混沌记忆系统的自适应反馈控制和轨迹跟踪控制

研究了混沌记忆系统的自适应反馈控制和基于反馈线性化的轨迹跟踪控制问题.首先,通过绘制系统的时域波形图和混沌吸引子图验证系统的复杂的动力学行为;然后,分别应用自适应反馈控制方法和基于反馈线性化的轨迹跟踪控制方法设计控制器,对系统施加控制;最后,通过数值仿真验证控制器的有效性.     

Stabilizing periodic orbits of fractional order chaotic systems via linear feedback theory

In this paper stabilizing unstable periodic orbits (UPO) in a chaotic fractional order system is studied. Firstly, a technique for finding unstable periodic orbits in chaotic fractional order systems is stated. Then by applying this technique to the fractional van der Pol and fractional Duffing systems as two demonstrative examples, their unstable periodic orbits are found. After that, a method is presented for stabilization of the discovered UPOs based on the theories of stability of linear integer order and fractional order systems. Finally, based on the proposed idea a linear feedback controller is applied to the systems and simulations are done for demonstration of controller performance.     

Feedback control and adaptive control of the energy resource chaotic system

In this paper, the problem of control for the energy resource chaotic system is considered. Two different method of control, feedback control (include linear feedback control, non-autonomous feedback control) and adaptive control methods are used to suppress chaos to unstable equilibrium or unstable periodic orbits. The Routh–Hurwitz criteria and Lyapunov direct method are used to study the conditions of the asymptotic stability of the steady states of the controlled system. The designed adaptive controller is robust with respect to certain class of disturbances in the energy resource chaotic system. Numerical simulations are presented to show these results.     

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.     

Control of discrete-time chaotic systems via feedback linearization

An approach for controlling discrete-time chaotic systems by feedback linearization is proposed. This method can not only stabilize unstable periodic orbits embedded in a strange attractor, but also can be applied even if the real trajectory is far from the target one. A Hénon map with different operation conditions is implemented to demonstrate the feasibility of the proposed method.     

Chaos control using small-amplitude damping signals of the extended Duffing equation

This paper examines the application of a simple feedback controller to eliminate the chaotic behavior in a controlled extended Duffing system. The main idea is to regulate the chaotic motion of an extended Duffing system around less complex attractors, such as equilibrium points and periodic orbits. The proposed feedback controller is composed by a high-pass filter and a saturator, so its implementation is quite simple and can be made on the basis of measured signals. The affectivity of the proposed feedback control strategy is illustrated by means of numerical simulations.     

Complex dynamic behaviour of an economic cycle model

In this paper, we investigate the dynamics of a nonlinear economic cycle model. The necessary and sufficient conditions are given to guarantee the existence and stability of the fixed point. It is also shown that the system undergoes a Neimark–Sacker bifurcation by using center manifold theorem and bifurcation theory. Furthermore, Marotto’s chaos is proved when certain conditions are satisfied. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviour, such as the period-10, -16, -20 orbits, attracting invariant cycles, quasi-periodic orbits, 10-coexisting chaotic attractors, and boundary crisis. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.     

Global finite-time stabilization of switched high-order rational power nonlinear systems

This works is concerned with the finite-time optimal stabilization problem for a class of switched non-strict-feedback nonlinear systems whose powers are possibly different positive odd rational numbers in the sense the powers of each subsystem might differ from others. It is well known that high-order nonlinear systems are intrinsically challenging as feedback linearization and backstepping method successfully developed for low-order systems fail to work. To this purpose, the nested saturation homogeneous controller is constructively devised to achieve global finite-time stability. Furthermore, the corresponding design parameters are optimized by minimizing a well-defined cost function, and thus an optimal controller being independent of switching signals is obtained. Simulation results are eventually provided to validate the effectiveness of the proposed control scheme.     

Output-feedback synchronization of chaotic systems based on sum-of-squares approach

This paper investigates the synchronization of chaotic systems using an output feedback polynomial controller. As only output system states are considered, it makes the controller design and system analysis more challenging compared to the full-state feedback control schemes. To study the system stability and synthesize the output feedback polynomial controller, Lyapunov stability theory is employed. Sufficient stability conditions are derived in terms of sum of squares (SOS) conditions to guarantee the system stability and aid the controller synthesis. A genetic algorithm-based SOS technique is proposed to find the solution to the SOS conditions and the parameter values of the output feedback polynomial controller. A simulation example is employed to illustrate the effectiveness of the proposed approach.     

Stability analysis of fractional order neutral-type systems considering time varying delays,nonlinear perturbations,and input saturation

This article investigates the robust stability of fractional order neutral-type systems involving nonlinear perturbations and time varying delays in the presence of input saturation. Design criteria, expressed in terms of linear matrix inequalities, are derived with the aid of the Lyapunov Krasovskii functional for the state feedback controller. Based on the cone complementarity linearization method, an optimization problem is also formulated for finding the controller gains subject to maximizing the domain of attraction. The main results are confirmed by numerical simulations.     

Complex dynamic behaviors of a discrete-time predator-prey system

The dynamics of a discrete-time predator-prey system is investigated in detail in this paper. It is shown that the system undergoes flip bifurcation and Hopf bifurcation by using center manifold theorem and bifurcation theory. Furthermore, Marotto''s chaos is proved when some certain conditions are satisfied. Numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as the period-6, 7, 8, 10, 14, 18, 24, 36, 50 orbits, attracting invariant cycles, quasi-periodic orbits, nice chaotic behaviors which appear and disappear suddenly, coexisting chaotic attractors, etc. These results reveal far richer dynamics of the discrete-time predator-prey system. Specifically, we have stabilized the chaotic orbits at an unstable fixed point using the feedback control method.     

混沌时滞随机神经网络同步控制

研究了一类混沌时滞随机神经网络同步控制问题．采用更具一般性的时滞反馈控制器,通过巧妙地构造Lyapunov数,分别得到了均方指数同步和均方渐近同步两个判别准则．仿真例子表明,新准则是有效的．     

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.     

Synchronization of a four-dimensional energy resource system via linear control

Synchronization of a four-dimensional energy resource system is investigated. Four linear control schemes are proposed to synchronize energy resource chaotic system via the back-stepping method. We use simpler controllers to realize a global asymptotical synchronization. In the first three schemes, the sufficient conditions for achieving synchronization of two identical energy resource systems using linear feedback control are derived by using Lyapunov stability theorem. In the fourth scheme, the synchronization condition is obtained by numerical method, in which only one state variable controller is contained. Finally, four numerical simulation examples are performed to verify these results.     

一类超混沌离散系统的控制

研究一类超混沌离散系统的控制问题.基于局部线性化建立了时变线性反馈控制律.采用Liapunov直接法估计了控制律可以有效作用的邻域范围.分别给出了应用该控制律解决不稳定周期轨道的镇定问题和任意给定周期轨道的追踪问题的算例.     

Synchronization and anti-synchronization of chaotic oscillators under input saturation

This paper addresses the design of simple state feedback controllers for synchronization and anti-synchronization of chaotic oscillators under input saturation and disturbance. By employing sector condition, linear matrix inequality (LMI)-based sufficient conditions are derived to design (global or local) controllers for chaos synchronization. The proposed local synchronization strategy guarantees a region of stability in terms of difference between states of the master–slave systems. This region of stability can be enlarged by means of an LMI-based optimization algorithm, through which asymptotic synchronization of chaotic oscillators can be ensured for a large difference in their initial conditions. Further, a novel LMI-based robust control strategy is developed, for local synchronization of input-constrained chaotic oscillators, by providing an upper bound on synchronization error in terms of disturbance and initial conditions of chaotic systems. Moreover, the proposed robust state feedback control methodology is modified to provide an inaugural treatment for robust anti-synchronization of chaotic systems under input saturation and disturbance. The results of the proposed methodologies are verified through numerical simulations for synchronization and anti-synchronization of the master–slave chaotic Chua’s circuits under input saturation.     

Synchronization of non-autonomous chaotic systems with time-varying delay via delayed feedback control

In this paper, we investigate the synchronization of non-autonomous chaotic systems with time-varying delay via delayed feedback control. Using a combination of Riccati differential equation approach, Lyapunov-Krasovskii functional, inequality techniques, some sufficient conditions for exponentially stability of the error system are formulated in form of a solution to the standard Riccati differential equation. The designed controller ensures that the synchronization of non-autonomous chaotic systems are proposed via delayed feedback control and intermittent linear state delayed feedback control. Numerical simulations are presented to illustrate the effectiveness of these synchronization criteria.     

Chaos synchronization of an energy resource system based on linear control

Synchronization of an energy resource system is investigated. Three linear control schemes are proposed to synchronize a chaotic energy resource system via the back-stepping method. This can be viewed as an improvement to the existing results of Tian et al. (2006) [14]. Because we use simpler controllers to realize a global asymptotical synchronization. In the first two schemes, the sufficient conditions for achieving synchronization of two identical energy resource systems using linear feedback control are derived by using Lyapunov stability theorem. In the third scheme, the synchronization condition is obtained by numerical method, in which only one state variable controller is contained. Finally, three numerical simulation examples are performed to verify these results.     

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.     

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.