calculate the control action based on ME strategy. The method is examined on some standard chaotic maps with error feedback and delayed feedback forms. Considering the fact that the optimization is online, simulation results show very good effectiveness of the presented technique in controlling chaos.     

Chaos control and synchronization of a modified chaotic system

By replacing a quadratic nonlinear term in Lü system with a piecewise linear signum (PWL) function, a new simplified three-dimensional piecewise continuous autonomous system (a modified Lü system) is introduced. The qualitative properties of the modified Lü system are studied. Based on these properties, the feedback control law is applied to suppress chaos to one of the three equilibria. Several different synchronized methods, such as the active control, one way coupling by active control, and the adaptive active control are applied to achieve the state synchronization of two identical modified Lü systems. These results show that after the simplification, the modified Lü system can still keep the basic and typical nonlinear phenomena. Compared with the original Lü system, the modified Lü system has a lot of advantages, by which the modified Lü system can be more easily implemented by theoretical analysis, and more practicable made by secret communications.     

Implementation of dynamic programming for chaos control in discrete systems

In this paper the control of discrete chaotic systems by designing linear feedback controllers is presented. The linear feedback control problem for nonlinear systems has been formulated under the viewpoint of dynamic programming. For suppressing chaos with minimum control effort, the system is stabilized on its first order unstable fixed point (UFP). The presented method also could be employed to make any desired nth order fixed point of the system, stable. Two different methods for higher order UFPs stabilization are suggested. Afterwards, these methods are applied to two well-known chaotic discrete systems: the Logistic and the Henon Maps. For each of them, the first and second UFPs in their chaotic regions are stabilized and simulation results are provided for the demonstration of performance.     

State feedback and output feedback control of a class of nonlinear systems with delayed measurements

This paper is concerned with the problem of state feedback and output feedback control of a class of nonlinear systems with delayed measurements. This class of nonlinear systems is made up of continuous-time linear systems with nonlinear perturbations. The nonlinearity is assumed to satisfy a global Lipschitz condition and the time delay is assumed to be time-varying and have no restriction on its derivative. On the basis of the Lyapunov–Krasovskii approach, sufficient conditions for the existence of the state feedback controller and the output feedback controller are derived in terms of linear matrix inequalities. Methods of calculating the controller gain matrices are also presented. Two numerical examples are given to illustrate the effectiveness of the proposed methods.