A method of continuous time approximation of delayed dynamical systems

This paper presents a new method to analyze response of linear and nonlinear dynamical systems with time delay. The method proposes a continuous time approximation of the delayed portion of the response. This leads to a high and finite dimensional state space formulation of the time-delayed system. The advantage of the current method lies in that the resulting finite dimensional state equations are in the standard state space form, making all the existing analysis methods and control design tools for linear and nonlinear dynamical systems amenable to the current approach. The method can also handle multiple independent time delays in a natural way. One- and two-dimensional dynamical systems with time delay are used to demonstrate the effectiveness of the method.     

Steady state bifurcation of a periodically excited system under delayed feedback controls

This paper investigates the steady state bifurcation of a periodically excited system subject to time-delayed feedback controls by the combined method of residue harmonic balance and polynomial homotopy continuation. Three kinds of delayed feedback controls are considered to examine the effects of different delayed feedback controls and delay time on the steady state response. By means of polynomial homotopy continuation, all the possible steady state solutions corresponding the third-order superharmonic and second-subharmonic responses are derived analytically, i.e. without numerical integration. It is found that the delayed feedback changes the bifurcating curves qualitatively and possibly eliminates the saddle-node bifurcation during resonant. The delayed position-velocity coupling and the delayed velocity feedback controls can destabilize the steady state responses. Coexisting periodic solutions, period-doubling bifurcation and even chaos are found in these control systems. The neighborhood of the periodic solutions is verified numerically in the phase portraits. The various effects of time delay on the steady state response are investigated. Many new phenomena are observed.     

Controlling chaos in a nonlinear pendulum using an extended time-delayed feedback control method

Chaos control is employed for the stabilization of unstable periodic orbits (UPOs) embedded in chaotic attractors. The extended time-delayed feedback control uses a continuous feedback loop incorporating information from previous states of the system in order to stabilize unstable orbits. This article deals with the chaos control of a nonlinear pendulum employing the extended time-delayed feedback control method. The control law leads to delay-differential equations (DDEs) that contain derivatives that depend on the solution of previous time instants. A fourth-order Runge–Kutta method with linear interpolation on the delayed variables is employed for numerical simulations of the DDEs and its initial function is estimated by a Taylor series expansion. During the learning stage, the UPOs are identified by the close-return method and control parameters are chosen for each desired UPO by defining situations where the largest Lyapunov exponent becomes negative. Analyses of a nonlinear pendulum are carried out by considering signals that are generated by numerical integration of the mathematical model using experimentally identified parameters. Results show the capability of the control procedure to stabilize UPOs of the dynamical system, highlighting some difficulties to achieve the stabilization of the desired orbit.     

Fuzzy delayed output feedback synchronization for time-delayed chaotic systems

In this paper, we propose a new fuzzy delayed output feedback synchronization (FDOFS) method for time-delayed chaotic systems. Based on Lyapunov–Krasovskii theory, T–S fuzzy model, and delayed feedback control scheme, the FDOFS controller is designed and an analytic expression of the controller is shown. The proposed controller can guarantee asymptotical synchronization of both drive and response systems. The FDOFS controller can be obtained by solving the linear matrix inequality (LMI) problem. A numerical example for time-delayed Lorenz system is presented to demonstrate the validity of the proposed FDOFS method.     

Analytical approximations for the periodic motion of the Duffing system with delayed feedback

In this study, the homotopy analysis method is developed to give periodic solutions of delayed differential equations that describe time-delayed position feedback on the Duffing system. With this technique, some approximate analytical solutions of high accuracy for some possible solutions are captured, which agree well with the numerical solutions in the whole time domain. Two examples of dynamic systems are considered, focusing on the periodic motions near a Hopf bifurcation of an equilibrium point. It is found that the current technique leads to higher accurate prediction on the local dynamics of time-delayed systems near a Hopf bifurcation than the energy analysis method or the traditional method of multiple scales.     

Digital modeling and control of multiple time-delayed distributed power grid

Distributed power grid (DPG) control systems are so highly interconnected that the effects of local disturbances as well as transmission time delays can be amplified as they propagate through a complex network of transmission lines. These effects deteriorate control performance and could possibly destabilize the overall system. In this paper, a new approximated discretization method and digital design for DPG control systems with multiple state, input and output delays as well as a generalized bilinear transformation method are presented. Based on a procedure for the generation of impulse response data, the multiple fractional/integer time-delayed continuous-time system is transformed to a discrete-time model with multiple integer time delays. To implement the digital modeling, the singular value decomposition (SVD) of a Hankel matrix together with an energy loss level is employed to obtain an extended discrete-time state space model. Then, the extended discrete-time state space model of the DPG control system is reformulated as an integer time-delayed discrete-time system by computing its observable canonical form. The proposed method can closely approximate the step response of the original continuous time-delayed DPG control system by choosing various energy loss levels. For completeness, an optimal digital controller design for the DPG control system and a generalized bilinear transformation method with a tunable parameter are also provided, which can re-transform the integer time-delayed discrete-time model to its continuous-time model. Illustrative examples are given to demonstrate the effectiveness of the developed method.     

On a new time-delayed feedback control of chaotic systems

In this paper, using the idea of the successive dislocation feedback method, a new time-delayed feedback control method called the successive dislocation time-delayed feedback control (SDTDFC) is designed. Firstly, the idea of SDTDFC is introduced. Then some analytic sufficient conditions of the chaos control from the SDTDFC approach are derived for stabilization. Finally, some established results are further clarified via a case study of the Lorenz system with the numerical simulations.     

Exponential stability analysis and impulsive tracking control of uncertain time-delayed systems

In this paper, we study exponential stability and tracking control problems for uncertain time-delayed systems. First, sufficient conditions of exponential stability for a class of uncertain time-delayed systems are established by employing Lyapunov functional methods and algebraic matrix inequality techniques. Furthermore, tracking control problems are investigated in which an uncertain linear time-delayed system is used to track the reference system. Sufficient conditions for solvability of tracking control problems are obtained for the cases that the system state is measurable and non-measurable, respectively. When the state is measurable, we design an impulsive control law to achieve the tracking performance. When the state information is not directly available from measurement, an impulsive control law based on the measured output will be used. Finally, numerical examples are presented to illustrate the effectiveness and usefulness of our results.     

Numerical methods for the stability of time-periodic hybrid time-delay systems with applications

This paper extends two numerical methods for the stability analysis of a class of time-periodic hybrid time-delay systems. In particular, the pseudospectral tau and spectral element methods are extended to hybrid systems. The analyzed delay-differential equation involves delayed terms with both continuous and piecewise constant arguments, in other words, it involves delays both without and with zero-order hold, respectively. The analyzed class of hybrid systems can be used to describe time-periodic hereditary processes subjected to digital feedback control. The proposed numerical algorithms are applied to the mathematical models of haptic systems and milling processes subjected to digital feedback control.     

A new time-discretization for delay multiple-input nonlinear systems using the Taylor method and first order hold

A new discretization method is proposed for multi-input-driven nonlinear continuous systems with time-delays, based on a combination of the Taylor series expansion and the first-order hold (FOH) assumption. The mathematical structure of the new discretization scheme is explored. On the basis of this structure, the sampled-data representation of the time-delayed multi-input nonlinear system is derived. First the new approach is applied to nonlinear systems with two inputs, and then the delayed multi-input general equation is derived. The resulting time discretization method provides a finite-dimensional representation for multi-input nonlinear systems with time-delays, thereby enabling the application of existing controller design techniques to such systems. The performance of the proposed method is evaluated using a nonlinear system with time-delays (maneuvering an automobile). Various sampling rates, time-delay values and control inputs are considered to evaluate the proposed method. The results demonstrate that the proposed discretization scheme can meet the system requirements even when using a large sampling period with precision limitations. The discretization results of the FOH method are also compared with those of the zero order hold (ZOH) method. The precision of the FOH method in the discretization procedure combined with the Taylor series expansion is much higher than that of the ZOH method except in the case of constant inputs.     

Response of Duffing system with delayed feedback control under combined harmonic and real noise excitations

This paper presents a procedure for predicting the response of Duffing system with delayed feedback bang–bang control under combined harmonic and real noise excitations by using the stochastic averaging method. First, the time-delayed feedback bang–bang control force is expressed approximately in terms of the system state variables without time delay. Then the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method. Finally, the response of the system is obtained by solving the Fokker–Plank–Kolmogorov (FPK) equation associated with the averaged Itô equations. It is shown that the time delay in feedback control can deteriorate the control effectiveness and cause bifurcation of stochastic jump of Duffing system. The validity of the proposed method is confirmed by digital simulation.     

Optimal Feedback Control for Linear Systems with Input Delays Revisited

The design problem of optimal feedback control for linear systems with input delays is very important in many engineering applications. Usually, the linear systems with input delays are firstly converted into linear systems without delays, and then all the design procedures are based on the delay-free linear systems. In this way, the feedback controllers are not designed in terms of the original states. This paper presents some new closed-form formula in terms of the original states for the delayed optimal feedback control of linear systems with input delays. We firstly reveal the essential role of the input delay in the optimal control design of the linear system with a single input delay: the input delay postpones the action of the optimal control only. Based on this fact, we calculate the delayed optimal control and find that the optimal state can be represented by a simple closed-form formula, so that the delayed optimal feedback control can be obtained in a simple way. We show that the delayed feedback gain matrix can be “smaller” than that for the controlled system with zero input delay, which implies that the input delay can be considered as a positive factor. In addition, we give a general formula for the delayed optimal feedback control of time-variant linear systems with multiple input delays. To show the effectiveness and advantages of the main results, we present five illustrative examples with detailed numerical simulation and comparison.     

时延混成系统的切换控制器合成

如何设计安全、可靠的信息物理融合系统是计算机科学和控制理论面临的一个重大挑战.时延现象在信息物理融合系统中普遍存在,时延对系统的稳定性、安全性和控制性能具有实质性影响.但是在已有时延系统验证和控制器合成的工作中往往忽略时延因素,这会导致在不考虑时延情况下能保证稳定和安全的系统在实际运行时因为时延原因而不再稳定和安全.因为时延使得系统的行为演化不仅与当前状态有关,还依赖于系统的历史状态,所以时延混成系统的验证和控制合成更加困难.本文研究信息物理融合系统在考虑时延情形下切换控制器合成问题,提出基于不变式生成技术的控制器合成方法.首先,利用谱分析和线性化技术将时延系统的微分不变式生成问题归结为有界时间的可达集计算问题;然后,提出基于抽象精化的算法计算时延系统有界时间可达集的上近似;最后,实现本文算法并使用实例验证该方法的有效性.     

Bursting mechanism in a time-delayed oscillator with slowly varying external forcing

This paper investigates the generation of complex bursting patterns in the Duffing oscillator with time-delayed feedback. We present the bursting patterns, including symmetric fold–fold bursting and symmetric Hopf–Hopf bursting when periodic forcing changes slowly. We make an analysis of the system bifurcations and dynamics as a function of the delayed feedback and the periodic forcing. We calculate the conditions of fold bifurcation and Hopf bifurcation as well as its stability related to external forcing and delay. We also identify two regimes of bursting depending on the magnitude of the delay itself and the strength of time delayed coupling in the model. Our results show that the dynamics of bursters in delayed system are quite different from those in systems without any delay. In particular, delay can be used as a tuning parameter to modulate dynamics of bursting corresponding to the different type. Furthermore, we use transformed phase space analysis to explore the evolution details of the delayed bursting behavior. Also some numerical simulations are included to illustrate the validity of our study.     

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

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

Continuous feedback control of perturbed mechanical systems

The problem of synthesizing continuous controls for a Lagrangian scleronomic mechanical system is investigated, on the assumption that the system is subject to uncontrollable bounded perturbations and that the vector of control forces is bounded in norm. a feedback control law is assumed, making it possible to steer the system to a given rest state in a finite time. The approach employed is based on methods of the theory of stability of motion. An implicitly given Lyapunov function is used to construct the control law and justify the construction. The existence of such a function is proved and its properties established. Results of a numerical simulation of the dynamics of various mechanical systems controllable in accordance with the proposed law are presented. It is shown that, for a point mass moving along a horizontal straight line, the control algorithm qualitatively approximates to time-optimal control.     

An indirect Lyapunov approach to robust stabilization for a class of linear fractional-order system with positive real uncertainty

The paper is concerned with the problem of the robust stabilization for a class of fractional order linear systems with positive real uncertainty under Riemann–Liouville (RL) derivatives. Firstly, by utilizing the continuous frequency distributed model of the fractional integrator, the fractional order system is expressed as an infinite dimensional integral order system. And via using indirect Lyapunov approach and linear matrix inequality techniques, sufficient condition for robust asymptotic stability of the fractional order systems and design methods of the state feedback controller are presented. Secondly, by using matrixs singular value decomposition technique the static output feedback controller and observer-based controller for asymptotically stabilizing the fractional order systems are derived. Finally, the validity of the proposed methods are demonstrated by numerical examples.     

Stochastic stability of Duffing–Mathieu system with delayed feedback control under white noise excitation

The asymptotic Lyapunov stability with probability one of Duffing–Mathieu system with time-delayed feedback control under white-noise parametric excitation is studied. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method and the expression for the Lyapunov exponent of the linearized averaged Itô equations is derived. Finally, the effects of time delay in feedback control on the Lyapunov exponent and the stability of the system are analyzed. Meanwhile, the stability conditions for the system with different time delays are also obtained. The theoretical results are well verified through digital simulation.     

Cascades-based synchronization of hyperchaotic systems: Application to Chen systems

We discuss the cascaded-based controlled synchronization method for hyperchaotic systems. The control approach is based on analysis tools for cascaded time-varying systems. That is, the closed-loop system takes the form of two subsystems which are interconnected in a manner that the state of one system enters into another but without feedback loop. The advantage of such construction is that the controller is largely simplified relative to other design methods such as backstepping. We apply the method to Chen’s hyperchaotic system and show that global synchronization is achieved via linear control. Also, we assume that only three instead of four control inputs are available. The method is tested in numerical simulations.     

Synchronization of chaotic dynamical network with unknown generally time-delayed couplings via a simple adaptive feedback control

In this paper, based on the principle of functional differential equations, a simple adaptive feedback controller is proposed to synchronize the network with unknown generally time-delayed coupling functions. Unlike the other control schemes, we design the controllers by the output of each node. The advantage of this scheme is that we need not to know the concrete structure of coupling functions or a solution of node system in advance. Moreover, an illustrative example is considered and numerical simulations are performed to demonstrate the effectiveness of this control scheme. The numerical simulation results show that our method is valid and robust against the weak noise.