Time-delayed feedback control design beyond the odd-number limitation

Unstable delayed feedback control to change sign of coupling strength for weakly coupled limit cycle oscillators

Weakly coupled limit cycle oscillators can be reduced into a system of weakly coupled phase models. These phase models are helpful to analyze the synchronization phenomena. For example, a phase model of two oscillators has a one-dimensional differential equation for the evolution of the phase difference. The existence of fixed points determines frequency-locking solutions. By treating each oscillator as a black-box possessing a single input and a single output, one can investigate various control algorithms to change the synchronization of the oscillators. In particular, we are interested in a delayed feedback control algorithm. Application of this algorithm to the oscillators after a subsequent phase reduction should give the same phase model as in the control-free case, but with a rescaled coupling strength. The conventional delayed feedback control is limited to the change of magnitude but does not allow the change of sign of the coupling strength. In this work, we present a modification of the delayed feedback algorithm supplemented by an additional unstable degree of freedom, which is able to change the sign of the coupling strength. Various numerical calculations performed with Landau-Stuart and FitzHugh-Nagumo oscillators show successful switching between an in-phase and anti-phase synchronization using the provided control algorithm. Additionally, we show that the control force becomes non-invasive if our objective is stabilization of an unstable phase difference for two coupled oscillators.

Horizons of cybernetical physics

The subject and main areas of a new research field-cybernetical physics-are discussed. A brief history of cybernetical physics is outlined. The main areas of activity in cybernetical physics are briefly surveyed, such as control of oscillatory and chaotic behaviour, control of resonance and synchronization, control in thermodynamics, control of distributed systems and networks, quantum control.This article is part of the themed issue 'Horizons of cybernetical physics'.

Generic stabilizability for time-delayed feedback control

Time-delayed feedback control is one of the most successful methods to discover dynamically unstable features of a dynamical system in an experiment. This approach feeds back only terms that depend on the difference between the current output and the output from a fixed time T ago. Thus, any periodic orbit of period T in the feedback-controlled system is also a periodic orbit of the uncontrolled system, independent of any modelling assumptions. It has been an open problem whether this approach can be successful in general, that is, under genericity conditions similar to those in linear control theory (controllability), or if there are fundamental restrictions to time-delayed feedback control. We show that, in principle, there are no restrictions. This paper proves the following: for every periodic orbit satisfying a genericity condition slightly stronger than classical linear controllability, one can find control gains that stabilize this orbit with extended time-delayed feedback control. While the paper's techniques are based on linear stability analysis, they exploit the specific properties of linearizations near autonomous periodic orbits in nonlinear systems, and are, thus, mostly relevant for the analysis of nonlinear experiments.

Relation between the extended time-delayed feedback control algorithm and the method of harmonic oscillators

In a recent paper [Phys. Rev. E 91, 012920 (2015)] Olyaei and Wu have proposed a new chaos control method in which a target periodic orbit is approximated by a system of harmonic oscillators. We consider an application of such a controller to single-input single-output systems in the limit of an infinite number of oscillators. By evaluating the transfer function in this limit, we show that this controller transforms into the known extended time-delayed feedback controller. This finding gives rise to an approximate finite-dimensional theory of the extended time-delayed feedback control algorithm, which provides a simple method for estimating the leading Floquet exponents of controlled orbits. Numerical demonstrations are presented for the chaotic Rössler, Duffing, and Lorenz systems as well as the normal form of the Hopf bifurcation.

Dynamics of dc bus networks and their stabilization by decentralized delayed feedback

The present paper deals with the dynamics of bus networks, which consist of several identical dc bus systems connected by resistors. It is analytically guaranteed that the stability of a stand-alone dc bus system is equivalent to that of the networks, independent of the number of bus systems and the network topology. In addition, we show that a decentralized delayed-feedback control can stabilize an unstable operating point embedded within the networks. Moreover, this stabilization does not depend on the number of bus systems or the network topology. A systematic procedure for designing the controller is presented. Finally, the validity of the analytical results is confirmed through numerical examples.