Time-delayed Feedback Research Articles

Time-delay Feedback Control of Fractional Chaotic Rössler Oscillator

In this research, we explore the utilization of the time-delayed feedback method to stabilize unstable steady states and aperiodic orbits within the chaotic fractional-order Rössler Oscillator. Employing the time-delay feedback control algorithm, we identify specific parameter ranges enabling the successful stabilization of unstable equilibria, considering variations in both feedback gain and time delay. Unlike previous research works where Caputo and Riemann-Liouville characterization of fractional derivatives are used, we are using Grünwald-Letnikov (GL) characterization because of its simplicity and ease of implementation and demonstrating the stability using the analytical and numerical analysis and plots of eigenvalues. Additionally, our analysis highlight the effectiveness of a sinusoidally modulated time delay in the control law, significantly expanding the stability region of steady states beyond the capabilities of the traditional time-delayed feedback scheme with a constant delay. Furthermore, the analysis of eigenvalues before and after applying the control strategy offers tangible insights into the system's stability dynamics.

Controlling nonlinearity and memory by feedback delay time in delay-based reservoir computing

Controlling nonlinearity and memory by feedback delay time in delay-based reservoir computing

Time-delayed feedback control for chaotic systems with coexisting attractors

<abstract><p>This study investigated the Hopf bifurcation of the equilibrium point of chaotic systems with coexisting attractors under the time-delayed feedback control. First, the equilibrium point and Hopf bifurcation of chaotic systems with coexisting attractors were analyzed. Second, the chaotic systems were controlled by time-delayed feedback, the transversality condition of Hopf bifurcation at the equilibrium point was discussed, and the time-delayed value of Hopf bifurcation at the equilibrium point was obtained. Lastly, the correctness of the theoretical analysis was verified by using the numerical results.</p></abstract>

Resonance analysis and time-delay feedback controllability for a fractional horizontal nonlinear roller system

<p>In this paper, we investigated the nonlinear vibration characteristics and time-delay feedback controllability of a fractional horizontal roll system, which is described by a fractional Duffing-van der Pol oscillator under an external harmonic excitation. We focused on the resonance of fractional roller systems and conducted corresponding vibration control. The amplitude-frequency equations of primary resonance and superharmonic resonance were obtained using the multiple scale method. The amplitude-frequency characteristic curves of the system with different parameters were presented, and the influence of system parameters on the curves was analyzed. In addition, the time-delay feedback controller was designed to control the parameter excitation vibration. The numerical simulation results have verified the effectiveness of the time-delay controller in eliminating the jumping and hysteresis phenomena of the rolling system. The comparisons of approximate analytical solution and numerical solution was fulfilled, and the result certifies the correctness and satisfactory precision of the approximately analytical solution. The analysis results provide certain theoretical guidance for the vibration reduction of the horizontal nonlinear roller system.</p>

High-speed parallel processing with photonic feedforward reservoir computing.

High-speed photonic reservoir computing (RC) has garnered significant interest in neuromorphic computing. However, existing reservoir layer (RL) architectures mostly rely on time-delayed feedback loops and use analog-to-digital converters for offline digital processing in the implementation of the readout layer, posing inherent limitations on their speed and capabilities. In this paper, we propose a non-feedback method that utilizes the pulse broadening effect induced by optical dispersion to implement a RL. By combining the multiplication of the modulator with the summation of the pulse temporal integration of the distributed feedback-laser diode, we successfully achieve the linear regression operation of the optoelectronic analog readout layer. Our proposed fully-analog feed-forward photonic RC (FF-PhRC) system is experimentally demonstrated to be effective in chaotic signal prediction, spoken digit recognition, and MNIST classification. Additionally, using wavelength-division multiplexing, our system manages to complete parallel tasks and improve processing capability up to 10 GHz per wavelength. The present work highlights the potential of FF-PhRC as a high-performance, high-speed computing tool for real-time neuromorphic computing.

Design and implementation for the state time-delay and input saturation compensator of gas turbine aero-engine control system

In order to obtain desired steady-state and transient performance of a gas turbine aero-engine during its operation, a usual way is to control the engine to operate in a manner close to its physical limits. The thermodynamic performance of a gas turbine aero-engine varies greatly within a wide working envelope, affected by factors such as engine performance degradation, actuator response lag and combustion time-delay. When the engine works in extreme conditions, for example rapid acceleration, some phenomena such as over-speed, over-temperature and over-pressure may occur. For this reason, various restrictions are often set in the engine control system to ensure safety during its operation, which are often realized by the form of saturation. In this paper, a compensation structure for gas turbine aero-engine state time-delay and input saturation is proposed. At first, the problems frequently occurred in the gas turbine aero-engine control system are investigated. And the gas turbine aero-engine control model is derived based on a thermodynamic non-linear model and its linearization model is established, together with its designed compensation structure. Differing from the existing technologies, a combination of dynamic system input saturation feedback and static state time-delay feedback is adopted in the proposed model. Then, the saturated nonlinearity and time-delay characteristic are introduced as convex combinations in the stability analysis. The local stability conditions of the closed-loop system are obtained by using Lyapunov–Krasovskii functionals and linear matrix inequalities (LMIs). Finally, hardware-in-loop (HIL) experiments are conducted to verify the feasibility of the proposed compensation method.

Postural control deficits due to bilateral pyramidal tract lesions exemplified by hereditary spastic paraplegia (HSP) originate from increased feedback time delay and reduced long-term error corrections.

Pyramidal tract lesions determine the clinical syndrome of Hereditary Spastic Paraplegia (HSP). The clinical impairments of HSP are typically exemplified by their deficits in mobility, leading to falls and injuries. The first aim of this study was to identify the cause for postural abnormalities caused by pyramidal tract lesions in HSP. The second aim was to specify the effect of treadmill training for postural abnormalities. We examined nine HSP patients before and after treadmill training, as well as nine healthy control subjects during perturbed and unperturbed stance. We found that HSP was associated with larger sway amplitudes and velocities. Body excursions following platform tilts were larger, and upper body excursions showed a phase lead. Model-based analysis detected a greater time delay and a reduced long-term error correction of postural reactions in the center of mass. HSP patients performed significantly better in clinical assessments after treadmill training. In addition, treadmill training reduced sway amplitudes and body excursions, most likely by increasing positional and velocity error correction gain as a compensatory mechanism, while the time delay and long-term error correction gain remained largely unaffected. Moreover, the upper body's phase lead was reduced. We conclude that HSP leads to very specific postural impairments. While postural control generally benefits from treadmill training, the effect seems to mainly rely on compensatory mechanisms, whereas the original deficits are not affected significantly.

Efficacy of a class of resonant nonlinear controllers of fractional-order for adaptive vibration control—Analysis, simulations and experiments

The efficacy of different class of resonant controllers has been successfully established for active vibration control. This class of controllers utilize a second-order filter fed back with the structural response (displacement, velocity or acceleration) and the control signal is synthesized as the linear/nonlinear function of the filter response. However, the performance of the controller largely depends on the tuning of the filter frequency to the structural natural frequency. Thus, for structural systems where the natural frequency and the excitation frequency are uncertain, one requires to tune the filter frequency adaptively. The present paper introduces two computationally efficient, fractional-order nonlinear adaptive resonant controllers with acceleration feedback for vibration control. The control signal is synthesized as the linear/nonlinear function of the filter variable and its fractional order derivative. Two parameters, namely the filter frequency and the control gain, are adaptively varied to control the phase and amplitude of the control signal, respectively. The objective of frequency adaptation is to maintain the phase of the control signal in the quadrature of the system displacement/acceleration. Theoretical analysis of the system is performed by the method of multiple scales under the assumption of slowly varying parameters. The results are verified by both numerical simulations and experiments. The theoretical and experimental results demonstrate the robustness of the proposed adaptive control against uncertainties in excitation force and system parameters. It is comprehensively shown that the fractional-order control is superior to the integer-order control already studied in previous works. Moreover, the effect of feedback time-delay is also studied, and it is shown that the fraction-order control can offset the detrimental effects of time-delay on the frequency response of the system.

Time-delayed Feedback Control of Chaotic Systems Based on Fractional Order Integration

Time-delayed Feedback Control of Chaotic Systems Based on Fractional Order Integration

161 A System Dynamics Model of Bovine Respiratory Disease Epidemiology and Prevention Strategies in an Integrated Beef Production System

Abstract Bovine respiratory disease (BRD) is a complex multifactorial disease that results in more than $1 billion in economic losses for the United States beef cattle industry, accounting for approximately 30% of the total antimicrobial use (AMU) in food animal production systems. The prevalence and production losses due to BRD vary significantly between different production systems, and is influenced by the host, pathogen, and environmental risk factors. These multiple risk factors together have a cumulative effect and will increase the prevalence of the disease if no preventive strategies are implemented in the system. Moreover, the increased AMU to treat BRD results in antimicrobial resistance, further complicating the issue. A linear, event-based solution to address such a complex problem will fail to consider the feedback loops and time delays in the system and may cause unintended consequences that might make the problem even more complex. Hence, we propose a more comprehensive system dynamic model to simulate an integrated beef production system with respect to BRD and predict the dynamic behavior of the system under different preventive strategies currently available to the beef industry. Our conceptual model accounts for host risk factors (auxiliary variables in the model) like age, nutritional status, daily weight gain, prior exposure to pathogens, and genetics. The environmental risk factors such as air quality, transportation, temperature humidity index, overcrowding, and source of animals were also included in the model. The various preventative strategies to reduce the incidence of BRD, including vaccination, biosecurity, adequate colostrum, and nutrition, were also added to the model. The key outcome variables considered were net profitability and AMU. The model simulates multiple scenarios by leveraging the auxiliary variables and can provide information about the system’s net profitability and total AMU. The model explains the interconnected effect of various risk factors associated with BRD in multiple complex scenarios and helps understand the combined effect of adopting various preventive strategies and host-environmental factors in controlling BRD. The initial result from our dynamic model suggests that using a combination of preventive measures and avoiding the selected host-environmental risk factors can minimize AMU while improving profit. The long-term goal of our modeling effort is to enable the beef system stakeholders to make informed decisions and strategies to mitigate the impact of BRD, thereby contributing to improved profitability. Through a systems-thinking approach, our model can contribute to addressing the challenges in BRD management currently faced by the US-based integrated beef production systems.

Square waves and Bykov T-points in a delay algebraic model for the Kerr-Gires-Tournois interferometer.

We study theoretically the mechanisms of square wave formation of a vertically emitting micro-cavity operated in the Gires-Tournois regime that contains a Kerr medium and that is subjected to strong time-delayed optical feedback and detuned optical injection. We show that in the limit of large delay, square wave solutions of the time-delayed system can be treated as relative homoclinic solutions of an equation with an advanced argument. Based on this, we use concepts of classical homoclinic bifurcation theory to study different types of square wave solutions. In particular, we unveil the mechanisms behind the collapsed snaking scenario of square waves and explain the formation of complex-shaped multistable square wave solutions through a Bykov T-point. Finally, we relate the position of the T-point to the position of the Maxwell point in the original time-delayed system.

On the Optimal Cubic Feedback Gain and Time Delay of Quasi-Zero Stiffness Vibration Isolators: Analytical and Numerical Study

On the Optimal Cubic Feedback Gain and Time Delay of Quasi-Zero Stiffness Vibration Isolators: Analytical and Numerical Study

Fractional Order Derivative and Time-Delay Feedback Enabled Stochastic Resonance for Bearing Fault Diagnosis

The benefits of noise can be found in nonlinear systems where a type of resonances can inject the noise into systems to enhance weak signals of interest, including stochastic resonance, vibrational resonance, and chaotic resonance. Such benefits of noise can be improved further by adding some items into the nonlinear systems. Considering the time-dependent memory of fractional-order derivative and time-delay feedback which makes the nonlinear systems take advantage of their historical information and makes the output of nonlinear systems affect the input by feedback control, therefore, we attempt to design the model of stochastic resonance (SR) enhanced by both fractional-order derivative and time-delay feedback. Among them, fractional-order derivative and time delay would reinforce the memory of nonlinear systems for historical information and feedback would use the output of systems to control the systems precisely. Therefore, we hope that their advantages would be fused to improve the weak signal detection performance of SR further. Then, it would be applied to bearing fault diagnosis and compared with that without fractional-order derivative and time-delay feedback and even other diagnostic methods. The experimental results indicate that the SR enhanced by fractional-order derivative and time-delay feedback where a local signal-to-noise ratio is designed as the objective function to optimize these tuning parameters of the proposed method could enhance early fault signature of bearings and outperform that without fractional-order derivative and time-delay feedback and even infogram method.

An underdamped and delayed tri-stable model-based stochastic resonance

Stochastic resonance (SR) is investigated in an underdamped tri-stable potential system driven by Gaussian colored noise and a periodic excitation, where both displacement and velocity time-delayed states feedback are considered. It is challenging to study SR in a second-order delayed multi-stable system analytically. In this paper, the improved energy envelope stochastic average method is developed to derive the analytical expressions of stationary probability density (SPD) and spectral amplification. The effects of noise intensity, damping coefficient, and time delay on SR are analyzed. The results show that the shapes of joint SPD can be adjusted to the desired structure by choosing the time delay and feedback gains. For fixed time delay, the SR peak is increased for negative displacement or velocity feedback gain. Meanwhile, the SR peak is decreased while the optimal noise intensity increases with increasing correlation time of noise. The Monte Carlo simulations (MCS) confirm the effectiveness of the theoretical results.

Complete and partial time-delay signature suppression in a laser array.

We model dynamics of a quantum dot based micropillar laser array subject to the time-delayed optical feedback. The global coupling provided by the feedback generates a rich set of various instabilities including chaotic regimes with strong time-delay signature in the autocorrelation function. We demonstrate that the dispersion of the array coupling phases leads to effective suppression of the time-delay signature due to the dispersion of the system's internal timescales. We find that the transition to the complete suppression of the time-delay signature appears via a chimera state where highly correlated and non-correlated laser outputs coexist. The degree of correlation in the chimera state depends on the coupling phase dispersion.

Response and Bifurcation of Fractional Duffing Oscillator under Combined Recycling Noise and Time-Delayed Feedback Control

Response and bifurcation of fractional Duffing oscillator under recycling noise and time-delayed feedback control are investigated. Firstly, based on the principle of the minimum mean square error and small time-delayed approximation and linearize the cubic stiffness term, the fractional derivative is equivalent to a linear combination of damping and restoring forces, and the original system is simplified into an equivalent integer order system. Secondly, the Itô differential equations and one-dimensional Markov process are obtained according to stochastic averaging method, and the stochastic stability and stochastic bifurcation of the system are analyzed. Lastly, through joint probability density function diagram and the stationary probability density function diagram, the stochastic bifurcation behavior of system under the different time-delay, fractional order and noise intensity are discussed respectively, the validity of the theory and the occurrence of bifurcation phenomenon are verified.

Effects of time-delayed vibration absorber on bandwidth of beam for low broadband vibration suppression

The effects of time-delayed vibration absorber (TDVA) on the dynamic characteristics of a flexible beam are investigated. First, the vibration suppression effect of a single TDVA on a continuous beam is studied. The first optimization criterion is given, and the results show that the introduction of time-delayed feedback control (TDFC) is beneficial to improving the vibration suppression at the anti-resonance band. When a single TDVA is used, the anti-resonance is located at a specific frequency by the optimum design of TDFC parameters. Then, in order to obtain low-frequency and broad bands for vibration suppression, multiple TDVAs are uniformly distributed on a continuous beam, and the relationship between the dynamic responses and the TDFC parameters is investigated. The obtained relationship shows that the TDVA has a significant regulatory effect on the vibration behavior of the continuous beam. The effects of the number of TDVAs and the nonlinearity on the bandgap variation are discussed. As the multiple TDVAs are applied, according to the different requirements on the location and bandwidth of the effective vibration suppression band, the optimization criteria for the TDFC parameters are given, which provides guidance for the applications of TDVAs in practical projects such as bridge and aerospace.

Stabilization of the Kawahara–Kadomtsev–Petviashvili equation with time-delayed feedback

Results of stabilization for the higher order of the Kadomtsev-Petviashvili equation are presented in this manuscript. Precisely, we prove with two different approaches that under the presence of a damping mechanism and an internal delay term (anti-damping) the solutions of the Kawahara–Kadomtsev–Petviashvili equation are locally and globally exponentially stable. The main novelty of this work is that we present the optimal constant, as well as the minimal time, that ensures that the energy associated with this system goes to zero exponentially.

Constraint of Air–Sea Interaction Significant to Skillful Predictions of North Pacific Climate Variations

Abstract The Pacific decadal oscillation (PDO) is the most dominant decadal climate variability over the North Pacific and has substantial global impacts. However, the interannual and decadal PDO prediction skills are not satisfactory, which may result from the failure of appropriately including the North Pacific midlatitude air–sea interaction (ASI) in the initialization for climate predictions. Here, we present a novel initialization method with a climate model to crack this nutshell and achieve successful PDO index predictions up to 10 years in advance. This approach incorporates oceanic observations under the constraint of ASI, thus obtaining atmospheric initial conditions (ICs) consistent with oceanic ICs. During predictions, positive atmospheric feedback to sea surface temperature changes and time-delayed negative ocean circulation feedback to the atmosphere over the North Pacific play essential roles in the high PDO index prediction skills. Our findings highlight a great potential of ASI constraints during initialization for skillful PDO predictions. Significance Statement The Pacific decadal oscillation is a prominent decadal climate variability over the North Pacific. However, accurately predicting the Pacific decadal oscillation remains a challenge. In this study, we use an advanced initialization method where the oceanic observations are incorporated into a climate model constrained by air–sea interactions. We can successfully predict the Pacific decadal oscillation up to 10 years in advance, which is hardly achieved by the state-of-the-art climate prediction systems. Our results suggest that the constraint of air–sea interaction during initialization is important to skillful predictions of the climate variability on decadal time scales.

Bifurcation control of solid angle car-following model through a time-delay feedback method

Bifurcation control of solid angle car-following model through a time-delay feedback method