Pull-in instability of a typical electrostatic MEMS resonator and its control by delayed feedback

Pull-in instability of the electrostatic microstructures is a common undesirable phenomenon which implies the loss of reliability of micro-electromechanical systems. Therefore, it is necessary to understand its mechanism and then reduce the phenomenon. In this work, pull-in instability of a typical electrostatic MEMS resonator is discussed in detail. Delayed position feedback and delayed velocity feedback are introduced to suppress pull-in instability, respectively. The thresholds of AC voltage for pull-in instability in the initial system and the controlled systems are obtained analytically by the Melnikov method. The theoretical predictions are in good agreement with the numerical results. It follows that pull-in instability of the MEMS resonator can be ascribed to the homoclinic bifurcation inducing by the AC and DC load. Furthermore, it is found that the controllers are both good strategies to reduce pull-in instability when their gains are positive. The delayed position feedback controller can work well only when the delay is very short and AC voltage is low, while the delayed velocity feedback will be effective under a much higher AC voltage and a wider delay range.     

混沌系统延迟反馈控制的理论与实验研究

综述了近年来控制混沌的延迟反馈控制技术------DFC控制的相关进展,总结了延迟反馈控制在不动点和不稳定周期轨道镇定方面的局限性和可控性研究的理论成果,介绍了延迟反馈控制在电子线路和磁弹性梁混沌控制方面的实验,并对延迟反馈控制技术未来的研究方向和发展前景进行了预测和展望。      

碰摩转子映射系统的延迟反馈混沌控制

将转子的碰摩映射地擦边轨道附近进行局部化,并通过实验数据去拟合局部映射,然后采用变量延迟反馈控制法对该系统进行控制,通过选取合适的控制增益,将转子系统的碰磨运动镇定到周期1转道上,从而实现对碰磨转子系统混沌运动的控制。     

Stability, bifurcation and chaos of a delayed oscillator with negative damping and delayed feedback control

This paper presents a detailed analysis on the dynamics of a delayed oscillator with negative damping and delayed feedback control. Firstly, a linear stability analysis for the trivial equilibrium is given. Then, the direction of Hopf bifurcation and stability of periodic solutions bifurcating from trivial equilibrium are determined by using the normal form theory and center manifold theorem. It shows that with properly chosen delay and gain in the delayed feedback path, this controlled delayed system may have stable equilibrium, or periodic solutions, or quasi-periodic solutions, or coexisting stable solutions. In addition, the controlled system may exhibit period-doubling bifurcation which eventually leads to chaos. Finally, some new interesting phenomena, such as the coexistence of periodic orbits and chaotic attractors, have been observed. The results indicate that delayed feedback control can make systems with state delay produce more complicated dynamics.     

Modeling,nonlinear dynamic analysis and control of fractional PMSG of wind turbine

This study aims to reveal the laws of the relationship between fractional-order system and integer-order system. Meanwhile, delayed feedback control is introduced to control the fractional-order PMSG (permanent magnet synchronous generator) model of a wind turbine. First, the fractional-order mathematical model of PMSG is established. Next, numerical simulations under different system orders are given and the system dynamic behaviors are analyzed in detail. Then, the delayed feedback control method is introduced to control the fractional-order PMSG and the control results when different parameters vary are analyzed. Complex dynamics are presented and some interesting phenomena are discovered. It is found that the system order influences the dynamics of the system in many aspects such as chaos pattern, bifurcation behavior, period window, shape and size of strange attractor. The delayed time, feedback gain, feedback limitation, system order can obviously influence the control result except the initial state of the system. Moreover, the feedback limitation has a minimum to successfully control the system to stable states and the system order also has a maximum to do so.     

The effect of time-delayed feedback controller on an electrically actuated resonator

This paper presents a study of the effect of a time-delayed feedback controller on the dynamics of a Microelectromechanical systems (MEMS) capacitor actuated as a resonator by DC and AC voltage loads. A linearization analysis is conducted to determine the stability chart of the linearized system equations as a function of the time delay period and the controller gain. Then the method of multiple-scales is applied to determine the response and stability of the system for small vibration amplitude and voltage loads. It is shown that negative time-delay feedback control gain can lead to unstable responses, even if AC voltage is relatively small compared to the DC voltage. On the other hand, positive time delay can considerably strengthen the system stability even in fractal domains. We also show how the controller can be used to control damping in MEMS, increasing or decreasing, by tuning the gain amplitude and delay period. Agreements among the results of a shooting technique, long-time integration, basin of attraction analysis with the perturbation method are achieved.     

非自治时滞反馈控制系统的周期解分岔和混沌

研究时滞反馈控制对具有周期外激励非线性系统复杂性的影响机理,研究对应的线性平衡态失稳的临界边界,将时滞非线性控制方程化为泛函微分方程,给出由Hopf分岔产生的周期解的解析形式．通过分析周期解的稳定性得到周期解的失稳区域,使用数值分析观察到时滞在该区域可以导致系统出现倍周期运动、锁相运动、概周期运动和混沌运动以及两条通向混沌的道路：倍周期分岔和环面破裂．其结果表明,时滞在控制系统中可以作为控制和产生系统的复杂运动的控制“开关”．     

Controlling Chaos in Porous Media Convection by Using Feedback Control

A method of using feedback control to promote or suppress the transition to chaos in porous media convection is demonstrated in this article. A feedback control suggested by Mahmud and Hashim (Transp Porous Media, doi:10.1007/s11242-009-9511-1, 2010) is used in the present article to provide a comparison between an analytical expression for the transition point to chaos and numerical results. In addition, it is shown that such a feedback control can be applied as an excellent practical means for controlling (suppressing or promoting) chaos by using a transformation made by Magyari (Transp Porous Media, doi:10.1007/s11242-009-9511-1, 2010). The latter shows that Mahmud and Hashim (Transp Porous Media, doi:10.1007/s11242-009-9511-1, 2010) model can be transformed into Vadasz-Olek’s model (Transp Porous Media 37(1):69–91, 1999a) through a simple transformation of variables implying that the main effect the feedback control has on the solution is equivalent to altering the initial conditions. The theoretical and practical significance of such an equivalent alteration of the initial conditions is presented and discussed.     

时变切换时滞反馈镇定混沌系统不稳定周期轨线

为提高经典时滞反馈控制镇定不稳定周期轨线的效果, 扩大受控周期轨线的稳定区域, 本文基于时变切换策略对经典时滞反馈控制进行改进, 提出了时变切换时滞反馈控制. 时变切换时滞反馈控制的控制信号仅在特定的时段中存在, 而在其他时段上不存在控制信号, 这与经典时滞反馈控制中具有固定的控制信号是不同的. 通过实例分析, 研究了时变切换时滞反馈控制在镇定不稳定周期轨线中的具体性能. 以反馈增益系数为变量, 计算受控周期轨线的最大条件Lyapunov指数, 得到了受控周期轨线的稳定区域随切换频率变化的关系曲线. 结果表明, 随着切换频率增大, 受控周期轨线的稳定区域呈现非平滑地变化. 当选取恰当的切换频率时, 时变切换时滞反馈控制的稳定区域显著大于经典时滞反馈控制的稳定区域. 在混沌控制的工程实践中, 控制信号常常受到一定的限制. 要实现对目标周期轨线的稳定控制, 就需要受控周期轨线具有足够大的稳定区域. 因此, 与经典时滞反馈控制相比, 本文提出的时变切换时滞反馈控制具有更广泛的应用前景.      

Chaos detection and parameter identification in fractional-order chaotic systems with delay

The paper first applies the 0–1 test for chaos to detecting chaos exhibited by fractional-order delayed systems. The results of the test reveal that there exists chaos in some fractional-order delayed systems with specific parameter values, which coincides with previous reports based on the phase portrait. In addition, it is very important to identify exactly the unknown specific parameters of fractional-order chaotic delayed systems in chaos control and synchronization. Thus, a method for parameter identification of fractional-order chaotic delayed systems based on particle swarm optimization (PSO) is presented. By treating the orders as parameters, the parameters and orders are identified through minimizing an objective function. PSO can efficiently find the optimal feasible solution of the objective function. Finally, numerical simulations on fractional-order chaotic logistic delayed system and fractional-order chaotic Chen delayed system show that the proposed method has effective performance of parameter identification.     

Research on the price game and the application of delayed decision in oligopoly insurance market

In the oligopoly insurance market, we assumed that some oligarchs make two-period delay decisions in bounded rationality and expectation, respectively, and others make decisions with bounded rationality without the condition of delay. There also exist two cases in which only one oligarch makes a delayed decision and two oligarchs make delayed decisions at the same time. Based on the analysis of these situations, we established the corresponding dynamic price game models. We then performed a numerical simulation to the complexity state of the system with different conditions such as stability, bifurcation, and chaos, and analyzed the profits of different oligarchs when the system is in different states. The results showed that when only one oligarch makes a delayed decision, with the decrease in the price weight of period t and increase in that of periods t?1 and t?2, the system??s stable region in the direction of the price adjustment of the oligarch with a delayed decision gets smaller. However, when there are two oligarchs with a delayed decision in the system, in the case where the delay parameters of oligarch 1 remain unchanged and the price parameters of different periods of oligarch 2 change, the system??s stable region in the direction of the price adjustment of oligarch 1 does not have the obvious change as that when only one oligarch makes a delayed decision. This showed that the sensibility of one oligarch in the direction of its own price adjustment is lower than other oligarchs. In addition, in the same system with delay and when the system is in chaos, the total profit of the oligarchs is obviously less than that when the system is in a stable state. However, the use of a delayed decision may not enhance the oligarch??s competitive advantages. Finally, the variable feedback control method is used to effectively control the chaos in the system.     

混沌的抑制研究进展综述

综述了抑制混沌的４种主要方法:加随机噪声、加周期摄动、加动力吸振器和加输出变量反馈．概括了各种方法的一般形式,列举了各种方法应用的例子,还指出一些尚待研究的问题．      

Chaos control in a pendulum system with excitations and phase shift

Melnikov methods are used for suppressing homoclinic and heteroclinic chaos of a pendulum system with a phase shift and excitations. This method is based on the addition of adjustable amplitude and phase-difference of parametric excitation. Theoretically, we give the criteria of suppression of homoclinic and heteroclinic chaos, respectively. Numerical simulations are given to illustrate the effect of the chaos control in this system. Moreover, we calculate the maximum Lyapunov exponents (LEs) in parameter plane, and study how to vary the maximum LE when the parametric frequency varies.     

Delayed feedback control and bifurcation analysis of Rossler chaotic system

In this paper, from the view of stability and chaos control, we investigate the Rossler chaotic system with delayed feedback. At first, we consider the stability of one of the fixed points, verifying that Hopf bifurcation occurs as delay crosses some critical values. Then, for determining the stability and direction of Hopf bifurcation we derive explicit formulae by using the normal-form theory and center manifold theorem. By designing appropriate feedback strength and delay, one of the unstable equilibria of the Rossler chaotic system can be controlled to be stable, or stable bifurcating periodic solutions occur at the neighborhood of the equilibrium. Finally, some numerical simulations are carried out to support the analytic results.     

结构可靠度分析FORM迭代算法的混沌控制

利用混沌控制原理对FORM收敛失败进行控制. 理清了全局性和局部性两类混沌反馈 控制各种方法的内在联系,说明稳定转换法和自适应调节法属于全局混沌反馈控制 方法,自适应调节法可视为稳定转换法的特例. 参 数调节混合法不过是松弛牛顿法的另一种表达形式,它们都属于局部混沌反馈控制方法. 阐 明了混沌反馈控制表达式与工程力学收敛控制迭代算法的对应关系. 也揭示了这些迭代算法 收敛控制措施的功效和局限性. 提出了一个以稳定转换法为主联合松弛牛顿法的混 沌反馈控制方法,对可靠度分析FORM迭代算法实现了周期振荡、分岔和混沌控制.     

Chaos control,spiral wave formation,and the emergence of spatiotemporal chaos in networked Chua circuits

In this paper, a certain kind of intermittent scheme is used to control the chaos in a single chaotic Chua circuit to reach an arbitrary orbit. Furthermore, it is confirmed to be effective in suppressing spatiotemporal chaos and a spiral wave in the networks of Chua circuits with nearest-neighbor connections. The controllable and measurable variable is sampled, and the linear error between the sampled variable and the selected thresholds is fed back into the system only if the sampled variable exceeds the thresholds; otherwise, the system will develop itself without any external perturbation. In experiments, the control scheme could be realized by using the Heavside function. In the case of one single chaotic Chua circuit, the chaotic state can be controlled to reach an arbitrary n-periodical orbit (n=1,2,3,5,6,…) with appropriate feedback intensity and thresholds. It is argued that this scheme could explain the mechanism of what is called phase compression. Then the phase compression scheme is used to control a spiral wave and spatiotemporal chaos in a network of Chua circuits with 256×256 sites. The numerical simulation results confirm its effectiveness when appropriate upper and bottom thresholds are used by monitoring the measurable output voltages of the chaotic circuit in one site of the network.     

An Impulsive Multi-delayed Feedback Control Method for Stabilizing Discrete Chaotic Systems

An impulsive multi-delayed feedback control strategy to control the period-doubling bifurcations and chaos in an n dimensional discrete system is proposed. This is an extension of the previous result in which the control method is applicable to the one-dimensional case. Then the application of the control method in a discrete prey–predator model is studied systematically, including the dynamics analysis on the prey–predator model with no control, the bifurcations analysis on the controlled model, and the bifurcations and chaos control effects illustrations. Simulations show that the period-doubling bifurcations and the resulting chaos can be delayed or eliminated completely. And the periodic orbits embedded in the chaotic attractor can be stabilized. Compared with the existed methods, a milder condition is needed for the realization of the proposed method. The condition may be considered as a generic case and we may state that almost all periodic orbits can be stabilized by the proposed method. Besides, the idea of impulsive control makes the implementation of the proposed control method easy. The impulsive interval is embodied in the analytical expression of the stability condition, hence can be chosen qualitatively according to the real needs, which is an extension of the existed related results. The introduction of multi-delay enlarges the domain of the control parameters and makes the selection of the control parameters have many choices, and hence become flexible.     

MULTI-MODE OSCILLATIONS AND HAMILTON ENERGY FEEDBACK CONTROL OF A CLASS OF MEMRISTOR NEURON$^{\bf 1)}$

根据法拉第电磁感应定律,在离子穿越细胞膜或者在外界电磁辐射下,细胞内外的电生理环境会产生电磁感应效应,继而会影响神经元的电活动行为. 基于此,本文考虑电磁感应影响下的 Hindmarsh-Rose (HR) 神经元模型,研究了其混合模式振荡放电特征,并设计一个 Hamilton 能量反馈控制器,将其控制到不同的周期簇放电状态. 首先,通过理论分析发现磁通 HR 神经元系统的 Hopf 分岔使其平衡点的稳定性发生了改变,并产生极限环,进而研究了 Hopf 分岔点附近膜电压的放电特征. 基于双参数数值仿真发现该系统具有丰富的分岔结构,在不同的参数平面上存在倍周期分岔、伴有混沌的加周期分岔、无混沌的加周期分岔以及共存的混合模式振荡. 最后,为了有效控制膜电压的混合模式振荡,利用亥姆霍兹理论计算出磁通 HR 神经元系统的 Hamilton 能量函数并设计 Hamilton 能量反馈控制器,通过数值仿真分析了膜电压在不同反馈增益下的簇放电状态,发现该控制器能够有效地控制膜电压到不同的周期簇放电模式. 本文的研究结果为探究电磁感应下神经元的分岔结构及其能量控制领域提供了有用的理论支撑.     

一类忆阻神经元的电活动多模振荡及Hamilton 能量反馈控制

根据法拉第电磁感应定律,在离子穿越细胞膜或者在外界电磁辐射下,细胞内外的电生理环境会产生电磁感应效应,继而会影响神经元的电活动行为. 基于此,本文考虑电磁感应影响下的 Hindmarsh-Rose (HR) 神经元模型,研究了其混合模式振荡放电特征,并设计一个 Hamilton 能量反馈控制器,将其控制到不同的周期簇放电状态. 首先,通过理论分析发现磁通 HR 神经元系统的 Hopf 分岔使其平衡点的稳定性发生了改变,并产生极限环,进而研究了 Hopf 分岔点附近膜电压的放电特征. 基于双参数数值仿真发现该系统具有丰富的分岔结构,在不同的参数平面上存在倍周期分岔、伴有混沌的加周期分岔、无混沌的加周期分岔以及共存的混合模式振荡. 最后,为了有效控制膜电压的混合模式振荡,利用亥姆霍兹理论计算出磁通 HR 神经元系统的 Hamilton 能量函数并设计 Hamilton 能量反馈控制器,通过数值仿真分析了膜电压在不同反馈增益下的簇放电状态,发现该控制器能够有效地控制膜电压到不同的周期簇放电模式. 本文的研究结果为探究电磁感应下神经元的分岔结构及其能量控制领域提供了有用的理论支撑.      

一类平面自治非线性时滞控制系统的多稳态和混沌

众所周知,平面自治系统即使具有光滑非线性存在,系统也不会出现复杂的动力学行为。本文研究这样的系统存在时滞时,时滞量对系统的动力学行为的影响。通过对一个平面自治非线性系统引入时滞反馈,得到数学模型。利用泛函分析和平均法建立系统平衡态随时滞量变化的失稳机理,研究表明：时滞量平面自治系统动力学行为的影响是本质的．时滞量不但可以使系统出现Hopf分岔,产生周期振动。而且还可以使系统出现多稳态的周期运动或周期吸引子,这些共存的吸引子相碰是导致系统复杂的动力学行为,包括概周期和混沌运动。