Time-delayed Feedback Control Method Research Articles

Bifurcation Analysis and Chaos Control of Carrier Phase-Shifted Cascaded H-Bridge Inverter

Cascaded H-bridge inverters, which belong to strongly nonlinear time-varying system with complex dynamics, have been widely used in high-voltage and high-power engineering fields. In this paper, a carrier phase-shifted sinusoidal pulse-width modulation cascaded H-bridge inverter is taken as an example with piecewise analysis to establish the discrete iterative mathematical model. Furthermore, the nonlinear dynamic behavior of this inverter system is investigated using folding diagrams, spectrum diagrams, bifurcation diagrams, Lyapunov exponents, etc. We find that this inverter can exhibit multiperiod bifurcation, tangential bifurcation and paroxysmal chaos within certain ranges of control coefficients and other circuit parameters, which will influence the system’s stability seriously. On the basis of this discovery, the sinusoidal time-delay feedback control method is proposed. The difference between the load current of the controlled object and its own delay is used to form a feedback term, which is then fed into the sine function and proportional link to obtain the control term. Moreover, the Jacobian matrix of this feedback inverter system is derived, and the constraints on the feedback control coefficient are obtained based on the Jury criterion. Simulation results have verified the effectiveness of such a sinusoidal time-delay feedback control method, which can improve the stability and anti-interference of the system. The research can be used to provide some reference and guidance for the circuit design, debugging and control of cascaded H-bridge inverters.

The Complexities in the R&D Competition Model with Spillover Effects in the Supply Chain

This study aims to investigate the research and development (R&D) competition within the supply chain, focusing on two aspects: R&D competition at the manufacturing level and competition in pricing strategies. This paper establishes a dynamic game model of R&D competition, comprising two manufacturers and two retailers, with both manufacturers exhibiting bounded rationality. The key findings are as follows: (1) an increase in the adjustment speed positively affects the chaotic nature of the R&D competition system, leading to a state of disorder. This chaotic state has adverse implications for manufacturing profitability. (2) The spillover effect exhibits a positive relationship with the level of chaos in the R&D competition system. A greater spillover effect contributes to a more turbulent environment, which subsequently impacts the profitability of manufacturers. (3) R&D cost parameters exert a positive influence on the stability of the R&D competition system. When the system reaches a state of equilibrium, an escalation in the R&D cost parameters poses a threat to manufacturer profitability. (4) Retailer costs play a detrimental role in the stability of the R&D competition system. As retailer costs increase, there is a decline in R&D levels, thereby diminishing manufacturer profitability. (5) To mitigate the chaotic state, we propose the implementation of the time-delayed feedback control (TDFC) method, which reflects a more stable state in the R&D competition system.

Control of chaos with time-delayed feedback based on deep reinforcement learning

The time-delayed feedback control method, as one of popular methods for chaos control, is noninvasive and flexible for various dynamical systems from different fields of science and technology. In the method, however, an appropriate choice of the feedback gain is challenging, which requires the explicit mathematical model of the controlled system for stability analysis. Additionally, another limitation of the method is the so-called odd number limitation. The two problems restrict its application to practical situations. Fortunately, a technique called deep reinforcement learning is capable of learning the controlled environment (the dynamical system), by continually interacting with the controlled system, which makes it possible to obtain the right feedback gain without the requirement of the accurate mathematical model of the system. Hence, in this work, a time-delayed feedback control method based on deep reinforcement learning is put forward to solve the two problems. Compared with the traditional time-delayed feedback control, the proposed method, as a data-driven method due to the combination with deep reinforcement learning, offers a time-varying feedback gain according to the well-trained policy learned by the deep reinforcement learning algorithm. It maintains the non-invasive property of the time-delayed feedback control, but expands the operating range due to the time-varying feedback gain overcoming the odd number limitation. With numerical simulations, the proposed method is successfully applied to three different kinds of systems, the discrete logistic map, the non-autonomous Duffing oscillator and the autonomous Lorenz system.

Switching between periodic orbits in impact oscillator by time-delayed feedback methods

An ability to exchange between different attractors can bring adaptability and new functionalities to an engineering system. While nonlinear controllers are widely used to stabilize on a preferred orbit, there is only a few which can exchange back and forth between two or more stable periodic responses. This work proposes a new variation of the original Time-Delayed Feedback (TDF) control method capable to effectively switch back and forth between periodic orbits having only a limited knowledge of the system dynamics. The stable or unstable periodic responses of the system that are inside or outside a chaotic attractor can be targeted. The proposed control method named here as the Fractional Time-Delayed Feedback Control (FTDF) is tested numerically and experimentally using a newly developed impact oscillator rig. Various responses including impacting and non-impacting orbits, high period attractors and chaos are considered. The main advantages of the FTDF over the TDF are presented by showcase scenarios where the TDF alone cannot safely perform the exchange between orbits.

Simulation and Experimental Research on Time Varying Delay Feedback Control Under Random Excitation

In this paper, the dynamic vibration absorber considering time delay is taken as the research object, and the effect of time delay on the vibration characteristics of the dynamic vibration absorber is studied by combining theory and experiment. In this paper, a time-varying time-delay feedback control method is proposed. Firstly, the instantaneous frequency of complex excitation is obtained by wavelet analysis. The instantaneous frequency determines the critical time-delay parameters of the dynamic vibration absorber with time-delay as the time-delay control quantity. Then, the optimal fixed time-delay feedback gain is calculated by improved particle swarm optimization. Finally, the time-varying delay gain parameter controls the time-delay dynamic vibration absorber. The vibration reduction effect of this approach under complicated excitation is investigated using a 2-DOF vibration system with a time-delay dynamic vibration absorber as an example. Based on simulation calculation, the bench test of the control strategy is carried out, and the influence of the control strategy on the control dynamic characteristics of the primary system is analyzed. The simulation and experimental results are consistent, laying the groundwork for this paper’s research on time-varying delay control.

Time delay feedback control for milling chatter suppression by reducing the regenerative effect

Chatter often occurs in the milling process and results in poor surface quality, which limits machining efficiency. In order to suppress milling chatter, a time delay feedback control method is proposed and achieved in milling chatter suppression. The controller is designed by solving a Lyapunov function containing time delay and performed in the active control system. Different from traditional real time feedback control methods, this method suppresses milling chatter by reducing the regenerative effect with higher energy efficiency. Simulated andexperimental results indicate the critical depth of milling improves nearly four times with the proposed controller. In addition, the energy cost in the time delay feedback controller is about one-fifth of that in the real time feedback controller. The small actuating force requirement makes active chatter suppression possible in practical manufacture.

Station-keeping for a solar sail during lander/probe deployment using feedback control

Due to its propellantless nature, a solar sail can provide the primary propulsion system for a high energy mission, such as that of a multiple asteroid rendezvous. Upon arrival at an asteroid, it is often desirable to interact with the surface of the body, such as for sample extraction. The deployment of a lander from a solar sail carries the difficulty of an instantaneous, and sometimes considerable, change to the system dynamics at the point of separation. This paper investigates the effects of changing sail performance during the release of multiple “ChipSat” probes as well as a large MASCOT-type lander and the control of the sail into a positional hold at an equilibrium point or periodic orbit. In one scenario, 20 ChipSat probes are released, with one-hour spacing between each release. The sail is then controlled to maintain the sailcraft close to the initial deployment point. The performance of a Linear Quadratic Regulator (LQR) is compared with maintaining a fixed sail attitude after deployment. In a second scenario, at the point of separation of the larger MASCOT-type lander, there will be a considerable instantaneous change in the sail characteristic acceleration, as opposed to the gradual small change for the staggered deployment of the small ChipSat probes. It is shown that the Time-Delayed Feedback Control (TDFC) method is effective in controlling the orbit of the sail after this deployment. The sail converges to an orbit in the same region of phase space when deployment is made from both a lower and higher inclination orbit.

Stability, Bifurcation, and Chaos Control of Two-Sided Market Competition

Benefitting from the popular uses of internet technologies, two-sided market has been playing an increasing prominent role in modern times. Users and developers can interact with each other through two-sided platforms. The two-sided market structure has been investigated profoundly. Through building a dynamics two-sided market model with bounded rational, stability conditions of the two-sided market competition system are presented. With the help of bifurcation diagram, Lyapunov exponent, and strange attractor, the stability of the two-sided market competition model is simulated. At last, we use the time-delayed feedback control (TDFC) method to control the chaos. Our main results are as follows: (1) when the adjustment speed of two-sided increases, the system becomes bifurcation, and chaos state happens finally. When the system is stable, the consumer fee is positive while developer fee is negative. (2) When the user externality increases, the stable area of the system increases, and the difference in user externality leads the whole system more stable. When the system is stable, the developer fee decreases. (3) The stable area becomes larger when developer externality increases; when the system is stable, the user fee becomes lower and developer fee becomes higher when developer externality increases. (4) The TDFC method is presented for controlling the chaos; we find that the system becomes more stable under the TDFC method.

Complex Dynamics in a Closed-Loop Supply Chain with Risk Aversion and Fairness Concerns Under Supply Disruption

In this paper, we consider a closed-loop supply chain (CLSC) consisting of two suppliers, one manufacturer, one risk-averse retailer and one fair-caring third-party in the presence of supply disruption. We focus on establishing a dynamic Stackelberg game model with bounded rational expectation and analyzing the game evolution process. The effects of key parameters on the Nash equilibrium solutions and their stability are investigated, as well as the complex dynamical behaviors of the CLSC system are explored by using the stability region, bifurcation graph, the largest Lyapunov exponent (LLE), strange attractors, etc. Moreover, the performance of channel members under different values of parameters is researched by utilizing the (average) expected profits or utilities index. The analysis results reveal that the excessive fast adjustment speed of the manufacturer will lead to the system losing stability and falling into chaos. Also, the retailer’s risk aversion and the third party’s fairness concerns have a destabilization effect on the Nash equilibrium point, while the possibility of supply disruption has different effects on the scope of the adjustment speed of decision variables of the manufacturer. Furthermore, in most cases, an over the top adjustment speed of the manufacturer is disadvantageous to all the channel members for more expected profits, but the third-party can achieve a better performance when the system is in periodic state. Finally, the time-delay feedback control method is proposed to eliminate the system chaos.

Time feedback control in a modified Sprott E model

This paper is concerned with chaos control for a modified Sprott E system. Applying time-delayed feedback control method, we establish some new conditions to control chaotic behaviour of modified S...

Time feedback control in a modified Sprott E model

This paper is concerned with chaos control for a modified Sprott E system. Applying time-delayed feedback control method, we establish some new conditions to control chaotic behaviour of modified Sprott E system. With the aid of local stability analysis, we theoretically prove the occurrences of Hopf bifurcation. Computer simulations are implemented to support analytical results. Finally, a brief conclusion is included.

State-dependent act-and-wait time-delayed feedback control algorithm

Time-delayed feedback control (TDFC) method is one of the most popular experimental tools used for stabilization of unstable periodic orbits in nonlinear dynamical systems. The theory of the method is difficult since systems controlled by time-delayed feedback evolve in an infinite-dimensional function space. Here we modify the TDFC by appending the feedback loop with a state-dependent switch. The switch closes and opens the feedback loop in the dependence of the phase of the output signal, which is measured by a specially designed phase detector. This modification represents a further development of the act-and-wait concept applied to time-delayed feedback controllers. The concept allows to reduce the phase space dimension of time-delayed feedback systems and to significantly simplify the theory. In previous works, the time-delayed feedback loop was tuned on and off periodically, independently of the state of the system. We discuss the advantages of our algorithm with respect to previously considered modifications. The efficacy of the proposed algorithm is numerically demonstrated with autonomous Rössler and Lorenz systems as well as a nonautonomous Duffing oscillator.

Surge Vibration-Induced Nonlinear Behavior Regulation of Power Amplifier for Magnetic Bearing in a 315 kW Centrifugal Compressor

The severe vibration induced by surge and rotating stall is an obstacle to the stability of a magnetically suspended centrifugal compressor (MSCC). In order to suppress the severe vibration caused by surge instability, this paper focuses on compressor surge performance improvements enabled by power amplifier control improvements which result in increased dynamic load capacity (DLC) of the systems axial thrust magnetic bearing. A complete discrete-time model of the active magnetic bearing (AMB) power amplifier, composed of three piecewise linear intervals, is developed. A comprehensive view of the dynamic evolution process from stable state to bifurcation for the power amplifier is also analyzed. In order to stabilize the unstable periodic orbits in the power amplifier, a time-delay feedback control (TDFC) method is introduced to enhance the stability of the power amplifier, while the MSCC is subjected to the surge instability. Simulation results show that the stable region of the power amplifier is extended significantly using the TDFC method. Finally, the experimental investigations performed by an MSCC test rig demonstrate the effectiveness of the proposed solution under the conditions of modified surge and mild surge.

Spatial static bifurcation of 2-D discrete dynamical systems and its control}{Spatial static bifurcation of 2-D discrete dynamical systems and its control

As bifurcation control has remained a hot research field in recent years, a unified time-delayed feedback control method is adopted in this paper to control the spatial static bifurcation of 2-D discrete dynamical systems. By transferring the existing bifurcation or by producing a new fork-shaped, trans-critical, or saddle-node bifurcation, this method determines and then controls the spatial static bifurcation of 2-D discrete dynamical systems.

Bifurcation control for a Duffing oscillator with delayed velocity feedback

In this paper, a Duffing oscillator model with delayed velocity feedback is considered. Applying the time delayed feedback control method and delayed differential equation theory, we establish some criteria which ensure the stability and the existence of Hopf bifurcation of the model. By choosing the delay as bifurcation parameter and analyzing the associated characteristic equation, the existence of bifurcation parameter point is determined. We found that if the time delay is chosen as a bifurcation parameter, Hopf bifurcation occurs when the time delay is changed through a series of critical values. Some numerical simulations show that the designed feedback controllers not only delay the onset of Hopf bifurcation, but also enlarge the stability region for the model.

Experimental bifurcation control of a parametric pendulum

The aim of the study is to maintain the desired period-1 rotation of the parametric pendulum over a wide range of the excitation parameters. Here the Time-Delayed Feedback control method is employed to suppress those bifurcations, which lead to loss of stability of the desired rotational motion. First, the nonlinear dynamic analysis is carried out numerically for the system without control. Specifically, bifurcation diagrams and basins of attractions are computed showing co-existence of oscillatory and rotary attractors. Then numerical bifurcation diagrams are experimentally validated for a typical set of the system parameters giving undesired bifurcations. Finally, the control has been implemented and investigated both numerically and experimentally showing a good qualitative agreement.

Dynamics in a Cournot investment game with heterogeneous players

In this paper we concern the investment process in a duopoly game played by heterogeneous players. A discrete and dynamic system is built for the case that a boundedly rational player adjusts its investment decision by the locally marginal profit and a naïve player chooses its strategy according to the opponent’s action in the previous period. By stability analysis of the system, we show that the boundary equilibrium is unstable and obtain the stability conditions for the interior equilibrium. Numerical simulations are used to provide evidence for the influence of the model parameters on the system stability and on the complicated behaviors in the system evolution. It is shown that the system with varying model parameters may drive to chaos and the loss of stability may be caused by period doubling bifurcations or Neimark–Sacker bifurcations. It is also shown that the time-delayed feedback control method can be used to keep the system from instability and chaos. All the numerical simulations show that the capital depreciation rate has great influence on the system evolution: a smaller depreciation rate has a stronger stabilization effect on the survival of the system and makes the system easier to control from chaos.

A Feedback Control of Fluctuations in Simple Molecular Dynamics

Suppression of diffusion processes generated by fluctuations in small-scale systems is crucial for its manipulation especially in nanotechnology. In this work, we focus on thermal fluctuations and their diffusion effect in a model of molecular dynamics. First, we propose time-delayed feedback control method instead of decreasing temperature which is a conventional way of controlling thermal fluctuations. We investigate numerically an effect of delayed-feedback on random walks in discrete time systems and find that diffusion processes of the controlled random walk is suppressed with increasing delay time. Second, we apply the proposed method to simple models of molecular dynamics which are composed of a single particle as well as two flocculated ones with the Lennard-Jones potential. Numerical simulations of the molecular dynamics show the effectiveness of the proposed method for suppressing diffusion processes observed in molecular dynamics.

Optimal control of nonlinear vibration resonances of single-walled nanotube beams

An optimal time-delay feedback control method is provided to mitigate the primary resonance of a single-walled carbon nanotube (SWCNT) subjected to a Lorentz force excited by a longitudinal magnetic field. The nonlinear governing equations of motion for the SWCNT under longitudinal magnetic field are derived and the modulation equations are obtained by using the method of multiple scales. The regions of the stable feedback gain are worked out by using the stability conditions of eigenvalue equation. Taking the attenuation ratio as the objective function and the stable vibration regions as constrained conditions, the optimal control parameters are worked out by using minimum optimal method. The optimal controllers are designed to control the dynamic behaviors of the nonlinear vibration systems. It is found that the optimal feedback gain obtained by the optimal method can enhance the control performance of the primary resonance of SWCNT devices.

On the chaos control of the Qi system

This paper is devoted to the problem of controlling chaos in the Qi system. A time-delayed feedback control method is applied to suppress chaos to unstable equilibria or unstable periodic orbits. Using a local stability analysis, we theoretically prove that the Hopf bifurcation occurs. Some numerical simulations are carried out to support the theoretical predictions. Finally, main conclusions are drawn.