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。      

Efficient parametric excitation walking with delayed feedback control

In passive dynamic walking proposed by McGeer, mechanical energy lost by heel strike is restored by transporting potential energy to kinetic energy as walking down a slope. When energy input is larger as a slope is steeper, the bifurcation of a walking cycle occurs. In the parametric excitation walking, which is to realize passive dynamic-like walking on the level ground, the bifurcation of a walking cycle has also been observed when walking speed is fast. Recently, Asano et al. have shown that bifurcation exerts an adverse influence upon walking performance by using a rimless wheel model. In this paper, we apply the delayed feedback control (DFC), originally used in chaos control, to parametric excitation walking to suppress bifurcation. We show in numerical simulation that the proposed method makes period-two walking to period-one walking, and improves energy efficiency. In addition, the proposed method can generate a sustainable gait in the region where a biped robot cannot walk without DFC. The analyses using a Poincaré map reveal that period-one walking with DFC corresponds to an unstable periodic orbit and reveal that a robot model in this paper satisfies the sufficient condition of applicability of DFC.     

Time delay Duffing’s systems: chaos and chatter control

The effect of a delay feedback control (DFC), realized by displacement in the Duffing oscillator, for parameters which generate strange chaotic Ueda attractor is investigated in this paper. First, the classical Duffing system without time delay is analysed to find stable and especially unstable periodic orbits which can be stabilized by means of displacement delay feedback. The periodic orbits are found with help of the continuation method using the AUTO97 software. Next, the DFC is introduced with a time delay and a feedback gain parameters. The proper time delay and feedback gain are found in order to destroy the chaotic attractor and to stabilize the periodic orbit. Finally, chatter generated by time delay component is suppressed with help of an external excitation.     

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.     

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.     

An adaptive delayed feedback control method for stabilizing chaotic time-delayed systems

This paper addresses a new adaptive delayed feedback control technique for stabilizing a class of chaotic time-delayed systems with a variable parameter. In the proposed scheme, the feedback gain of a delayed feedback controller is automatically tuned according to an adaptation law in order to stabilize unstable fixed points of the system. Such a mechanism provides a way to cope with unexpected changes in the parameters of the system. The adaptation algorithm is constructed based on the Lyapunov?CKrasovskii??s stability theorem. The control technique provides the advantages of increased stability and optimality, adaptability to the changes in the parameters, high privacy, simplicity, and noninvasiveness. The effectiveness of the control scheme is demonstrated using numerical simulations for a well-known chaotic time-delayed system.     

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.     

Experimental study of delayed positive feedback control for a flexible beam

Recently, some researches indicate that positive feedback can benefit the control if appropriate time delay is intentionally introduced into control system. However, most work is theoretical one but few are experimental. This paper presents theoretical and experimental studies of delayed positive feedback control technique using a flexible beam as research object. The positive feedback weighting coefficient is designed by using the optimal control method. The available time delay is determined by analyzing the maximal real part of characteristic roots of the system. A DSP-based experiment system is introduced. Simulation and experimental results indicate that the delayed positive feedback control may effectively reduce the beam vibration if time delay is appropriately selected.     

Exponential synchronization of chaotic Lur’e systems with delayed feedback control

The exponential synchronization problem is studied in this paper for a class of chaotic Lur’e systems by using delayed feedback control. An augmented Lyapunov functional based approach is proposed to deal with this issue. A delay-dependent condition is established such that the controlled slave system can exponentially synchronize with the master system. It is shown that the delayed feedback gain matrix and the exponential decay rate can be obtained by solving a set of linear matrix inequalities. The decay coefficient can be also easily calculated. Finally, as an example, the Chua’s circuit is used to illustrate the effectiveness of the developed approach and the improvement over some existing results.     

Stabilizing the unstable periodic orbits of a chaotic system using model independent adaptive time-delayed controller

In this paper we deal with the control of chaotic systems. Knowing that a chaotic attractor contains a myriad of unstable periodic orbits (UPO’s), the aim of our work is to stabilize some of the UPO’s embedded in the chaotic attractor and which have interesting characteristics. First, using the input-to-state linearization method in conjunction with a time-delayed state feedback, we design a control signal that can achieve stabilization. Next, an adaptive time-delayed state feedback is proposed which shows at once efficiency and simplicity and circumvents the construction complexity of the first controller. Finally, we propose a reduced order sliding mode observer to estimate the necessary states for the design of an adaptive time delayed state feedback controller. This last controller has one main advantage, it in fact achieves UPO stabilization without using the system model. The efficacy of the proposed methods is illustrated by numerical simulations onto Chua’s system.     

Constrained predictive synchronization of discrete-time chaotic Lur’e systems with time-varying delayed feedback control

This paper presents a predictive synchronization method for discrete-time chaotic Lur’e systems with input constraints by using time-varying delayed feedback control. Based on the model predictive control scheme, a delay-dependent stabilization criterion is derived for the synchronization of chaotic systems that is represented by Lur’e systems with input constraints. By constructing a suitable Lyapunov–Krasovskii functional and combining with a reciprocally convex combination technique, a delay-dependent stabilization condition for synchronization is obtained via linear matrix inequality (LMI) formulation. The control inputs are obtained by solving a min-max problem subject to cost monotonicity, which is expressed in terms of LMIs. The effectiveness of the proposed method will be verified throughout a numerical example.     

Stabilization of discrete-time chaotic systems via improved periodic delayed feedback control based on polynomial matrix right coprime factorization

Delayed feedback control, proposed by Pyragas,is one of useful control methods of stabilizing unstable periodic orbits of chaotic systems without their exact information. This paper discusses an improved periodic delayed feedback control for the stabilization of the discrete-time chaotic systems. A technique to solve the generalized Sylvester matrix equation, based on polynomial matrix right coprime factorization, is introduced in order to give an explicit solution to the periodic gains of the closed-loop system. Moreover, some examples are demonstrated to show the effectiveness of the proposed method.     

Stabilization control for offshore steel jacket platforms with actuator time-delays

This paper is concerned with the stabilization control for the offshore steel jacket platforms subject to wave-induced force. Two state feedback stabilization control schemes are proposed to reduce the vibration amplitudes of the systems. One scheme is that for the systems without actuator time-delay, a state feedback controller is designed. Compared with the nonlinear controller, both the control force and the vibration amplitudes of the systems under the state feedback controller are much reduced; and compared with the dynamic output feedback controller and the integral sliding mode controller, the required control force under the state feedback controller are significantly reduced. The other scheme is that based on the integral inequality approach, a delay-dependent state feedback controller, which can be solved by using the cone complementarity algorithm, is developed to control the systems with actuator time-delays. Compared with the state feedback controller, the delay-dependent state feedback controller is less conservative with actuator time-delays. In addition, it is capable of improving the control performance of the offshore platforms significantly, which are illustrated by simulation results.     

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.     

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。     

The impact of delayed feedback on the pulsating oscillations of class-B lasers

Class-B laser systems can be described by a set of slow-fast differential equations with a small parameter, and they exhibit pulsating oscillations in common. In this paper, the impact of the delayed feedback on the pulsating solutions is investigated. At first, a careful analysis of the local stability and bifurcation shows the existence of a series of Hopf bifurcations and double Hopf bifurcations when the strength of the delayed feedback or the delay increases, by means of stability switches. Then, an application of the geometric singular perturbation theory reveals the evolutional mechanism of the pulsating solutions from transients to stationary states, and the effect of the delayed feedback upon the pulsating solutions is examined.     

Multiplicity-induced optimal gains of an inverted pendulum system under a delayed proportional-derivative-acceleration feedback

This paper studies the stabilization to an inverted pendulum under a delayed proportional-derivative-acceleration (PDA) feedback, which can be used to understand human balance in quiet standing. The closed-loop system is described by a neutral delay differential equation (NDDE). The optimal feedback gains (OFGs) that make the exponential decaying rate maximized are determined when the characteristic equation of the closed-loop has a repeated real root with multiplicity 4. Such a property is called multiplicity-induced dominancy of time-delay systems, and has been discussed intensively by many authors for retarded delay differential equations (RDDEs). This paper shows that multiplicity-induced dominancy can be achieved in NDDEs. In addition, the OFGs are delay-dependent, and decrease sharply to small numbers correspondingly as the delay increases from zero and varies slowly with respect to moderate delays. Thus, the inverted pendulum can be well-stabilized with moderate delays and relatively small feedback gains. The result might be understandable that the elderly with obvious response delays can be well-stabilized with a delayed PDA feedback controller.     

Resonances of a Harmonically Forced Duffing Oscillator with Time Delay State Feedback

The paper presents analytical and numerical studies of the primary resonance and the 1/3 subharmonic resonance of a harmonically forced Duffing oscillator under state feedback control with a time delay. By using the method of multiple scales, the first order approximations of the resonances are derived and the effect of time delay on the resonances is analyzed. The concept of an equivalent damping related to the delay feedback is proposed and the appropriate choice of the feedback gains and the time delay is discussed from the viewpoint of vibration control. In order to numerically solve the problem of history dependence prior to the start of excitation, the concepts of the Poincaré section and fixed points are generalized. Then, a modified shooting scheme associated with the path following technique is proposed to locate the periodic motion of the delayed system. The numerical results show the efficacy of the first order approximations of the resonances.     

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