Delayed feedback control of unstable steady states with high-frequency modulation of the delay

Adaptive delayed feedback control for stabilizing unstable steady states

Delayed feedback control is a commonly used control method for stabilizing unstable periodic orbits and unstable steady states. The present paper proposes an adaptive tuning delay time rule for delayed feedback control focused on stabilizing unstable steady states. The rule is designed to slowly vary the delay time, increasing the difference between the past and current states of dynamical systems, which induces the delay time to automatically fall into the stability region. We numerically confirm that the tuning rule works well for the Stuart-Landau oscillator, FitzHugh-Nagumo model, and Lorenz system.

Laminar chaos in systems with quasiperiodic delay

A type of chaos called laminar chaos was found in singularly perturbed dynamical systems with periodic time-varying delay [Phys. Rev. Lett. 120, 084102 (2018)]0031-900710.1103/PhysRevLett.120.084102. It is characterized by nearly constant laminar phases, which are periodically interrupted by irregular bursts, where the intensity levels of the laminar phases vary chaotically from phase to phase. In this paper, we demonstrate that laminar chaos can also be observed in systems with quasiperiodic delay, where we generalize the concept of conservative and dissipative delays to such systems. It turns out that the durations of the laminar phases vary quasiperiodically and follow the dynamics of a torus map in contrast to the periodic variation observed for periodic delay. Theoretical and numerical results indicate that introducing a quasiperiodic delay modulation into a time-delay system can lead to a giant reduction of the dimension of the chaotic attractors. By varying the mean delay and keeping other parameters fixed, we found that the Kaplan-Yorke dimension is modulated quasiperiodically over several orders of magnitudes, where the dynamics switches quasiperiodically between different types of high- and low-dimensional types of chaos.

Robust stabilization at uncertain equilibrium by output derivative feedback control

The problem of stabilizing dynamic systems with unknown or uncertain equilibrium states is studied. Derivative control schemes are proposed for both state and output feedback for the local stabilization at the true equilibrium states. The case where norm-bounded uncertainties are present in the system model is considered to derive robust stability conditions in linear matrix inequality form. The proposed control solutions drive the closed-loop system exponentially to its equilibrium states as shown via simulation on the chaotic Rössler and Lorenz attractors, when the equilibrium states are unknown and model uncertainties are present. A practical example involving a magnetic levitation system, in which two disks are to be levitated at an unknown magnetic equilibrium, demonstrates the effectiveness of the output derivative feedback controller.

Laminar chaos in experiments and nonlinear delayed Langevin equations: A time series analysis toolbox for the detection of laminar chaos

Recently, it was shown that certain systems with large time-varying delay exhibit different types of chaos, which are related to two types of time-varying delay: conservative and dissipative delays. The known high-dimensional turbulent chaos is characterized by strong fluctuations. In contrast, the recently discovered low-dimensional laminar chaos is characterized by nearly constant laminar phases with periodic durations and a chaotic variation of the intensity from phase to phase. In this paper we extend our results from our preceding publication [Hart, Roy, Müller-Bender, Otto, and Radons, Phys. Rev. Lett. 123, 154101 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.154101], where it is demonstrated that laminar chaos is a robust phenomenon, which can be observed in experimental systems. We provide a time series analysis toolbox for the detection of robust features of laminar chaos. We benchmark our toolbox by experimental time series and time series of a model system which is described by a nonlinear Langevin equation with time-varying delay. The benchmark is done for different noise strengths for both the experimental system and the model system, where laminar chaos can be detected, even if it is hard to distinguish from turbulent chaos by a visual analysis of the trajectory.

Control of amplitude chimeras by time delay in oscillator networks

We investigate the influence of time-delayed coupling in a ring network of nonlocally coupled Stuart-Landau oscillators upon chimera states, i.e., space-time patterns with coexisting partially coherent and partially incoherent domains. We focus on amplitude chimeras, which exhibit incoherent behavior with respect to the amplitude rather than the phase and are transient patterns, and we show that their lifetime can be significantly enhanced by coupling delay. To characterize their transition to phase-lag synchronization (coherent traveling waves) and other coherent structures, we generalize the Kuramoto order parameter. Contrasting the results for instantaneous coupling with those for constant coupling delay, for time-varying delay, and for distributed-delay coupling, we demonstrate that the lifetime of amplitude chimera states and related partially incoherent states can be controlled, i.e., deliberately reduced or increased, depending upon the type of coupling delay.

Act-and-wait time-delayed feedback control of nonautonomous systems

Act-and-wait modification of a time-delayed feedback control (TDFC) algorithm is proposed to stabilize unstable periodic orbits in nonautonomous dynamical systems. Due to periodical switching on and off the control perturbation, an infinite-dimensional function space of the TDFC system is reduced to the finite-dimensional state space. As a result the number of Floquet exponents defining the stability of the controlled orbit remains the same as for the control-free system. The values of these exponents can be effectively manipulated by the variation of control parameters. We demonstrate the advantages of the modification for the chaotic nonautonomous Duffing oscillator with diagonal and nondiagonal control matrices. In both cases very deep minima of the spectral abscissa of Floquet exponents have been attained. The advantage of the modification is particularly remarkable for the nondiagonal coupling; in this case the conventional TDFC fails, whereas the modified version works.

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.

Synchronization of networks of oscillators with distributed delay coupling

This paper studies the stability of synchronized states in networks, where couplings between nodes are characterized by some distributed time delay, and develops a generalized master stability function approach. Using a generic example of Stuart-Landau oscillators, it is shown how the stability of synchronized solutions in networks with distributed delay coupling can be determined through a semi-analytic computation of Floquet exponents. The analysis of stability of fully synchronized and of cluster or splay states is illustrated for several practically important choices of delay distributions and network topologies.

Synchronization-desynchronization transitions in complex networks: an interplay of distributed time delay and inhibitory nodes

We investigate the combined effects of distributed delay and the balance between excitatory and inhibitory nodes on the stability of synchronous oscillations in a network of coupled Stuart-Landau oscillators. To this end a symmetric network model is proposed for which the stability can be investigated analytically. It is found that beyond a critical inhibition ratio, synchronization tends to be unstable. However, increasing distributional widths can counteract this trend, leading to multiple resynchronization transitions at relatively high inhibition ratios. The extended applicability of the results is confirmed by numerical studies on asymmetrically perturbed network topologies. All investigations are performed on two distribution types, a uniform distribution and a Γ distribution.

Amplitude death in oscillator networks with variable-delay coupling

We study the conditions of amplitude death in a network of delay-coupled limit cycle oscillators by including time-varying delay in the coupling and self-feedback. By generalizing the master stability function formalism to include variable-delay connections with high-frequency delay modulations (i.e., the distributed-delay limit), we analyze the regimes of amplitude death in a ring network of Stuart-Landau oscillators and demonstrate the superiority of the proposed method with respect to the constant delay case. The possibility of stabilizing the steady state is restricted by the odd-number property of the local node dynamics independently of the network topology and the coupling parameters.

Analysis of a dc bus system with a nonlinear constant power load and its delayed feedback control

This paper tackles a destabilizing problem of a direct-current (dc) bus system with constant power loads, which can be considered a fundamental problem of dc power grid networks. The present paper clarifies scenarios of the destabilization and applies the well-known delayed-feedback control to the stabilization of the destabilized bus system on the basis of nonlinear science. Further, we propose a systematic procedure for designing the delayed feedback controller. This controller can converge the bus voltage exactly on an unstable operating point without accurate information and can track it using tiny control energy even when a system parameter, such as the power consumption of the load, is slowly varied. These features demonstrate that delayed feedback control can be considered a strong candidate for solving the destabilizing problem.