Delayed Feedback Control Research Articles

On the analysis of the tristable vibration isolation system with delayed feedback control under parametric excitation

In this paper, the delayed feedback control of a tristable vibration isolation system (TVIS) under the stochastic parametric excitation is investigated. Firstly, theoretical solutions of the frequency response equation that governs the dynamical performance of the TVIS are derived. Meanwhile, the stability conditions are obtained. The controlled performance and response properties induced by the time delay, the feedback gains and the excitation amplitude are discussed. The existence of the time delay can considerably suppress the vibration amplitude. Particular phenomena induced by the time delay are observed. In addition, the first-order and second-order steady-state moments of the TVIS are obtained based on the Itô stochastic differential equations under the stochastic parametric excitation. The phenomenon of the time delay island about steady-state moments is found. The stationary probability density is discussed for the parametric resonance by means of the finite difference method. The variation of the noise bandwidth and the time delay can induce the transition of the stationary probability density between trivial solutions and non-trivial solutions. Overall, the response mechanism of the TVIS under the stochastic parametric excitation is revealed.

Stability and Bifurcation Analysis of a Discrete Singular Bioeconomic System

The main concern of this paper is to discuss stability and bifurcation analysis for a class of discrete predator-prey interaction with Holling type II functional response and harvesting effort. Firstly, we establish a discrete singular bioeconomic system, which is based on the discretization of a system of differential algebraic equations. It is shown that the discretized system exhibits much richer dynamical behaviors than its corresponding continuous counterpart. Our investigation reveals that, in the discretized system, two types of bifurcations (i.e., period-doubling and Neimark–Sacker bifurcations) can be studied; however, the dynamics of the continuous model includes only Hopf bifurcation. Moreover, the state delayed feedback control method is implemented for controlling the chaotic behavior of the bioeconomic model. Numerical simulations are presented to illustrate the theoretical analysis. The maximal Lyapunov exponents (MLE) are computed numerically to ensure further dynamical behaviors and complexity of the model.

Intermittent delay stabilization of complex-valued stochastic complex network

In this paper, the stabilization issue for complex-valued stochastic Markovian switching complex network with time delay and time-varying coupling structure (CSMNDC) is investigated via intermittent delay feedback control. Different from intermittent control based on current state in previous work, a class of intermittent control on the basis of past state is designed for the first time, by combining with the advantages of aperiodically intermittent control and delay feedback control. Then, some sufficient conditions are derived to guarantee the exponential stability in mean square of CSMNDC based on Lyapunov method, graph theory as well as some techniques of inequalities. In particular, the stabilization of networks is studied on complex space directly without splitting their real and imaginary parts by using complex generalized Itô’s formula. Additionally, both delay feedback control and aperiodically intermittent control are employed to solve the stabilization issue for CSMNDC here. Whereafter, the stabilization issue of complex-valued stochastic Markovian switching Cohen-Grossberg neural network with time delay and time-varying coupling structure is researched as a practical application of our theoretical results. Ultimately, a numerical example is presented to verify the validity and effectiveness of the theoretical results.

Hopf bifurcation in a generalized Logistic reaction–diffusion population model with instantaneous and delayed feedback

This article considers a generalized Logistic reaction–diffusion population model with mixed instantaneous and delayed feedback control and subject to the homogeneous Dirichlet boundary condition on the one-dimensional bounded spatial domain (0,π). Existence and asymptotic expression of the small enough positive steady state bifurcating from the zero solution are given in virtue of the implicit function theorem. By choosing the delay τ as the bifurcation parameter and analyzing in detail the associated eigenvalue problem, the local asymptotic stability and the existence of Hopf bifurcation of the small enough positive steady state are obtained. Particularly, the normal form for Hopf bifurcations is derived according to the normal form method and the center manifold theory for partial functional differential equations (PFDEs). It is demonstrated that when the delayed feedback is dominant, the model can undergo a forward Hopf bifurcation at the positive steady state as the delay crosses through monotonically increasing a sequence of critical values and all the bifurcating periodic solutions are locally orbitally asymptotically stable on the center manifold. Specifically, the bifurcating periodic solutions from the first bifurcation value are orbitally asymptotically stable in the entire phase space. The reasonability of the main analytical conclusions is also verified by carrying out some numerical simulations.

Controlling grazing-induced multistability in a piecewise-smooth impacting system via the time-delayed feedback control

Grazing events may create coexisting attractors and cause complex dynamics in piecewise-smooth dynamical systems. This paper studies the control of grazing-induced multistability in a soft impacting oscillator by using the time-delayed feedback control. The control switches from one of the coexisting attractors to a desired one to suppress complex dynamics near grazing events. We use path-following (continuation) techniques for non-smooth dynamical systems to investigate robustness of the controller and the parameter dependence of the controlled system. In particular, several newly developed computational methods are used, including a numerical method for analysing non-smooth delay equations and a method for calculating the Lyapunov exponents and the grazing point estimation. Numerical simulations demonstrate that the delayed feedback controller is effective, and a proper selection of the control gain and delay time can simplify the complex dynamics of the system near grazing.

Delay feedback stabilisation of stochastic differential equations driven by G-Brownian motion

This paper aims to design the feedback control based on past states to stabilise a class of nonlinear stochastic differential equations driven by G-Brownian motion. By building up the connection between the delay feedback control and the control function of no-delay, sufficient conditions are established in terms of size of the time delay ensuring both the pth moment exponential stability and quasi-surely exponential stability of the delay feedback controlled system. Moreover, methods for determining the upper bound of the length of the time delay and further designing the delay feedback controller are provided. A numerical example is presented to demonstrate our new theory.

Feedback Control for a Diffusive and Delayed Brusselator Model: Semi-Analytical Solutions

This paper describes the stability and Hopf bifurcation analysis of the Brusselator system with delayed feedback control in the single domain of a reaction–diffusion cell. The Galerkin analytical technique is used to present a system equation composed of ordinary differential equations. The condition able to determine the Hopf bifurcation point is found. Full maps of the Hopf bifurcation regions for the interacting chemical species are shown and discussed, indicating that the time delay, feedback control, and diffusion parameters can play a significant and important role in the stability dynamics of the two concentration reactants in the system. As a result, these parameters can be changed to destabilize the model. The results show that the Hopf bifurcation points for chemical control increase as the feedback parameters increase, whereas the Hopf bifurcation points decrease when the diffusion parameters increase. Bifurcation diagrams with examples of periodic oscillation and phase-plane maps are provided to confirm all the outcomes calculated in the model. The benefits and accuracy of this work show that there is excellent agreement between the analytical results and numerical simulation scheme for all the figures and examples that are illustrated.

Stabilization in Distribution by Delay Feedback Control for Hybrid Stochastic Differential Equations

This article is concerned with the design of a feedback control based on past states in order to make a given unstable hybrid stochastic differential equation (SDE) to be stable in distribution (stabilization in distribution). This is the first article in this direction. Under the global Lipschitz condition on the coefficients of the given unstable hybrid SDE, we will show that the stabilization in distribution can be achieved by linear delay feedback controls. In particular, we discuss how to design feedback controls in two structure cases: state feedback and output injection.

Further results on stabilization of stochastic differential equations with delayed feedback control under $ G $-expectation framework

<p style='text-indent:20px;'>In this paper we establish a comparison approach to study stabilization of stochastic differential equations driven by <inline-formula><tex-math id="M2">\begin{document}$ G $\end{document}</tex-math></inline-formula>-Brownian motion with delayed (<inline-formula><tex-math id="M3">\begin{document}$ G $\end{document}</tex-math></inline-formula>-SDDEs for short) feedback control. This theory also extends to a general range of moment order and brings more choices of <inline-formula><tex-math id="M4">\begin{document}$ p $\end{document}</tex-math></inline-formula>. Finally, a simple example is proposed to demonstrate the applications of our theory.</p>

Study of anti-swing control of ship cranes based on time delay feedback

2 tons 3 meters ship crane as the research object, establish the complete mathematical model of ship crane, and design the time-delay feedback control algorithm to eliminate payload swing on the basis of the built mathematical model. First, a high-precision potentiometer is used to measure the in-plane and out-of-plane oscillation of the payload. Measuring devices for the rotary angle and variation amplitude angle are created, and the anti-swing control hardware system is built by the sensing unit and the High-Speed Data Acquisition hardware system. Secondly, the control software on the VB platform is developed using the time-delay feedback algorithm. Experimentally study the effect of time-delayed feedback controller on payload swing elimination, and use induction method to get the optimal control parameters of the time-delayed feedback control algorithm. At last, the AMEsim-ADAMS co-simulation platform was built to evaluate the payload sway of the ship crane under simulated working conditions. The results show that: the hardware and software systems for anti-swing control built-in the paper can give the real-time and the exact payload swing angles. The method of adding time-delay feedback control signal to the slewing operation signal can achieve better anti-swing effects than the others. At the same time, the delayed feedback control algorithm still has a nice control effect in the case of virtual ship deck motion.

Bifurcation and control of a planktonic ecological system with double delays by delayed feedback control

In this paper, a delayed feedback controller with the delay-dependent coefficient is introduced into a multiple delay phytoplankton-zooplankton system. For uncontrolled system, choosing delays as the bifurcation parameters, we prove that Hopf bifurcation can occur when the delays change and cross some values. Then, the delays are still chosen as the bifurcation parameters to research the dynamic behaviors of the controlled system. Under this control mechanism, the onset of Hopf bifurcation can be delayed by selecting the appropriate control parameters and the stability domain can be extended as feedback gain (the decay rate) decreases (increases), and the influence of the decay rate cannot be ignored. Furthermore, using the crossing curve methods, the stable changes of equilibrium in two delay plane can be obtained. Some numerical simulations are given to verify the correctness and validity of the delayed feedback controller in the bifurcation control.

Exponential stabilization by delay feedback control for highly nonlinear hybrid stochastic functional differential equations with infinite delay

Given an unstable hybrid stochastic functional differential equation, how to design a delay feedback controller to make it stable? Some results have been obtained for hybrid systems with finite delay. However, the state of many stochastic differential equations are related to the whole history of the system, so it is necessary to discuss the feedback control of stochastic functional differential equations with infinite delay. On the other hand, in many practical stochastic models, the coefficients of these systems do not satisfy the linear growth condition, but are highly nonlinear. In this paper, the delay feedback controls are designed for a class of infinite delay stochastic systems with highly nonlinear and the influence of switching state.

Enlarging the stable domain of controlled high-period oscillations with transient extended delayed feedback control

A transient extended delayed feedback control method was developed by modifying the original extended delayed feedback control method with a transient control technique to enlarge the stable domain of the controlled high-period oscillations. The control force is applied to the system only when the system is in the controlling region, which is different from the static control force of the original extended delayed feedback control. The specific control performance of the transient extended delayed feedback control was analyzed through case studies. The relationships between the stable domain of the controlled high-period oscillations and the controlling regions were obtained by calculating the largest magnitude of the Floquet multipliers. It is shown that the stable domain changes in a non-smooth manner with the variation of the controlling region. In particular, the transient extended delayed feedback control can stabilize high-period unstable oscillations in a much larger stable domain than that of the original extended delayed feedback control when the controlling region is properly chosen.

Stabilization by delay feedback control for highly nonlinear switched stochastic systems with time delays

AbstractThis paper firstly discusses the existence and boundedness of unique global solution for highly nonlinear switched stochastic systems (SSSs) with time delays, where the switching signal is deterministic switching signal. Although the existence and boundedness of unique global solution can be obtained under local Lipschitz condition and Khasminskii‐type condition, highly nonlinear SSSs with time delays may not be stable. Therefore, a delay feedback control function can be designed for studying the stability of highly nonlinear SSSs with time delays, whose coefficients do not satisfy the linear growth condition. By using the Mathematical induction, Lyapunov function approach and the average dwell time method, H∞ stability and the pth moment exponential stability are studied for highly nonlinear SSSs with time delays. An explanatory example is given to illustrate the effectiveness of the main results.

Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivatives

<p style='text-indent:20px;'>The goal of this paper is to provide systematic approaches to study the feedback control systems governed by fractional impulsive delay evolution equations involving Caputo fractional derivatives in separable reflexive Banach spaces. This work is a continuation of previous work. We firstly give an existence result of mild solutions for the equations by applying the Banach's fixed point theorem and the Leray-Schauder alternative fixed point theorem. Next, by using the Filippove theorem and the Cesari property, we obtain the existence result of feasible pairs for the feedback control system. Finally, some applications are given to illustrate our main results.</p>

Bifurcation control of a minimal model of marine plankton interaction with multiple delays

Plankton blooms and its control is an intriguing problem in ecology. To investigate the oscillatory nature of blooms, a two-dimensional model for plankton species is considered where one species is toxic phytoplankton and other is zooplankton. The delays required for the maturation time of zooplankton, the time for phytoplankton digestion and the time for phytoplankton cells to mature and release toxic substances are incorporated and the delayed model is analyzed for stability and bifurcation phenomena. It proves that periodic plankton blooms can occur when the delay (the sum of the above three delays) changes and crosses some threshold values. The phenomena described by this mechanism can be controlled through the toxin release rates of phytoplankton. Then, a delay feedback controller with the coefficient depending on delay is introduced to system. It concludes that the onset of the bifurcation can be delayed as negative feedback gain (the decay rate) decreases (increases). Some numerical examples are given to verify the effectiveness of the delay feedback control method and the existence of crossing curve. These results show that the oscillatory nature of blooms can be controlled by human behaviors.

LED cooling by PSJA using delayed feedback control in linear region

LED cooling by PSJA using delayed feedback control in linear region

A chaos control method of single-phase H-bridge photovoltaic inverter

There are complex nonlinear behaviors such as bifurcation and chaos in a single-phase H-bridge photovoltaic inverter under proportional integral control, which will increase the harmonic content of the output current greatly and reduce the stability of system operation and reliability of power supply. There are the problems suffering the complicated modeling and difficulty in determining the control coefficients in existing chaos control methods. The exponential delay feedback control is a further development of the delay feedback control, which has the advantages of requiring no precise mathematical model of the system and simple implementation. However, our research shows that when the exponential delay feedback control is directly applied to the system, the feedback intensity cannot be controlled, which will bring too big a disturbance to the system. Based on it, an improved exponential delayed feedback control method is proposed in this paper. Firstly, a feedback signal is formed by the difference between the output current of the system and its own delay, then the feedback signal is used to obtain the control signal through an exponential link, a subtraction link and an proportion link, and the control signal is applied to the controlled system in the form of a feedback. At the same time, the discrete mapping model of the system is established and its Jacobian matrix expression is determined. Finally, the limiting conditions of the feedback control coefficient of the control signal are derived based on the stability criterion, and the control is applied to the system. In order to verify the control effect of this method, a lot of simulation experiments are conducted. The results show that the problems that the exponential delay feedback cannot control the feedback strength and causes excessive disturbance to the system will be effectively solved by this method. When the bifurcation parameters vary greatly, the chaos behaviors in the system will be suppressed effectively, the stable operating domain of the system will be expanded greatly and the harmonic content of the output current will be reduced.

DESENVOLVIMENTO DE SOFTWARE E INTERFACE PARA O CONTROLE POR RETROALIMENTAÇÃO ATRASADA EM SISTEMAS ELETROQUÍMICOS

DEVELOPMENT OF SOFTWARE AND INTERFACE FOR DELAYED FEEDBACK CONTROL IN ELECTROCHEMICAL SYSTEMS. In this paper, we show the development of a LabVIEW algorithm able to apply the delayed feedback control on oscillatory electrochemical reactions. The coupling of the control and the experiments was carried out by a real-time apparatus, measuring the faradaic current and, in sequence, applying a calculated circuit voltage as an analog output. The setup was used to control the oscillatory Cu/Sn electrodeposition dynamics, used as a model-system, and the results clearly demonstrate the control accuracy. Overall, the code may be a great tool for the controlled synthesis of nanomaterials in different oscillatory electrodeposition reactions, as it offers the ability to control the composition and structure of the deposit by fine tuning the dynamic properties.

Time-Delayed Sampled-Data Feedback Control of Differential Systems Undergoing Hopf Bifurcation

In this paper, time-delayed sampled-data feedback control technique is used to asymptotically stabilize a class of unstable delayed differential systems. Through the analysis for the distribution change of eigenvalues, an effective interval of the control parameter is obtained for a given sampling period. Here an indirect strategy is taken. Specifically, the system of continuous-time delayed feedback control is studied first by Hopf bifurcation theory. And then, the result and implicit function theorem are used to analyze the system of time-delayed sampled-data feedback control with a sufficiently small sampling period. Considering the practical criterion for the size of sampling period, the upper bound of sampling period is estimated. Finally, an application example, an unstable Mackey–Glass model, is asymptotically stabilized by introducing a blood transfusion item with time-delayed sampled-data feedback control. The blood transfusion speed and blood collection test period are derived from the main results. Some simulations and comparisons show the correctness and advantages of the main theoretical results.