Time-Delayed Feedback Control

Abstract

Chaos is found in greatest abundance wherever order is being sought. Chaos always defeats order because it is better organized. Terry Pratchett In the seminal work by Ott et al. [1], they demonstrated that small time-dependent changes of a parameter in a deterministic chaotic system can lead to periodic motion. Their findings are beyond classical control theory [2–5] and opened the field of chaos control which has become an aspect of increasing interest in nonlinear science [6, 7]. An especially powerful control scheme was introduced by Pyragas [8]. It is called time-delayed feedback control or time-delay autosynchronization and constructs a control force from the difference of the present state of a given system to its delayed value, i.e., s(t) -s(t -s). For proper choices of the time delay s, the control force vanishes if the state to be stabilized is reached. Thus, the method is noninvasive. This feedback scheme is easy to implement in an experimental setup and numerical calculation. It is capable of stabilizing fixed points as well as periodic orbits even if the dynamics are very fast. Furthermore, the Pyragas scheme has no need for a reference system since it generates the control force from information of the system itself. Also from a mathematical perspective it is an appealing instrument as the corresponding equations fall in the class of delay differential equations. This chapter provides a summary of the time-delayed feedback scheme which is investigated in the subsequent chapters of this thesis and includes basic concepts for its analysis. Thus, it can be seen as the central node in this thesis and connects the other parts, where time-delayed feedback is applied to different classes of dynamic systems. The chapter is organized as follows: In Sect. 2.1, I will intro-duce the general concept of time-delayed feedback control starting with the ori-ginal work by Pyragas [8]. Section 2.2 is devoted to extended time-delayed feedback invented by Socolar et al. [9]. This is an extension of the Pyragas scheme which will be used frequently in the subsequent chapters. Sections 2.3 and 2.4 P. Hövel, Control of Complex Nonlinear Systems with Delay, Springer Theses, DOI: 10.1007/978-3-642-14110-2_2, Ó Springer-Verlag Berlin Heidelberg 2010 11 cover special realizations and further extensions of time-delayed feedback control. These include different coupling schemes, control loop latency, filtering, and nonlocal feedback. Section 2.5 describes the concept of linear stability analysis in the presence of time delay. This technique will be used several times in this thesis. Section 2.6 deals with the formalism of transfer functions and provides an addi-tional perspective on the control mechanism. Finally, Sect. 2.7 concludes this chapter with an intermediate summary. 2.1 Control Method