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Register a new userDissipative systems with nonlocal delayed feedback control
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We present a linear model, which mimics the response of a spatially extended dissipative medium to a distant perturbation, and investigate its dynamics under delayed feedback control. The time a perturbation needs to propagate to a measurement point is captured by an inherent delay time (or latency). A detailed linear stability analysis demonstrates that a non-zero system delay acts destabilizing on the otherwise stable fixed point for sufficiently large feedback strengths. The imaginary part of the dominant eigenvalue is bounded by twice the feedback strength. In the relevant parameter space it changes discontinuously along specific lines when switching between branches of eigenvalues. When the feedback control force is bounded by a sigmoid function, a supercritical Hopf bifurcation occurs at the stability-instability transition. The frequency and amplitude of the resulting limit cycles respond to parameter changes like the dominant eigenvalue. In particular, they show similar discontinuities along specific lines. These results are largely independent of the exact shape of the sigmoid function. Our findings match well with previously reported results on a feedback-induced instability of vortex diffusion in a rotationally driven Newtonian fluid [M. Zeitz, P. Gurevich, and H. Stark, Eur. Phys. J. E 38, 22 (2015)]. Thus, our model captures the essential features of nonlocal delayed feedback control in spatially extended dissipative systems.
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In-phase synchronization is a special case of synchronous behavior when coupled oscillators have the same phases for any time moments. Such behavior appears naturally for nearly identical coupled limit-cycle oscillators when the coupling strength is greatly above the synchronization threshold. We investigate the general class of nearly identical complex oscillators connected into network in a context of a phase reduction approach. By treating each oscillator as a black-box possessing a single-input single-output, we provide a practical and simply realizable control algorithm to attain the in-phase synchrony of the network. For a general diffusive-type coupling law and any value of a coupling strength (even greatly below the synchronization threshold) the delayed feedback control with a specially adjusted time-delays can provide in-phase synchronization. Such adjustment of the delay times performed in an automatic fashion by the use of an adaptive version of the delayed feedback algorithm when time-delays become time-dependent slowly varying control parameters. Analytical results show that there are many arrangements of the time-delays for the in-phase synchronization, therefore we supplement the algorithm by an additional requirement to choose appropriate set of the time-delays, which minimize power of a control force. Performed numerical validations of the predictions highlights the usefulness of our approach.
Weakly coupled limit cycle oscillators can be reduced into a phase model using phase reduction approach, and the phase model itself is helpful to analyze a synchronization. For example, phase model of two oscillators is one-dimensional differential equation for the evolution of a phase difference, and an existence of fixed points determines frequency-locking solutions. By treating each oscillator as a black-box possessing a single-input 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, which applied to oscillator after the phase reduction gives the same phase model as of the control-free case, yet a coupling strength is rescaled. The conventional delayed feedback control is limited to change a magnitude but not a sign of the coupling strength. In this work we present 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 an anti-phase synchronization using provided control algorithm. Additionally we show that the control force becomes non-invasive if our objective is a stabilization of an unstable phase difference for two coupled oscillators.
The purpose of this paper is to extend J.C. Willems theory of dissipative systems to the quantum domain. This general theory, which combines perspectives from the quantum physics and control engineering communities, provides useful methods for analysis and design of dissipative quantum systems. We describe the interaction of the plant and a class of exosystems in general quantum feedback network terms. Our results include an infinitesimal characterization of the dissipation property, which generalizes the well-known Positive Real and Bounded Real Lemmas, and is used to study some properties of quantum dissipative systems. We also show how to formulate control design problems using quantum network models, which implements Willems `control by interconnection for open quantum systems. This control design formulation includes, for example, standard problems of stabilization, regulation, and robust control.
The stability of functional differential equations under delayed feedback is investigated near a Hopf bifurcation. Necessary and sufficient conditions are derived for the stability of the equilibrium solution using averaging theory. The results are used to compare delayed versus undelayed feedback, as well as discrete versus distributed delays. Conditions are obtained for which delayed feedback with partial state information can yield stability where undelayed feedback is ineffective. Furthermore, it is shown that if the feedback is stabilizing (respectively, destabilizing), then a discrete delay is locally the most stabilizing (resp., destabilizing) one among delay distributions having the same mean. The result also holds globally if one considers delays that are symmetrically distributed about their mean.
Control of the motion of cavity solitons is one the central problems in nonlinear optical pattern formation. We report on the impact of the phase of the time-delayed optical feedback and carrier lifetime on the self-mobility of localized structures of light in broad area semiconductor cavities. We show both analytically and numerically that the feedback phase strongly affects the drift instability threshold as well as the velocity of cavity soliton motion above this threshold. In addition we demonstrate that non-instantaneous carrier response in the semiconductor medium is responsible for the increase in critical feedback rate corresponding to the drift instability.
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