Time-delayed Feedback Control Research Articles

Dynamical analysis of a stay cable with a nonlinear energy sink and time-delayed feedback control

Stay cables exist commonly in engineering and can exhibit sustainable and large vibrations under external excitations. The unexpected oscillations always lead to the fatigue and damage of structures, resulting in the economical loss. This paper studies the vibration control of a stay cable system with a nonlinear vibration absorber and time delay control. The governing equations of motion of the stay cable system under harmonic excitations are given. The stability and different bifurcations of the trivial equilibrium point of the system are analyzed. The bifurcation diagrams with respect to the control parameters are illustrated. The frequency responses of the forced system are obtained by using the harmonic balance method. Numerical simulations are performed to validate the obtained results and various dynamical phenomena are shown, such as the multi-period oscillations, quasi-periodic responses and chaotic motions. The effects of the time-delayed feedback control on the nonlinear dynamics and vibration reduction of the system are extracted. It is shown that the proper values of the time delay and feedback gain can improve the vibration reduction performance of the system, such as the suppression of the resonance peaks and elimination of the complex responses.

Dynamic stability and control of vortex induced oscillations of tension leg platform tethers

Dynamic stability and control of sway oscillations of Tension Leg Platform (TLP) tether under regular vortex shedding are investigated. The TLP is modelled as a single degree of freedom simply supported uniform beam under pretension, assuming transverse oscillation to take place only in its fundamental natural mode. The tether is subjected to parametric excitation due to the heave motion of the TLP. The incremental harmonic balance (IHB) method is employed to investigate the period-1 and period-2 responses of the TLP as a function of the detuning parameter. The same system is then investigated under time-delayed state feedback control. Suitable control parameters are selected from the analysis of the local stability of the system equilibrium. The stability of solutions is analyzed by the semidiscretization method with the modified Floquet theory. The stable and periodic responses obtained by the IHB method are verified by directly integrating the equation of motion using the fifth-order Runge-Kutta algorithm at discrete points. Finally, the efficacy of the control scheme has been investigated in the plane of response amplitude versus excitation amplitude. The results show that appropriate selections of control parameters can effectively reduce the response amplitude to a great extent with the improved measure of stability.

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.

Time-delay feedback control of a suspended cable driven by subharmonic and superharmonic resonance

The longitudinal time-delay feedback control strategy is implemented to suppress the subharmonic and superharmonic responses of the suspended cable. Formulated based on the Hamilton variational principle and the longitudinal time-delay feedback strategy, the in-plane nonlinear differential equations for a suspended cable are formulated, and Galerkin method is utilized to transform the equations into delayed differential equations. By using the method of multiple scales, approximate solutions for the 1/2-subharmonic and second-order superharmonic resonance responses of the controlled suspended cable were derived. The effects of the time delay and control gain are obtained through numerical examples. The research results demonstrate that through the adjustment of control gain and time delay, resonance regions can be avoided, effectively suppressing large-amplitude vibrations in the suspended cable, thereby achieving favorable control performance.

Mobility, response and transport in non-equilibrium coarse-grained models

We investigate two different types of non-Markovian coarse-grained models extracted from a linear, non-equilibrium microscopic system, featuring a tagged particle coupled to underdamped oscillators. The first model is obtained by analytically ‘integrating out’ the oscillators and the second is based on a derivation using projection operator techniques. We observe that these two models behave very differently when the tagged particle is exposed to external harmonic potentials or pulling forces. Most importantly, we find that the analytic model has a well defined friction kernel and can be used to extract work, consistent with the microscopic system, while the projection model corresponds to an effective equilibrium model, which cannot be used to extract work. We apply the analysis to two popular non-equilibrium systems, time-delay feedback control and the active Ornstein–Uhlenbeck process. Finally, we highlight that our study could have important consequences for dynamic coarse-graining of non-equilibrium systems going far beyond the linear systems investigated in this manuscript.

Fractional order time-delayed feedback control of hysteresis dynamics in giant magnetostrictive actuators

Fractional order time-delayed feedback control of hysteresis dynamics in giant magnetostrictive actuators

Periodic oscillations in electrostatic actuators under time delayed feedback controller

In this paper, we prove the existence of two positive T-periodic solutions of an electrostatic actuator modeled by the time-delayed Duffing equation ẍ(t)+fD(x(t),ẋ(t))+x(t)=1−eV2(t,x(t),xd(t),ẋ(t),ẋd(t))x2(t),x(t)∈]0,∞[where xd(t)=x(t−d) and ẋd(t)=ẋ(t−d), denote position and velocity feedback respectively, and V(t,x(t),xd(t),ẋ(t),ẋd(t))=V(t)+g1(x(t)−xd(t))+g2(ẋ(t)−ẋd(t)),is the feedback voltage with positive input voltage V(t)∈C(R/TZ) for e∈R+,g1,g2∈R, d∈0,T. The damping force fD(x,ẋ) can be linear, i.e., fD(x,ẋ)=cẋ, c∈R+ or squeeze film type, i.e., fD(x,ẋ)=γẋ/x3, γ∈R+. The fundamental tool to prove our result is a local continuation method of periodic solutions from the non-delayed case (d=0). Our approach provides new insights into the delay phenomenon on microelectromechanical systems and can be used to study the dynamics of a large class of delayed Liénard equations that govern the motion of several actuators, including the comb-drive finger actuator and the torsional actuator. Some numerical examples are provided to illustrate our results.

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>

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

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

Theoretical and experimental characterization of non-Markovian anti-parity-time systems

Open systems with anti-parity-time ({{{{{{{mathcal{APT}}}}}}}}) or {{{{{{{mathcal{PT}}}}}}}} symmetry exhibit a rich phenomenology absent in their Hermitian counterparts. To date all model systems and their diverse realizations across classical and quantum platforms have been local in time, i.e., Markovian. Here we propose a non-Markovian system with anti-{{{{{{{mathcal{PT}}}}}}}}-symmetry where a single time-delay encodes the retention of memory, and experimentally demonstrate its consequences with two time-delay coupled semiconductor lasers. A transcendental characteristic equation with infinitely many eigenvalue pairs sets our model apart. We show that a sequence of amplifying-to-decaying dominant mode transitions is induced by the time delay in our minimal model. The signatures of these transitions quantitatively match results obtained from four, coupled, nonlinear rate equations for laser dynamics, and are experimentally observed as constant-width sideband oscillations in the laser intensity profiles. Our work introduces a paradigmatic non-Hermitian system with memory, paves the way for its realization in classical systems, and may apply to time-delayed feedback-control for quantum systems.

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.

Mechanism of time-delay feedback control of suspension damping with an annular vibration-absorbing structure

With the aim of enhancing both the ride comfort and the safety of the vehicle, we propose a new type of suspension with an annular vibration-absorbing structure, and establish a 3-DOF 1/4 vehicle model. The structure parameters and time-delay feedback control parameters are determined by particle swarm optimization algorithms, which take the root mean values of body acceleration, suspension dynamic deflection, and tire dynamic displacement as their optimization objectives. We analyze the stability of the suspension control system to ensure the stability of the time-delay control system through the Routh-Hurwitz stability criterion, characteristic root method, and stability switching method. Then, we compare and analyze the response characteristics of conventional suspension, new suspension without time-delay feedback control, and new suspension with time-delay feedback control under simple harmonic excitation and random excitation. The results show that the new suspension with time-delay feedback control has a significant damping effect on the body under the premise of ensuring the stability of the system.

Improvement master–slave robustly synchronous criteria of uncertain chaotic Lur’e systems via an augmented Lyapunov–Krasovskii functional

The robust synchronization of uncertain master–slave Lur’e systems based on time-delayed feedback control is further investigated. A less conservative synchronization stability criterion and a less conservative robust synchronization stability criterion than some recent published references are proposed via Lyapunov stability theory. First, an augmented delay-dependent Lyapunov–Krasovskii functional (LKF) is constructed, where some single integral terms are augmented to the integrand component of a single-integral subfunction. This not only increases some coupling information on some system variables but also increases the coupling information on some necessary variables contained in the inequality lemmas, which can make full use of the inequality lemmas and reduce the conservatism of the synchronization stability criterion. Second, to overcome the nonlinear phenomena in the synchronization criterion, the novel negative definite inequality equivalent transformation lemma is used to transform the nonlinear inequalities to the linear matrix inequalities (LMIs) equivalently, which can be easily solved by the MATLAB LMI-Toolbox. Finally, some common numerical examples are presented to show the effectiveness of the proposed approach.

Stabilization and Chaos Control of an Economic Model via a Time-Delayed Feedback Scheme

This paper addresses the problem of chaos control in an economic mathematical dynamical model. By regarding the control variables as the bifurcation parameters, the stability of equilibria and the existence of Hopf bifurcations of the relevance feedback system are investigated, and the criterion of controllability for the chaotic system is obtained based on a time-delayed feedback control technique. Furthermore, numerical simulations are provided to demonstrate the feasibility of our methods and results.

Quantized-Output-Based Exponential Stabilization of Fuzzy DPSs With Time-Varying Delay and Nonlinearities

The exponential stabilization of nonlinear distributed parameter systems (DPSs) with time-varying delay is studied in this paper. With the employment of the Takagi-Sugeno (T-S) fuzzy model, the nonlinear DPSs are described as a fuzzy parabolic partial differential equation (PDE). Then, with the consideration of the measurement output and control input information quantized by a kind of dynamic quantizers, fuzzy static and dynamic output feedback controllers are proposed. The time-delayed feedback controllers are also provided in the forms of static and dynamic. On the basis of the derived T-S fuzzy PDE model, Lyapunov-based control design strategies are proposed with respect to the effect of quantization. Moreover, by constructing appropriate Lyapunov functionals, sufficient conditions for designing the output feedback control gains and the quantizers' adjusting parameters which guarantee the exponential stability of the fuzzy closed-loop system are provided in terms of standard linear matrix inequalities (LMIs). Finally, we explore the output feedback controllers into the FitzHugh-Nagumo (FHN) equation and Fisher equation as applications. The efficiency of the design strategies is indicated through the simulation results.

Bifurcation Behavior Analysis and Stability Region Discrimination for Series-Parallel Architecture Electric Energy Router

Electric energy routers (EERs) can effectively deal with the energy management issues caused by the access of multi-sources and multi-loads. Different from the energy transmission form in the existing series architecture EER (SA-EER) for low-voltage distribution networks, a new series-parallel architecture EER (SPA-EER) has the advantages of high-power transmission capability and reactive power flexible operation. However, if controller parameters change beyond their critical stability boundaries, SPA-EER will produce slow- and fast-scale bifurcation behaviors, which are manifested as low- and high-frequency oscillations of port voltages or port currents in varying degrees. To avoid the system instability caused by controller parameter changes, how to discriminate the stability regions of controller parameters for SPA-EER is a key research content in this paper. To this end, the stroboscopic mapping discrete model of SPA-EER system is first established, and system bifurcation behaviors are analyzed based on bifurcation theory. Furthermore, the critical stability boundaries of controller parameters are discriminated by using Jacobian matrix, root locus and bifurcation diagram. After that, a time delay feedback control (TDFC) is employed to suppress system bifurcation behaviors. Finally, the correctness of bifurcation analysis and the effectiveness of TDFC are verified by a hardware-in-the-loop (HIL) experimental platform.

Broad bandgap active metamaterials with optimal time-delayed control

In this paper, a metamaterial with time-delayed feedback control for broadband vibration suppression is investigated. Genetic algorithm is used to optimize the time-delayed feedback controller to achieve low broadband vibration suppression and even full frequency vibration suppression. Time-delayed feedback controller is introduced between the visible and internal masses of the metamaterial. Time-delayed control tunes the bandgap of the metamaterial and achieves broadband tunable vibration suppression. To further broaden the bandgap, a supercell model with time-delayed control is established, and the time-delayed controller is optimized using the genetic algorithm to determine the optimal control gain and time delay. Two optimization schemes with different objectives are designed. Optimization I: Achieve the widest bandgap at the target attenuation rate. Optimization II: Achieve the maximum attenuation rate in the target frequency range. The results show that multiple bandgaps can be coupled to achieve the widest bandgap at the target attenuation rate. The controller is optimized over the full frequency band to achieve the maximum attenuation rate. Numerical results verify that the metamaterial with optimal controller can achieve vibration suppression over the full frequency range.

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.

Slow-Scale Nonlinear Control of a H-Bridge Photovoltaic Inverter

Aiming at the slow-scale nonlinear behaviors of a dual-loop control H-bridge photovoltaic inverter, a slow-scale nonlinear control method based on time-delay feedback control is proposed. Firstly, a feedback signal is formed by the difference between the current error signal and its own delay, and the feedback signal is made to pass through a proportion link so as to obtain a time-delay control signal. Then, based on the necessary conditions of the stability criterion, the value range of the delay feedback proportional coefficient is quickly determined, and the best delay time is chosen by determining the variation curve of the largest Lyapunov exponent. Finally, based on the double-loop PI regulation, the time-delay control signal is added to the original control signal to form the final control signal applied to the inverter. The results obtained show that the slow-scale nonlinear behaviors in the system can be absolutely eliminated, and the stable operation domain of the inverter can be remarkably expanded. When the sun irradiance and temperature change, the proposed control method still has a good control effect and shows strong robustness. In addition, the complex calculation in determining the control coefficient and delay time can be avoided by the method.