柔性梁受迫振动时滞变结构控制的试验研究

时滞反馈控制是一种利用时滞进行系统控制的策略,目前对该控制策略的研究多是在理论上进行探讨,少有试验研究报道. 以受简谐激励的柔性悬臂梁为对象,开展时滞反馈控制的试验研究,给出了一个多时滞控制律的设计方法. 首先给出悬臂梁系统含有时滞项的控制模态状态方程; 然后对方程进行离散化和一种特殊的状态变量增广,得到形式上不含有时滞项的标准差分方程; 最后使用离散变结构控制的方法设计控制律. 试验中采用压电片作为作动器和外界激励,应变片作为传感器,分别考虑单时滞和双时滞的情况,通过试验验证了时滞反馈控制的可行性和有效性.关键词:柔性悬臂梁;变结构控制;时滞;实验      

时滞状态正反馈在振动控制中的新特征

控制理论中广泛采用负反馈,而正反馈的应用不多, 一个重要原因是正反馈将系统的变化放大而使系统的稳定性变差. 如果反馈环节具有时滞, 那么正反馈未必使系统稳定性变差. 本文以线性振动系统为例, 采用稳定性切换方法和利用确定时滞系统稳定性的最大实部特征根, 详细研究了时滞状态正反馈在镇定系统不稳定运动和改善系统稳定性方面的作用. 我们发现,时滞位移正反馈明显优于时滞位移负反馈, 表现为: (1). 正反馈控制可以用较小的时滞去镇定不稳定运动和改善系统稳定性; (2). 正反馈控制可容许的时滞范围很大, 而负反馈控制的可容许时滞范围很小; (3). 正反馈对应的闭环系统的最大实部特征根的实部的极小值可显著小于负反馈对应的闭环系统的最大实部特征根的实部的极小值, 因而在相同的初始条件下, 正反馈作用下的闭环系统比之负反馈作用下的闭环系统可以更快地稳定到平衡点. 我们还发现, 对时滞速度反馈与时滞加速度反馈来说, 负反馈优于正反馈; 而对相同的反馈增益, 时滞位移正反馈优于时滞速度正反馈和时滞加速度正反馈. 关键字镇定,振动控制,时滞正反馈, 稳定性切换, 特征根      

ew Features of Positive Time-Delayed Feedbacks in Vibration Control

控制理论中广泛采用负反馈,而正反馈的应用不多, 一个重要原因是正反馈将系统的变化放大而使系统的稳定性变差. 如果反馈环节具有时滞, 那么正反馈未必使系统稳定性变差. 本文以线性振动系统为例, 采用稳定性切换方法和利用确定时滞系统稳定性的最大实部特征根, 详细研究了时滞状态正反馈在镇定系统不稳定运动和改善系统稳定性方面的作用. 我们发现,时滞位移正反馈明显优于时滞位移负反馈, 表现为: (1). 正反馈控制可以用较小的时滞去镇定不稳定运动和改善系统稳定性; (2). 正反馈控制可容许的时滞范围很大, 而负反馈控制的可容许时滞范围很小; (3). 正反馈对应的闭环系统的最大实部特征根的实部的极小值可显著小于负反馈对应的闭环系统的最大实部特征根的实部的极小值, 因而在相同的初始条件下, 正反馈作用下的闭环系统比之负反馈作用下的闭环系统可以更快地稳定到平衡点. 我们还发现, 对时滞速度反馈与时滞加速度反馈来说, 负反馈优于正反馈; 而对相同的反馈增益, 时滞位移正反馈优于时滞速度正反馈和时滞加速度正反馈. 关键字镇定,振动控制,时滞正反馈, 稳定性切换, 特征根     

Delayed full-state feedback control of airfoil flutter using sliding mode control method

This paper studies the delayed feedback control of flutter of a two-dimensional airfoil using a sliding mode control (SMC) method. The dynamic equation of airfoil flutter is firstly established using the Lagrange method, in which the cubic hardening spring nonlinearity of pitch stiffness is considered. Then, the state equation with time delay is transformed into a standard state equation with implicit time delay by a special integral transformation. Next a nonlinear time-delay controller is designed using the SMC method. Finally the effectiveness of the proposed controller is verified through numerical simulations. Simulation results indicate that time delay in the control system has significant influence on the control performance. Control failure may happen if time delay is not considered in control design. The time-delay controller proposed is effective in suppressing the airfoil flutter with either small or large control time delay.     

Delayed feedback control experiments on some flexible structures

In recent decades,studies on delayed system dynamics have attracted increasing attention and advances have been achieved in stability,nonlinearity,delay identification, delay elimination and application.However,most of the existing work is on the theoretical basis and little is on the experiment.This paper presents our experimental studies on delayed feedback control conducted in recent years with the focus on the discussion of a DSP-based delayed experiment system.Some phenomena in our delay experiments are discussed and a few topics of interest for further research are brought forward.     

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.     

Experimental studies on active control of a dynamic system via a time-delayed absorber

The traditional passive absorber is fully effective within a narrow and certain frequency band.To solve this problem,a time-delayed acceleration feedback is introduced to convert a passive absorber into an active one.Both the inherent and the intentional time delays are included.The former mainly comes from signal acquiring and processing,computing,and applying the actuation force,and its value is fixed.The latter is introduced in the controller,and its value is actively adjustable.Firstly,the mechanical model is established and the frequency response equations are obtained.The regions of stability are delineated in the plane of control parameters.Secondly,the design scheme of control parameters is performed to help select the values of the feedback gain and time delay.Thirdly,the experimental studies are conducted.Effects of both negative and positive feedback control are investigated.Experimental results show that the proper choices of control parameters may broaden the effective frequency band of vibration absorption.Moreover,the time-delayed absorber greatly suppresses the resonant response of the primary system when the passive absorber totally fails.The experimental results are in good agreement with the theoretical predictions and numerical simulations.     

Delayed feedback control based on the act-and-wait concept

This paper proposes a delayed feedback control (DFC) based on the act-and-wait concept, which reduces the dynamics of DFC systems to that of discrete-time systems. Based on this concept, a delayed feedback controller is designed for a prototype two-dimensional oscillator using a simple systematic procedure. This control has two advantages: the feedback delay time can be large and it can obtain deadbeat behavior. A numerical example using a double-scroll circuit model demonstrates these theoretical results.     

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.     

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。      

旋转运动柔性梁的时滞主动控制实验研究

对旋转运动柔性梁的时滞主动控制进行实验研究,验证时滞反馈控制的有效性. 实验中采用交流伺服电机带动柔性梁旋转运动,柔性梁上粘贴有压电作动器,用于控制梁的弹性振动. 实验研究考虑如下3种情况:(1)仅使用电机扭矩进行控制,电机扭矩存在时滞;(2)使用电机扭矩和压电作动器同时控制,仅压电作动器存在时滞;(3)使用电机扭矩和压电作动器同时控制,电机和压电作动器存在不同的时滞量. 重点通过实验验证时滞反馈控制的可行性和有效性.      

Non-linear vibrations of parametrically excited cantilever beams subjected to non-linear delayed-feedback control

Non-linear feedback control provides an effective methodology for vibration mitigation in non-linear dynamic systems. However, within digital circuits, actuation mechanisms, filters, and controller processing time, intrinsic time-delays unavoidably bring an unacceptable and possibly detrimental delay period between the controller input and real-time system actuation. If not well-studied, these inherent and compounding delays may inadvertently channel energy into or out of a system at incorrect time intervals, producing instabilities and rendering controllers’ performance ineffective. In this work, we present a comprehensive investigation of the effect of time delays on the non-linear control of parametrically excited cantilever beams. More specifically, we examine three non-linear cubic delayed-feedback control methodologies: position, velocity, and acceleration delayed feedback. Utilizing the method of multiple scales, we derive the modulation equations that govern the non-linear dynamics of the beam. These equations are then utilized to investigate the effect of time delays on the stability, amplitude, and frequency–response behavior. We show that, when manifested in the feedback, even the minute amount of delays can completely alter the behavior and stability of the parametrically excited beam, leading to unexpected behavior and responses that could puzzle researchers if not well-understood and documented.     

Response of quasi-integrable Hamiltonian systems with delayed feedback bang–bang control

The response of quasi-integrable Hamiltonian systems with delayed feedback bang–bang control subject to Gaussian white noise excitation is studied by using the stochastic averaging method. First, a quasi-Hamiltonian system with delayed feedback bang–bang control subjected to Gaussian white noise excitation is formulated and transformed into the Itô stochastic differential equations for quasi-integrable Hamiltonian system with feedback bang–bang control without time delay. Then the averaged Itô stochastic differential equations for the later system are derived by using the stochastic averaging method for quasi-integrable Hamiltonian systems and the stationary solution of the averaged Fokker–Plank–Kolmogorov (FPK) equation associated with the averaged Itô equations is obtained for both nonresonant and resonant cases. Finally, two examples are worked out in detail to illustrate the application and effectiveness of the proposed method and the effect of time delayed feedback bang–bang control on the response of the systems.     

Effect of delay combinations on stability and Hopf bifurcation of an oscillator with acceleration-derivative feedback

This paper presents new observations of delayed AD (acceleration-derivative) controller in active vibration control and in bifurcation control of a Duffing oscillator. Based on the stability analysis of the linear delayed oscillator, it is found that combination of the two delays in acceleration feedback and velocity feedback has a significant influence on the stable region in the parameter plane of the gains. By calculating the real part of the rightmost characteristic roots of the controlled oscillator with fixed delays, it is shown that a delayed acceleration feedback with positive gain can work much better than the corresponding delayed negative acceleration feedback, which is used in classic control theory. For given feedback gains, by calculating the critical delay values, it is shown that a delayed positive acceleration feedback can result in a much larger stable delay interval than the corresponding delayed negative acceleration feedback does. As an application of these results to a delayed Duffing oscillator with acceleration-derivative feedback, a delayed positive acceleration feedback can be well used to postpone the occurrence of Hopf bifurcation in the delayed nonlinear oscillators.     

Principal resonance of a Duffing oscillator with delayed state feedback under narrow-band random parametric excitation

The principal resonance of a Duffing oscillator with delayed state feedback under narrow-band random parametric excitation is studied by using the method of multiple scales and numerical simulations. The first-order approximations of the solution, together with the modulation equations of both amplitude and phase, are derived. The effects of the frequency detuning, the deterministic amplitude, the intensity of the random excitation and the time delay on the dynamical behaviors, such as stability and bifurcation, are studied through the largest Lyapunov exponent. Moreover, the appropriate choice of the feedback gains and the time delay is discussed from the viewpoint of vibration control. It is found that the appropriate choice of the time delay can broaden the stable region of the trivial steady-state solution and enhance the control performance. The theoretical results are well verified through numerical simulations.     

Stabilization of the rotating disk-beam system with a delay term in boundary feedback

In this article, the stabilization problem of a rotating disk-beam system is addressed. It is assumed that the flexible beam is free at one end, whereas the other end is attached to the center of the rotating disk whose angular velocity is time-varying. The proposed feedback law consists of a torque control which acts on the disk, whereas a delayed boundary force control is exerted at the free end of the beam. Thereafter, it is proved that the presence of such controls in the system guarantees the exponential stability of the system under a realistic smallness condition on the angular velocity of the disk as well as the feedback gain in the delay term. This result is illustrated by numerical examples.     

Design of active flutter suppression and wind-tunnel tests of a wing model involving a control delay

In this study, a delayed controller was designed for active flutter suppression of a three-dimensional wing model. The design of controller can be divided into two steps. At the first step, a short time delay was artificially introduced into the control loop and the dynamic equations of the aeroelastic system with delayed control were converted into a set of delay-free state-space equations by using a state transformation. At the second step, the control law was synthesized by using the theory of optimal control for the delay-free state-space equations. To demonstrate the performance of the delayed controller, the margin of time delay was studied. The numerical results showed that the delayed controller had good robustness with respect to the time delay. Moreover, the delayed controller was digitally implemented and tested for the three-dimensional wing model in NH-2 subsonic wind-tunnel. The experimental results illustrated that the critical flow speed of flutter instability of the wing model could be effectively increased from 36.5 m/s to 39 m/s.     

时滞耦合质量摆动力吸振器减振系统的等峰优化理论与实验

摆式调谐质量阻尼器因其便于安装、维修、更换,且经济实用,广泛应用于结构减振.它通过将摆的自振频率调谐到接近主系统的控制频率,使摆产生与主系统相反的振动,从而抑制或消除主系统的振动.本文通过对主系统无阻尼的被动减振系统和主系统有阻尼的时滞反馈主动减振系统进行多目标优化设计,实现了对主系统幅频响应曲线的等峰控制和共振峰与反共振峰差值的有效控制.首先,建立了时滞耦合质量摆动力吸振器减振系统的力学模型和振动微分方程,通过对主系统无阻尼的被动减振系统进行等峰优化,获得了减振系统的最优频率比和质量摆的最优阻尼比.对于主系统存在阻尼的被动减振系统,在该优化参数下主系统的幅频响应曲线等峰优化失效.其次,对于主系统存在阻尼的时滞反馈优化控制系统,采用CTCR方法得到了反馈增益系数和时滞的稳定区域.在保证系统稳定的前提下,通过调节反馈增益系数和时滞量两个控制参数能够实现对主系统幅频响应曲线的等峰控制.再次,对共振点处主系统振幅放大因子时滞敏感度和反馈增益系数敏感度进行分析,表明共振点幅值对反馈增益系数比对时滞更为敏感.最后,通过实验分别在频域和时域内对理论结果进行了验证.研究表明,通过采用时滞反馈对摆式调...     

索-梁组合结构主共振响应的时滞反馈控制

基于时滞加速度反馈控制策略对索-梁组合结构进行振动控制。根据Hamilton原理推导了索-梁组合结构非线性振动控制方程,运用多尺度法得到时滞反馈作用下索-梁组合结构主共振的一阶近似解,得出系统响应与控制参数的关系以及响应峰值和临界激励值与时滞参数的表达式。结果表明,主共振的响应存在多解和跳跃现象,调节控制增益和时滞值,可以有效抑制大幅振动。     

Stability and Hopf bifurcation of a two-dimensional supersonic airfoil with a time-delayed feedback control surface

The time-delayed feedback control for a supersonic airfoil results in interesting aeroelastic behaviors. The effect of time delay on the aeroelastic dynamics of a two-dimensional supersonic airfoil with a feedback control surface is investigated. Specifically, the case of a 3-dof system is considered in detail, where the structural nonlinearity is introduced in the mathematical model. The stability analysis is conducted for the linearized system. It is shown that there is a small parameter region for delay-independently stability of the system. Once the controlled system with time delay is not delay-independently stable, the system may undergo the stability switches with the variation of the time delay. The nonlinear aeroelastic system undergoes a sequence of Hopf bifurcations if the time delay passes the critical values. Using the normal form method and center manifold theory, the direction of the Hopf bifurcation and stability of Hopf-bifurcating periodic solutions are determined. Numerical simulations are performed to illustrate the obtained results.