A data-analysis method for identifying differential effects of time-delayed feedback forces and periodic driving forces in stochastic systems

In this work a method is developed for analyzing time series of periodically driven stochastic systems involving time-delayed feedback. The proposed data-analysis method yields dynamical models in terms of stochastic delay differential equations. On the basis of these dynamical models differential effects of driving forces and time-delayed feedback forces can be identified.     

Logical stochastic resonance in bistable system under α-stable noise

An autonomous stochastic system with nonlinear time-delayed feedback is investigated employing the stochastic simulation method. In the autonomous stochastic system with quadratic time-delayed feedback or under positive feedback, the nonlinear delay time fails to possess the role improving the noisy state of the system. In the autonomous stochastic system with cubic time-delayed feedback and under negative feedback, the nonlinear delay time can improve the noisy state, tuning the signal output, and generating incoherence and coherence maximization. We reveal a new kind of anti-coherence and coherence resonance phenomena induced by the nonlinear time delay in the autonomous stochastic system without external periodic force, discussing further the effects of the noise strength, the control parameter, and the feedback strength on anti-coherence and coherence resonance.     

Dual-anticipating, dual and dual-lag synchronization in modulated time-delayed systems

In this Letter, dual synchronization in modulated time delay system using delay feedback controller is proposed. Based on Lyapunov stability theory, we suggest a general method to achieve the dual-anticipating, dual, dual-lag synchronization of time-delayed chaotic systems and we find both its existing and sufficient stability conditions. Numerically it is shown that the dual synchronization is also possible when driving system contain two completely different systems. Effect of parameter mismatch on dual synchronization is also discussed. As an example, numerical simulations for the Mackey-Glass and Ikeda systems are conducted, which is in good agreement with the theoretical analysis.     

Correlation Functions of an Autonomous Stochastic System with Nonlinear Time Delays

The auto-correlation function and the cross-correlation of an autonomous stochastic system with nonlinear time-delayed feedback are investigated by using the stochastic simulation method. There are prominent differences between the roles of quadratic time-delayed feedback and cubic time-delayed feedback on the correlations of an autonomous stochastic system. Under quadratic time-delayed feedback, the nonlinear time delay fails to improve the noisy state of the autonomous stochastic system, the auto-correlation decreases monotonously to zero, and the cross-correlation increases monotonously to zero with the decay time. Under cubic time-delayed feedback, the nonlinear time delay can improve the noisy state of the autonomous stochastic system; the auto-correlation and the cross-correlation show periodical oscillation and attenuation, finally tending to zero with the decay time. Comparing the correlations of the system between with nonlinear time-delayed feedback and linear time-delayed feedback, we find that nonlinear time-delayed feedback lowers the correlation strength of the autonomous stochastic system.     

Improving temporal stability of stable cavitation activity of circulating microbubbles using a closed-loop controller based on pulse-length regulation

Stable cavitation (SC) has shown great potential for novel therapeutic applications. The spatiotemporal distribution of the SC activity of microbubbles circulating in a target region is not only correlated with the uniformity of treatment, but also with some undesirable effects. Therefore, it is important to achieve controllable and desirable SC activity in target regions for improved therapeutic efficiency and biosafety. This study proposes a closed-loop feedback controller based on pulse length (PL) regulation to improve the temporal stability of SC activity. Microbubbles circulating in a physiological flowing phantom were exposed to a 1 MHz focused transducer. The SC signals produced were initially received by another 7.5 MHz plane transducer, followed by high-speed signal acquisition and real-time processing. Based on the real-time-measured SC intensity excited by the current acoustic pulse, the proposed closed-loop feedback controller used three proportional coefficients to regulate the peak negative pressure (PNP) and PL of the next acoustic pulse during the acceleration and stable stages, respectively. The results show that the rise time and the temporal stability of the SC intensity of the microbubbles circulating in these two stages were improved significantly by the optimized proportional coefficients used in the proposed controller. Importantly, when compared with the traditional closed-loop feedback controller based on PNP regulation, the proposed closed-loop feedback controller based on PL regulation reduced the probability of a transition between stable and inertial cavitation, thus avoiding the risk of disadvantageous bioeffects in practical applications. These results demonstrate the effectiveness of the proposed PL-based closed-loop feedback controller and provide a feasible strategy for realization of controllable cavitation activity in applications.     

Control of unstable steady states in neutral time-delayed systems

We present an analysis of time-delayed feedback control used to stabilize an unstable steady state of a neutral delay differential equation. Stability of the controlled system is addressed by studying the eigenvalue spectrum of a corresponding characteristic equation with two time delays. An analytic expression for the stabilizing control strength is derived in terms of original system parameters and the time delay of the control. Theoretical and numerical results show that the interplay between thecontrol strength and two time delays provides a number of regions in the parameter space where the time-delayed feedback control can successfully stabilize an otherwise unstable steady state.     

基于时延补偿机理的网络化输出反馈控制器设计

本文考虑前向和反馈通道均存在网络时延的网络化控制系统,提出了一种新的动态输出反馈控制器的设计方法.针对状态可测和不可测两种情况,整个设计过程采用不同的时延补偿机理,来主动地消除网络时延的影响.同时,讨论了闭环网络化控制系统的稳定性.最后的仿真实例表明了该方法的有效性.     

The effect of time-delayed feedback on logical stochastic resonance

We examine the possibility of obtaining logic operation in a quartic-bistable system with linear time-delayed feedback subjected to Gaussian noise. The effect of time-delayed feedback on the effective potential well is investigated, and explicit numerical stimulation is conducted to study the influence of delay time and strength of the time-delayed feedback on the responses of the system. Although the response deteriorates slightly at low values of noise intensity with time-delayed feedback and the peak correct probability decreases from 100% when the delay time is too long, the reliability of obtaining the desired logic output is enhanced in the higher noise boundary with the help of moderate time-delayed feedback. We also found that increasing the linear factor of the system can shift the optimal noise intensity to a higher level.     

Chaos control via TDFC in time-delayed systems: The harmonic balance approach

This Letter deals with the problem of designing time-delayed feedback controllers (TDFCs) to stabilize unstable equilibrium points and periodic orbits for a class of continuous time-delayed chaotic systems. Harmonic balance approach is used to select the appropriate controller parameters: delay time and feedback gain. The established theoretical results are illustrated via a case study of the well-known Logistic model.     

Using synchronization for prediction of high-dimensional chaotic dynamics

We experimentally observe the nonlinear dynamics of an optoelectronic time-delayed feedback loop designed for chaotic communication using commercial fiber optic links, and we simulate the system using delay differential equations. We show that synchronization of a numerical model to experimental measurements provides a new way to assimilate data and forecast the future of this time-delayed high-dimensional system. For this system, which has a feedback time delay of 22 ns, we show that one can predict the time series for up to several delay periods, when the dynamics is about 15 dimensional.     

Vibration control by recursive time-delayed acceleration feedback

This paper presents a theoretical basis of time-delayed acceleration feedback control of linear and nonlinear vibrations of mechanical oscillators. The control signal is synthesized by an infinite, weighted sum of the acceleration of the vibrating system measured at equal time intervals in the past. The proposed method is shown to have controlled linear resonant vibrations, low-frequency non-resonant vibrations, primary and 1/3 subharmonic resonances of a forced Duffing oscillator. The concept of an equivalent damping and natural frequency of the system is also introduced. It is shown that a large amount of damping can be produced by appropriately selecting the control parameters. For some combinations of the control parameters, the effective damping factor of the system is shown to be inversely related to the time-delay in the small delay limit. Selection of the optimum control parameters for controlling the forced and free vibrations is discussed. It is shown that forced vibration is best controlled by unity recursive gain and smaller values of the time-delay parameter. However, the transient response can be optimally controlled by suitably selecting the time delay depending upon the gain. The delay values for the optimal forced response may be different from that required for the optimum transient response. When both are important, a suboptimal choice of the delay parameters with unity recursive gain is recommended.     

Network-based modelling and active control for offshore steel jacket platform with TMD mechanisms

The network-based modelling and active control for an offshore steel jacket platform with an active tuned mass damper mechanism is investigated. A network-based dynamic model of the offshore platform is first established. A network-based state feedback control scheme is developed. Under this scheme, the corresponding closed-loop system is modelled by a system with an artificial interval time-varying delay. Then, a delay-dependent stability criterion for the corresponding closed-loop system is derived. Based on this stability criterion, a sufficient condition on the existence of the network-based controller is obtained. It is found through simulation results that (i) both the oscillation amplitudes of the offshore platform and the required control force under the network-based state feedback controller are smaller than those under the nonlinear controller and the dynamic output feedback controller; (ii) the oscillation amplitudes of the offshore steel jacket platform under the network-based feedback controller are almost the same as the ones under the integral sliding mode controller, while the required control force by the former is smaller than the one by the latter.     

Medium Entropy Reduction and Instability in Stochastic Systems with Distributed Delay

Many natural and artificial systems are subject to some sort of delay, which can be in the form of a single discrete delay or distributed over a range of times. Here, we discuss the impact of this distribution on (thermo-)dynamical properties of time-delayed stochastic systems. To this end, we study a simple classical model with white and colored noise, and focus on the class of Gamma-distributed delays which includes a variety of distinct delay distributions typical for feedback experiments and biological systems. A physical application is a colloid subject to time-delayed feedback control, which is, in principle, experimentally realizable by co-moving optical traps. We uncover several unexpected phenomena in regard to the system’s linear stability and its thermodynamic properties. First, increasing the mean delay time can destabilize or stabilize the process, depending on the distribution of the delay. Second, for all considered distributions, the heat dissipated by the controlled system (e.g., the colloidal particle) can become negative, which implies that the delay force extracts energy and entropy of the bath. As we show here, this refrigerating effect is particularly pronounced for exponential delay. For a specific non-reciprocal realization of a control device, we find that the entropic costs, measured by the total entropy production of the system plus controller, are the lowest for exponential delay. The exponential delay further yields the largest stable parameter regions. In this sense, exponential delay represents the most effective and robust type of delayed feedback.     

Time-Delayed Feedback Control in a Single-Mode Laser System

The effects of time-delayed feedback control in a single-mode laser system is investigated. Using the small time delay approximation, the analytic expression of the stationary probability distribution function of the laser field is obtained. The mean, normalized variance and skewness of the steady-state laser intensity are calculated. It is found that the time-delayed feedback control can suppress the intensity fluctuation of the laser system. The numerical simulations are in good agreement with the approximate analytic results.     

Actuator saturation control of uncertain structures with input time delay

This paper presents a robust saturation control approach for active vibration attenuation of building structures involving parameter uncertainties and input time delay. The parameter uncertainties are described in both polytopic and norm-bounded forms and represent the variations of floor masses, stiffnesses and damping coefficients. The input time delay can be time-varying within a known bound. In terms of the feasibility of certain delay-dependent linear matrix inequalities (LMIs), a state feedback controller can be designed to guarantee the robust stability and performance of the closed-loop system in the presence of parameter uncertainties, actuator saturation, and input time delay. The effectiveness of the proposed approach is investigated by numerical simulations on the vibration control of a three-storey building structure subject to seismic excitation. It is validated that the designed robust saturation controller can effectively suppress the structural vibration and keep the system stability when there are parameter uncertainties and input time delay.     

Stochastic averaging of quasi-integrable Hamiltonian systems with delayed feedback control

A stochastic averaging method for quasi-integrable Hamiltonian systems with time-delayed feedback control is proposed. First, a quasi-integrable Hamiltonian system with delayed feedback control subjected to Gaussian white noise excitations is formulated and then transformed into Itô stochastic differential equations without time delay. Then, the averaged Itô stochastic differential equations for the system are derived and the stationary solution of the averaged Fokker–Planck–Kolmogorov (FPK) equation associated with the averaged Itô equations is obtained for both non-resonant and resonant cases. Finally, three examples are worked out in detail to illustrate the application and effectiveness of the proposed method and the effect of time delayed feedback control on the response of the systems.     

Delay control of coherence resonance in type-I excitable dynamics

We consider a simple paradigmatic system of type-I excitability subject to noise and time-delayed feedback. This system is governed by a global bifurcation, namely a saddle-node bifurcation on a limit cycle. In the absence of noise, delay can induce complex dynamics including multiple stable and unstable periodic orbits. Random fluctuations result in coherence resonance in dependence on the noise strength. We show that this effect can be enhanced by delayed feedback control with suitably chosen feedback strength and time delay.     

Nonlinear analysis,design and vibration isolation for a bilinear system with time-delayed cubic velocity feedback

This paper combines cubic nonlinearity and time delay to improve the performance of vibration isolation. Nonlinear dynamics properties, design methodology and isolation performance are studied for a piecewise bilinear vibration isolation system with the time-delayed cubic velocity feedback control. By the multi-scale perturbation method, the equivalent stiffness and damping are first defined to interpret the effect of feedback control loop on dynamics behaviours, such as frequency island phenomenon. Then, a design criterion is proposed to suppress the jump phenomenon induced by the saddle-node bifurcation. With the purpose of obtaining the desirable vibration isolation performance, stability conditions are obtained to find appropriate feedback parameters including gain and time delay. Last, the influence of the feedback parameters on vibration transmissibility is assessed. Results show that the strategy developed in this paper is practicable and feedback parameters are significant factors to alter dynamics behaviours, and more importantly, to improve the isolation effectiveness for the bilinear isolation system.     

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

Recently an act-and-wait modification of time-delayed feedback control has been proposed for the stabilization of unstable periodic orbits in nonautonomous dynamical systems (Pyragas and Pyragas, 2016 [30]). The modification implies a periodic switching of the feedback gain and makes the closed-loop system finite-dimensional. Here we extend this modification to autonomous systems. In order to keep constant the phase difference between the controlled orbit and the act-and-wait switching function an additional small-amplitude periodic perturbation is introduced. The algorithm can stabilize periodic orbits with an odd number of real unstable Floquet exponents using a simple single-input single-output constraint control.     

Continuous pole placement method for time-delayed feedback controlled systems

Continuous pole placement method is adapted to time-periodic states of systems with timedelay. The method is applied for finding an optimal control matrix in the problem ofstabilization of unstable periodic orbits of dynamical systems via time-delayed feedbackcontrol algorithm. The optimal control matrix ensures the fastest approach of a perturbedsystem to the stabilized orbit. An application of the pole placement method to systemswith time delay meets a fundamental problem, since the number of the Floquet exponents isinfinity, while the number of control parameters is finite. Nevertheless, we show thatseveral leading Floquet exponents can be efficiently controlled. The method is numericallydemonstrated for the Lorenz system, which until recently has been considered as a systeminaccessible for the standard time-delayed feedback control due to the odd-numberlimitation. The proposed optimization method is also adapted for an extended time-delayedfeedback control algorithm and numerically demonstrated for the Rössler system.