Delayed Feedback Method Research Articles

Linear stability analysis of a combustor model with a delayed feedback tube

We conducted a linear stability analysis of a laboratory-scale combustor to develop a delayed feedback method for mitigating combustion oscillations. We employed linearized hydrodynamic equations and a flame transfer function in the frequency domain to assess the stability under various feedback tube configurations, including changes in the length, radius, and connection position. The results of the stability analysis demonstrated a significant improvement in stability when the length of the feedback tube was 0.56 times the wavelength of the unstable mode. Furthermore, the tube radius had an optimal value for stabilizing the system. The connection position also played a crucial role in stability. The effective feedback tube conditions determined in the analysis were applied to a real combustor to demonstrate the successful suppression of oscillations experimentally. The suppression mechanism is discussed using the acoustic power based on the calculation results. The acoustic power profile of the system revealed that the oscillation suppression was not simply an addition of acoustic dissipation, but importantly, a reduction in acoustic power generation at the flame.

Расчет оптимальных параметров систем управления нелинейными динамическими процессами импульсных преобразователей напряжения

A technique for choosing the optimal control system parameters for DC voltage converters is proposed, based on the joint application of the theory of linear automatic control systems and the theory of nonlinear dynamic systems. A small-signal structural dynamic model of an open loop of an automatic control system based on a direct step-up voltage converter with a control system for nonlinear dynamic processes based on a delayed feedback method is considered. Applying this model makes it possible to carry out a scientifically substantiated choice of the control system parameters for nonlinear dynamic processes using the methods of linear automatic control system theory. Bode diagrams of an open loop system are calculated without additional control of nonlinear dynamic processes and with additional control. The efficiency of using control systems for nonlinear dynamic processes is shown, which allows eliminating undesirable dynamic modes without additional parametric synthesis of the controller and, consequently, without reducing the system performance as a whole. In addition, applying these methods allows adjusting the controller parameters to increase the system performance without switching the system to undesirable dynamic modes. The results obtained can be applied at the stage of designing wide-class DC voltage pulse converters

Controlling chaos for energy harvesting via digital extended time-delay feedback

Chaotic vibrations may appear in nonlinear energy harvesting systems, which can be problematic when using the recovered power, as it may require an extra expenditure of energy to rectify the voltage signal or reduce the harvesting process efficiency when charging the battery. Both cases can derail the energy harvester’s functionality. An alternative in this situation is to explore chaos control to stabilize the system dynamics so that the recovered voltage signal is regular and more suitable for use in the applications of interest. This paper address this problem employing an extended delayed feedback method that combines a displacement actuator and a digital controller to implement the control mechanism. The control strategy is mathematically formulated and tested in a bistable energy harvesting system that often operates in a chaotic regime. The controller shows itself capable of stabilizing the chaotic dynamics at a very low energetic cost.

Complex Dynamics Analysis of Closed-Loop Supply Chain considering CSR and Fairness Concerns

This paper constructs a supply chain model composed of a manufacturer and a recycler. The manufacturer’s CSR and the recycler’s fairness concerns are introduced to the benchmark model in turn, and the optimal decision-making problems under different models are studied and compared. The findings show that the manufacturer’s utility will increase and the recycler’s utility will decrease when the manufacturer undertakes CSR within a reasonable range. The optimal utility of manufacturers does not change, and the utility of the recycler is affected by the proportion of CSR undertaken by the manufacturer when the recycler considers fairness concerns. Based on the CSR and fairness model, this paper constructs a dynamic decision system of production quantity and eco-innovation effort. We analyze the influence of adjustment speed on the dynamic decision system and obtain the conditions required to maintain system stability. The research conclusion indicates that with the increase of adjusting parameters, the system gradually appears chaotic state from a stable state and the chaotic state of the system has a negative impact on the utility of manufacturer and recycler. In order to avoid chaos in the system, this paper uses the delayed feedback method to control the system.

Stabilizing Circadian Rhythms in Bipolar Disorder by Chaos Control Methods

Bipolar disorder (BD), which involves mood swings between mania and depression, is associated with multiple relapses during long-term treatment and high suicide and morbidity rates. In BD, the circadian rhythms, which are measured by daily mood scores and actigraphic records, are disturbed. Chronotherapy has emerged as a potential treatment for BD because it stabilizes the disturbed circadian rhythms and improves BD symptoms. Concrete treatments include light therapy and combination therapy (light therapy and drugs). However, some patients have difficulty adjusting to light therapy; inappropriate light and duration of treatment increase risks for inducing mixed states and the emergence of conditions, such as hypomania and autonomic hyperactivation. Therefore, it is important to devise methods for optimizing chronotherapy for BD. We aimed to develop feedback signals for the frontal cortex, which were based on the delayed feedback method as one of the chaos control methods, to stabilize the disturbed circadian rhythms of BD. Concrete procedures of this study are indicated as follows: first, circadian rhythms of BD are reproduced using the frontal cortex and hypothalamus neural system, which has been previously proposed. Second, the delayed feedback signal is developed by using bifurcation analysis. Third, the effect of delayed feedback signal is evaluated by index for complexity, and power spectrum under the condition with/without stochastic noise in feedback term. We found that application of the delayed feedback signal to the frontal cortical neural activity induces the periodic state of circadian rhythms from the disturbed complex and is feasible for treating BD. However, when increasing the influence of noise in feedback term, the stabilizing effect is diminished. In conclusion, we developed a stabilizing method for disturbed circadian rhythms of BD using the circadian neural systems. The present study highlights the potential usefulness of the chaos control method for treating BD.

Control of Continuous Time Chaotic Systems With Unknown Dynamics and Limitation on State Measurement

This paper has dedicated to study the control of chaos when the system dynamics is unknown and there are some limitations on measuring states. There are many chaotic systems with these features occurring in many biological, economical and mechanical systems. The usual chaos control methods do not have the ability to present a systematic control method for these kinds of systems. To fulfill these strict conditions, we have employed Takens embedding theorem which guarantees the preservation of topological characteristics of the chaotic attractor under an embedding named “Takens transformation.” Takens transformation just needs time series of one of the measurable states. This transformation reconstructs a new chaotic attractor which is topologically similar to the unknown original attractor. After reconstructing a new attractor its governing dynamics has been identified. The measurable state of the original system which is one of the states of the reconstructed system has been controlled by delayed feedback method. Then the controlled measurable state induced a stable response to all of the states of the original system.

Robust desynchronization of Parkinson's disease pathological oscillations by frequency modulation of delayed feedback deep brain stimulation.

The hyperkinetic symptoms of Parkinson’s Disease (PD) are associated with the ensembles of interacting oscillators that cause excess or abnormal synchronous behavior within the Basal Ganglia (BG) circuitry. Delayed feedback stimulation is a closed loop technique shown to suppress this synchronous oscillatory activity. Deep Brain Stimulation (DBS) via delayed feedback is known to destabilize the complex intermittent synchronous states. Computational models of the BG network are often introduced to investigate the effect of delayed feedback high frequency stimulation on partially synchronized dynamics. In this study, we develop a reduced order model of four interacting nuclei of the BG as well as considering the Thalamo-Cortical local effects on the oscillatory dynamics. This model is able to capture the emergence of 34 Hz beta band oscillations seen in the Local Field Potential (LFP) recordings of the PD state. Train of high frequency pulses in a delayed feedback stimulation has shown deficiencies such as strengthening the synchronization in case of highly fluctuating neuronal activities, increasing the energy consumed as well as the incapability of activating all neurons in a large-scale network. To overcome these drawbacks, we propose a new feedback control variable based on the filtered and linearly delayed LFP recordings. The proposed control variable is then used to modulate the frequency of the stimulation signal rather than its amplitude. In strongly coupled networks, oscillations reoccur as soon as the amplitude of the stimulus signal declines. Therefore, we show that maintaining a fixed amplitude and modulating the frequency might ameliorate the desynchronization process, increase the battery lifespan and activate substantial regions of the administered DBS electrode. The charge balanced stimulus pulse itself is embedded with a delay period between its charges to grant robust desynchronization with lower amplitudes needed. The efficiency of the proposed Frequency Adjustment Stimulation (FAS) protocol in a delayed feedback method might contribute to further investigation of DBS modulations aspired to address a wide range of abnormal oscillatory behavior observed in neurological disorders.

Numerical time-delayed feedback control in a neural network with delay

In this article, by a nonstandard finite-difference method we obtain the general time delayed feedback control numerical discrete scheme for a delayed neural network model. Firstly, the local stability of the equilibria point is discussed according to the Neimark–Sacker bifurcation theory. Then, from the point of view of control, for any step-size, a general time delayed feedback control numerical algorithm is introduced to delay the onset of the Neimark–Sacker bifurcation at a desired point by choosing appropriate control parameters. This controller can deal with the general system that the natural equilibrium cannot be given by analytic expression. Finally, numerical examples are provided to illustrate the theoretical results. The results show that the time delayed feedback numerical scheme is better than a polynomial function time delayed feedback method.

Further development of the attitude difference method for estimating deflections of the vertical in real time

An improvement of the attitude difference method (ADM) to estimate deflections of the vertical (DOV) in real time is described in this paper. The ADM without offline processing estimates the DOV with a limited accuracy due to the response delay. The proposed model selection-based self-adaptive delay feedback (SDF) method takes the results of the ADM as the a priori information, then uses fitting and extrapolation to estimate the DOV at the current epoch. The active region selection factor Fth is used to take full advantage of the Earth model EGM2008 and the SDF with different DOV exhibitions. The factors which affect the DOV estimation accuracy are analyzed and modeled. An external observation which is specified by the velocity difference between the global navigation satellite system (GNSS) and the inertial navigation system (INS) with DOV compensated is used to select the optimal model. The response delay induced by the weak observability of an integrated INS/GNSS to the violent DOV disturbances in the ADM is compensated. The DOV estimation accuracy of the SDF method is improved by approximately 40% and 50% respectively compared to that of the EGM2008 and the ADM. With an increase in GNSS accuracy, the DOV estimation accuracy could improve further.

Bifurcation analysis and chaos control of the modified Chua’s circuit system

From the view of bifurcation and chaos control, the dynamics of modified Chua’s circuit system are investigated by a delayed feedback method. Firstly, the local stability of the equilibria is discussed by analyzing the distribution of the roots of associated characteristic equation. The regions of linear stability of equilibria are given. It is found that there exist Hopf bifurcation and Hopf-zero bifurcation when the delay passes though a sequence of critical values. By using the normal form method and the center manifold theory, we derive the explicit formulas for determining the direction and stability of Hopf bifurcation. Finally, chaotic oscillation is converted into a stable equilibrium or a stable periodic orbit by designing appropriate feedback strength and delay. Some numerical simulations are carried out to support the analytic results.

Controlling a Shape Memory Alloy Two-Bar Truss Using Delayed Feedback Method

This work discusses the use of chaos control in smart structures. An archetypal model of a shape memory alloy (SMA) two-bar truss is treated. This system exhibits both constitutive and geometrical nonlinearities presenting a complex nonlinear dynamics response including either the snap-through or the chaotic behaviors. A constitutive model that presents a close agreement with experimental data is employed to describe the themomechanical SMA behavior. A variation of the continuous time-delayed feedback method is employed as a control strategy. This variable structure controller is applied to the stabilization of unstable periodic orbits of the SMA structure avoiding the snap-through behavior.

Experimental control for initiating and maintaining rotation of parametric pendulum

In this paper, the authors have studied experimentally the control methods of a parametric pendulum excited harmonically to initiate and maintain a period one rotation – the most superior response for energy harvesting. For initiating the period one rotation inherent in the system, first the bang-bang method is applied. Then a new method where velocity is monitored is proposed and applied and finally the time-delayed feedback method with multi-switching is considered. Ultimately the problem of maintaining the rotation of the pendulum is addressed. For first time, robustness and sensitivity of the latter method to change of frequency and amplitude of excitation and added noise are studied. Finally, it has been demonstrated how the delayed feedback method can be applied in a system of two pendula to ensure synchronized rotation.

Feedback control of unstable periodic orbits in equivariant Hopf bifurcation problems

Symmetry-breaking Hopf bifurcation problems arise naturally in studies of pattern formation. These equivariant Hopf bifurcations may generically result in multiple solution branches bifurcating simultaneously from a fully symmetric equilibrium state. The equivariant Hopf bifurcation theorem classifies these solution branches in terms of their symmetries, which may involve a combination of spatial transformations and temporal shifts. In this paper, we exploit these spatio-temporal symmetries to design non-invasive feedback controls to select and stabilize a targeted solution branch, in the event that it bifurcates unstably. The approach is an extension of the Pyragas delayed feedback method, as it was developed for the generic subcritical Hopf bifurcation problem. Restrictions on the types of groups where the proposed method works are given. After addition of the appropriately optimized feedback term, we are able to compute the stability of the targeted solution using standard bifurcation theory, and give an account of the parameter regimes in which stabilization is possible. We conclude by demonstrating our results with a numerical example involving symmetrically coupled identical nonlinear oscillators.

Chaotic oscillations in user-centric cognitive radio networks and their control

In the cognitive radio networks, spectrum resources are dynamically shared among the users for radio resource usage optimisation. However, depending on the distributed decision-making algorithms of the spectrum sharing procedure, undesirable chaotic switching may occur in the network. For chaotic non-linear systems, the chaos control theory has been proposed and applied to a wide variety of non-linear systems to stabilise such systems. In this paper, the delay feedback method for chaos control is applied to the chaotic phenomena in the cognitive radio networks. We show that our method can stabilise the chaotic oscillations in the user-centric cognitive radio networks. By comparisons with conventional parameter tuning methods, we confirm that our control approach is more efficient and faster to stabilise the cognitive radio systems.

On the control of chaos via fractional delayed feedback method

In this paper the problem of controlling unstable fixed points (in discrete systems) and periodic orbits (in continuous system) is investigated via a new scheme involving fractional derivatives. This method is based on applying feedback of measured states and using the period of fixed points and periodic orbits. In this method there is no need of information for fixed point and periodic orbits, just the period is enough. The effectiveness of this method is investigated via some demonstrative example.

Frequency adaptation in controlled stochastic resonance utilizing delayed feedback method: Two-pole approximation for response function

Stochastic resonance (SR) enhanced by time-delayed feedback control is studied. The system in the absence of control is described by a Langevin equation for a bistable system, and possesses a usual SR response. The control with the feedback loop, the delay time of which equals to one-half of the period (2π/Ω) of the input signal, gives rise to a noise-induced oscillatory switching cycle between two states in the output time series, while its average frequency is just smaller than Ω in a small noise regime. As the noise intensity D approaches an appropriate level, the noise constructively works to adapt the frequency of the switching cycle to Ω, and this changes the dynamics into a state wherein the phase of the output signal is entrained to that of the input signal from its phase slipped state. The behavior is characterized by power loss of the external signal or response function. This paper deals with the response function based on a dichotomic model. A method of delay-coordinate series expansion, which reduces a non-Markovian transition probability flux to a series of memory fluxes on a discrete delay-coordinate system, is proposed. Its primitive implementation suggests that the method can be a potential tool for a systematic analysis of SR phenomenon with delayed feedback loop. We show that a D-dependent behavior of poles of a finite Laplace transform of the response function qualitatively characterizes the structure of the power loss, and we also show analytical results for the correlation function and the power spectral density.

A Proof for the Existence of Chaos in Diffusively Coupled Map Lattices with Open Boundary Conditions

We first study how to make use of the Marotto theory to prove rigorously the existence of the Li‐Yorke chaos in diffusively coupled map lattices with open boundary conditions (i.e., a high‐dimensional discrete dynamical system). Then, the recent 0‐1 test for chaos is applied to confirm our theoretical claim. In addition, we control the chaotic motions to a fixed point with delay feedback method. Numerical results support the theoretical analysis.

Elimination of Breathing Spiral Waves in the Aliev–Panfilov Model

We study spiral waves in an annulus with the Aliev–Panfilov model. The spiral pattern exhibits a regular breathing motion in a certain parameter range. A time-periodic force is applied for the elimination of spiral waves. The defibrillation becomes most effective if the period of the sinusoidal force is suitably chosen near the resonance condition. Next, we investigate a delayed feedback method for the elimination of spiral waves. The feedback method is effective because the feedback force oscillates with the breathing period and the resonance effect works naturally.

Dual input radix 23 SDF IFFT/FFT processor for wireless multi-channel real sound speakers using time division duplex scheme

There has been an increased demand for realistic audio services because of the recent developments in multimedia techniques and contents. In addition, the rapid growth in wireless communications has brought about a new era of wireless multimedia applications and services such as the transmission of video, images, and data over wireless broadband access networks. Accordingly, orthogonal frequency division multiple access (OFDMA) modulation technique based on time division duplex (TDD) scheme is an attractive wireless communication technology for high data rate wireless access with multi-channel real sound system. For modulation and de-modulation of the OFDMA, inverse fast Fourier transform (IFFT) and FFT processor are used and these are most energy consuming operation in OFDMA communication systems. In this paper, we proposed a two input radix 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> single delay feedback (TIR2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> SDF) method to simultaneously use the IFFT and FFT in one FFT processor. We also described the architectures and the control signals for the proposed TIR2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> SDF. The experimental results show that the FFT processor based on proposed TIR2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> SDF reduce about 43.3% of the hardware complexity in the FPGA implementation.

Transport control in a deterministic ratchet system

We study the control of transport properties in a deterministic inertia ratchet system via the extended delay feedback method. A chaotic current of a deterministic inertia ratchet system is controlled to a regular current by stabilizing unstable periodic orbits embedded in a chaotic attractor of the unperturbed system. By selecting an unstable periodic orbit, which has a desired transport property, and stabilizing it via the extended delay feedback method, we can control transport properties of the deterministic inertia ratchet system. Also, we show that the extended delay feedback method can be utilized for separation of particles in the deterministic inertia ratchet system as a particle's initial condition varies.