A Twenty-Year Review of Time-Delay Feedback Control and Recent Developments
A Twenty-Year Review of Time-Delay Feedback Control and Recent Developments
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时滞反馈控制二十年回顾及最新进展
DOI：
10.15248/proc.1.683
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
Kestutis Pyragas
中科院分区：
文献类型：
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作者：
Kestutis Pyragas
延迟反馈控制（DFC）方法于1992年发明（今年是20周年）。根据Pyragas的原始论文[Phys. Lett. A，170，421，1992]，1500多篇论文专门或有关的DFC已出版。人们对该算法提出了许多不同的改进方案，并在DFC理论方面取得了重要成果。虽然这个理论是不平凡的，目前的机制DFC行动是相当好的理解，并建立算法的主要局限性。DFC已成功地实现了一些不同的物理性质的实验系统。这次谈话的目的是提出一个简短的审查的重要修改的DFC算法，重要的理论成果和实验实现在过去的二十年中取得的。最近的结果有关自适应修改的DFC和分析成果的基础上减少时滞系统的相位将被讨论。1.实验和理论结果的简要回顾DFC算法[1]是一种简单，鲁棒，有效的方法来稳定混沌系统中的不稳定周期轨道（UPO）。目前，它已成为混沌控制研究中最流行的方法之一[2]。该方法允许一个非侵入性的稳定的动力系统的UPO在这个意义上，当达到目标状态时，控制力消失。DFC算法是无参考的，并且利用从系统的当前状态s（t）和延迟目标轨道的一个周期τ的系统状态s（t-τ）之间的差Δ s（t）= s（t）-s（t-τ）获得的控制信号K Δ s（t）。在适当选择反馈强度K的情况下，UPO可以变得稳定。请注意，只有轨道的稳定性会改变，而轨道本身及其周期保持不变。控制系统可以被视为一个黑箱，因为该方法不需要任何确切的知识，无论是周期轨道的形式或系统的方程。这种方法对实验者特别有吸引力，因为人们不需要知道任何关于周期τ以外的目标轨道的信息。在快速动态系统中，DFC算法明显优于其他控制方法，因为它不需要任何实时计算机处理。DFC算法已在不同的实验系统中成功实现，包括电子混沌振荡器，机械振荡器，激光器，气体放电系统，电流驱动的离子声不稳定性，混沌泰勒-库埃特流，化学系统，高功率铁磁共振，直升机转子叶片和心脏系统。（参见：[3]至2006年进行审查）。Yamasue等人最近证明了DFC算法的一个重要实际应用。[4]。作者已经成功地实现了DFC方法在原子力显微镜，并设法稳定悬臂梁振荡。因此，它们去除了表面图像上的伪影。参考文献[5]最近考虑了DFC在分析实验系统周期态分叉中的另一个有趣应用。为了提高DFC的性能，已经提出了多种DFC修改方案（参见。[3]）。在这里，我们只提到最重要的修改，称为扩展的DFC（EDFC），它已在参考文献[6]中介绍。作者利用系统的许多先前状态的信息改进了一个原始的DFC方案。EDFC方案实现了具有更大不稳定度的UPO的稳定化[7，8]。由于延迟反馈导致了无穷多个自由度，使得DFC的理论研究比较困难。即使是线性分析这样的系统是复杂的，由于无穷多的Floquet指数表征控制轨道的稳定性。然而，在周期轨道的各种分叉附近已经开发了一些分析方法，例如倍周期分叉[9，10]，亚临界Hopf分叉[11，12，13]和Nejmark-Sacker（离散Hopf）分叉[14]。1997年Nakajima [15，9]证明了所谓的奇数限制，即任何具有大于1的奇数个真实的Floquet乘数的UPO都不能通过任何DFC技术稳定。这一限制已被普遍接受，并在文献中进行了深入讨论。然而，在2007年，Fiedler等人[16]通过一个简单的例子表明，这种限制通常不适用于自治系统（注意，对于非自治系统，它通常仍然有效）。最近，Hooton和2012年非线性理论及其应用国际研讨会NOLTA2012（2012年10月22日至26日，西班牙马略卡岛帕尔马）提出了自治系统极限的修改（更正）证明
The delayed feedback control (DFC) method has been invented in 1992 (this year is a 20th anniversary). Following the original paper by Pyragas [Phys. Lett. A, 170, 421, 1992], more than 1500 papers devoted or related to the DFC have been published. Many different modifications of the algorithm have been proposed, and significant achievements are attained in the theory of the DFC. Although this theory is non-trivial, currently the mechanism of the DFC action is rather well understood, and the main limitations of the algorithm are established. The DFC has been successfully implemented in a number of experimental systems of different physical nature. The aim of this talk is to present a brief review of important modifications of the DFC algorithm, significant theoretical results and experimental implementations attained during the past twenty years. The recent results concerning adaptive modifications of the DFC and analytical achievements based on phase reduction of time-delay systems will be discussed as well. 1. Brief review of experimental and theoretical results The DFC algorithm [1] is a simple, robust, and efficient method to stabilize unstable periodic orbits (UPOs) in chaotic systems. Nowadays, it becomes one of the most popular methods in the chaos control research [2]. The method allows a noninvasive stabilization of UPOs of dynamical systems in the sense that the control force vanishes when the target state is reached. The DFC algorithm is reference-free and makes use of a control signal K∆s(t) obtained from the difference ∆s(t) = s(t) − s(t − τ) between the current state s(t) of the system and the state of the system s(t−τ) delayed by one-period τ of the target orbit. The UPO may become stable under the appropriate choice of feedback strength K. Note that only the stability properties of the orbit are changed, while the orbit itself and its period remain unaltered. The controlled system can be treated as a black box, since the method does not require any exact knowledge of either the form of the periodic orbit or the system’s equations. The method is particularly appealing for experimentalists, since one does not need to know anything about the target orbit beyond its period τ. The DFC algorithm is notably superior to other control methods in fast dynamical systems, since it does not require any realtime computer processing. Successful implementation of the DFC algorithm has been attained in diverse experimental systems, including electronic chaotic oscillators, mechanical pendulums, lasers, gas discharge systems, a current-driven ion acoustic instability, a chaotic Taylor-Couette flow, chemical systems, high-power ferromagnetic resonance, helicopter rotor blades, and a cardiac system. (cf. [3] for review up to 2006). An important practical application of the DFC algorithm has been recently demonstrated by Yamasue et al. [4]. The authors have successfully implemented the DFC method in an atomic force microscope and managed to stabilize cantilever oscillations. As a result, they remove artifacts on a surface image. Another interesting application of the DFC for the analysis of bifurcations of periodic states in experimental systems has been recently considered in Ref. [5]. A reach variety of modifications of the DFC has been suggested in order to improve its performance (cf. [3]). Here we mention only the most important modification known as an extended DFC (EDFC), which has been introduced in Ref. [6]. The authors improved an original DFC scheme by using an information from many previous states of the system. The EDFC scheme achieves stabilization of UPOs with a greater degree of instability [7, 8]. The theory of DFC is difficult because the delayed feedback induces an infinite number of degrees of freedom. Even linear analysis of such systems is complicated due to the infinite number of Floquet exponents characterizing the stability of controlled orbits. Nevertheless, some analytical approaches have been developed in vicinity to various bifurcations of periodic orbits, such as the period doubling bifurcation [9, 10], the subcritical Hopf bifurcation [11, 12, 13] and the Nejmark-Sacker (discrete Hopf) bifurcation [14]. In 1997 Nakajima [15, 9] proved the so-called odd number limitation, which states that any UPOs with an odd number of real Floquet multipliers greater than unity can never be stabilized by any DFC technique. This limitation has been commonly accepted and intensively discussed in the literature. However, in 2007, Fiedler at al [16] have shown by a simple example that this limitation does not hold in general for autonomous systems (note that for non-autonomous systems it remains valid in general). Recently, a modified (corrected) proof of the limitation for autonomous systems has been presented by Hooton and 2012 International Symposium on Nonlinear Theory and its Applications NOLTA2012, Palma, Majorca, Spain, October 22-26, 2012