Using the delayed feedback control and saturation control to suppress the vibration of the dynamical system

In the present paper, the delayed feedback control is applied to suppress or stabilize the vibration of the primary system in a two degree-of-freedom dynamical system with parametrically excited pendulum. The case of a 1:2 internal resonance between pendulum and primary system is studied. The method of multiple scales is applied to obtain second-order approximations of the response of the system. The system stability and bifurcations of equilibrium point of the averaged equations are computed. It is shown that the delayed feedback control can be used to suppress the vibration or stabilize the system when the saturation control is invalid. The vibration of the primary system can be suppressed by the delayed feedback control when the original system is in the single-mode motion. The effect of gain and delay on the vibration suppression is discussed. As the delay varies at a fixed value of the gain, the vibration of the primary system can be suppressed at some values of the delay. The vibration suppression performance of the system is improved at a large value of the gain. The vibration of the primary system could be suppressed about 56% compared with the original system by choosing the appropriate values of gain and delay. The delayed feedback control also can be used to stabilize the system when the original system is unstable. The gain and delay could be chosen as the controlling parameters. Numerical simulation is agreement with the analytical solutions well.     

Delayed feedback control based on the act-and-wait concept

This paper proposes a delayed feedback control (DFC) based on the act-and-wait concept, which reduces the dynamics of DFC systems to that of discrete-time systems. Based on this concept, a delayed feedback controller is designed for a prototype two-dimensional oscillator using a simple systematic procedure. This control has two advantages: the feedback delay time can be large and it can obtain deadbeat behavior. A numerical example using a double-scroll circuit model demonstrates these theoretical results.     

Hopf bifurcation for a small-world network model with?parameters delay feedback control

To delay the onset of undesirable bifurcation, the bifurcation control has become a subject of intense research activities. In this paper, a small-world network model with the delay feedback is considered, in which the strength of feedback control is a nonlinear function of delay. With this controller, one can change the critical value of bifurcation, and thus enlarge the stable region. Moreover, by adding some proper slowly varying parts into the bifurcation parameters, the stability can be improved. Numerical results show that the dynamics of the small-world network model with the controller of delay-dependent parameters is quite different from that of a system with the controller of delay-independent parameters only.     

The effect of time-delayed feedback controller on an electrically actuated resonator

This paper presents a study of the effect of a time-delayed feedback controller on the dynamics of a Microelectromechanical systems (MEMS) capacitor actuated as a resonator by DC and AC voltage loads. A linearization analysis is conducted to determine the stability chart of the linearized system equations as a function of the time delay period and the controller gain. Then the method of multiple-scales is applied to determine the response and stability of the system for small vibration amplitude and voltage loads. It is shown that negative time-delay feedback control gain can lead to unstable responses, even if AC voltage is relatively small compared to the DC voltage. On the other hand, positive time delay can considerably strengthen the system stability even in fractal domains. We also show how the controller can be used to control damping in MEMS, increasing or decreasing, by tuning the gain amplitude and delay period. Agreements among the results of a shooting technique, long-time integration, basin of attraction analysis with the perturbation method are achieved.     

An adaptive delayed feedback control method for stabilizing chaotic time-delayed systems

This paper addresses a new adaptive delayed feedback control technique for stabilizing a class of chaotic time-delayed systems with a variable parameter. In the proposed scheme, the feedback gain of a delayed feedback controller is automatically tuned according to an adaptation law in order to stabilize unstable fixed points of the system. Such a mechanism provides a way to cope with unexpected changes in the parameters of the system. The adaptation algorithm is constructed based on the Lyapunov?CKrasovskii??s stability theorem. The control technique provides the advantages of increased stability and optimality, adaptability to the changes in the parameters, high privacy, simplicity, and noninvasiveness. The effectiveness of the control scheme is demonstrated using numerical simulations for a well-known chaotic time-delayed system.     

线性时滞系统的离散最优控制

介绍了对线性时滞系统进行最优控制的设计，将具有时滞控制的线性系统离散后引入增广状态向量。获得不显含时滞的差分方程，根据时滞量的两种分类情况采用连续和离散形式的性能指标函数导出了最优控制律。控制律包含当前状态和此前若干步状态向量的叠加，最优控制律直接从时滞方程中得到，可保证系统的稳定性，此方法亦适用于大时滞的情况。数值算例验证了控制策略的有效性。     

Time delay Duffing’s systems: chaos and chatter control

The effect of a delay feedback control (DFC), realized by displacement in the Duffing oscillator, for parameters which generate strange chaotic Ueda attractor is investigated in this paper. First, the classical Duffing system without time delay is analysed to find stable and especially unstable periodic orbits which can be stabilized by means of displacement delay feedback. The periodic orbits are found with help of the continuation method using the AUTO97 software. Next, the DFC is introduced with a time delay and a feedback gain parameters. The proper time delay and feedback gain are found in order to destroy the chaotic attractor and to stabilize the periodic orbit. Finally, chatter generated by time delay component is suppressed with help of an external excitation.     

Fractional-order PD control at Hopf bifurcations in a fractional-order congestion control system

In this paper, we address the problem of the bifurcation control of a delayed fractional-order dual model of congestion control algorithms. A fractional-order proportional–derivative (PD) feedback controller is designed to control the bifurcation generated by the delayed fractional-order congestion control model. By choosing the communication delay as the bifurcation parameter, the issues of the stability and bifurcations for the controlled fractional-order model are studied. Applying the stability theorem of fractional-order systems, we obtain some conditions for the stability of the equilibrium and the Hopf bifurcation. Additionally, the critical value of time delay is figured out, where a Hopf bifurcation occurs and a family of oscillations bifurcate from the equilibrium. It is also shown that the onset of the bifurcation can be postponed or advanced by selecting proper control parameters in the fractional-order PD controller. Finally, numerical simulations are given to validate the main results and the effectiveness of the control strategy.

     

Truncated predictor stabilization control for interconnected nonlinear systems with time-varying input delay

This paper deals with control design for interconnected nonlinear systems with time-varying input delay. Based on the truncated prediction of the system state over the delay period, the state feedback control law is constructed. In the framework of the Lyapunov–Krasovskii function, the stability equations of closed-loop system under state feedback law are established, and the feasibility of the controller is transformed into the problem of establishing a set of linear matrix inequality (LMI) conditions. Based on the Lyapunov stability theorem, it is proved that the closed-loop system is asymptotically stable. Finally, a simulation example is provided to demonstrate the effectiveness of the control scheme.

     

Effect of delay combinations on stability and Hopf bifurcation of an oscillator with acceleration-derivative feedback

This paper presents new observations of delayed AD (acceleration-derivative) controller in active vibration control and in bifurcation control of a Duffing oscillator. Based on the stability analysis of the linear delayed oscillator, it is found that combination of the two delays in acceleration feedback and velocity feedback has a significant influence on the stable region in the parameter plane of the gains. By calculating the real part of the rightmost characteristic roots of the controlled oscillator with fixed delays, it is shown that a delayed acceleration feedback with positive gain can work much better than the corresponding delayed negative acceleration feedback, which is used in classic control theory. For given feedback gains, by calculating the critical delay values, it is shown that a delayed positive acceleration feedback can result in a much larger stable delay interval than the corresponding delayed negative acceleration feedback does. As an application of these results to a delayed Duffing oscillator with acceleration-derivative feedback, a delayed positive acceleration feedback can be well used to postpone the occurrence of Hopf bifurcation in the delayed nonlinear oscillators.     

The alternating between complete synchronization and hybrid synchronization of hyperchaotic Lorenz system with time delay

The various cases of synchronization in two identical hyperchaotic Lorenz systems with time delay are studied. Based on Lyapunov stability theory, the sufficient conditions for achieving synchronization of two identical hyperchaotic Lorenz systems with time delay are derived, and a simple scheme only with a single linear controller is proposed. When the parameters in the response system are known, the alternating between complete synchronization and hybrid synchronization (namely, coexistence of antiphase and complete synchronization) is observed with the control feedback gain varying. Furthermore, when the parameters in the response system are unknown, for the same feedback controller, the complete synchronization and the hybrid synchronization can be obtained, respectively, as the associated parameters updated laws of the unknown parameters are chosen. Numerical simulation results are presented to demonstrate the proposed chaos synchronization scheme.     

Stabilization problems for a system with output feedback (review)

Algorithms for solving the problem of design of static output feedback controllers for stationary linear systems with continuous and discrete time are reviewed. The inverse problem is considered. The algorithms of synthesis of output feedback controllers are generalized to the case of a periodic discrete-time system. To solve such problems, it might be more natural to use an approach based on multi-criterion optimization. It is also shown that these algorithms can be used for the optimal stabilization of unstable systems with delay. In this connection, the parameters of a controller with given structure for a controlled unstable scalar system with delay are optimized. To this end, the system is first approximated by a system without delay, with the exponent approximated by a fractionally rational function. Since the structure of the controller is given, the quality of approximation is estimated as the difference (in the space of controller coefficients) between the stability domains of the original and approximating systems. At the next stage, the gain coefficients of the controller for the reduced system are optimized. The efficiency of the thus synthesized controller is assessed through mathematical modeling of a system with delay whose feedback loop is defined by the gain coefficients found. The approach is illustrated by stabilizing an inverted simple pendulum with a proportional–derivative controller with delay. The problem of synthesis of a robust controller for this example is considered. Some examples of designing a robust controller, including for a third-order system in which the delay rather than some parameter is uncertain are presented     

控制存在延时的建筑结构地震作用下的最优控制方法

直接从时滞微分方程进行控制律设计,对控制存在延时的建筑结构在地震作用下的最优控制方法进行了研究。在控制时滞量为采样周期的整数倍和非整数倍的两种情况下,通过采用零阶保持器,将包含时滞的连续系统转化为形式上不包含时滞的标准离散线性系统,然后进行控制律的设计。所得出的控制律表达式中,除了含有当前的状态反馈外,还包含有前若干步控制项的线形组合。最后对某三自由度结构模型进行了仿真计算,结果表明,延时对控制效果有较大的影响,延时并非愈短愈好。     

Stabilizing control of Hopf bifurcation in the Hodgkin–Huxley model via washout filter with linear control term

It is a significant issue to control bifurcation because many neuronal diseases have close relevance to bifurcation of neuron system. Some studies have been done on bifurcation control in the Hodgkin–Huxley (HH) model, but there is no clear mathematical criterion for bifurcation stabilization. In this paper, according to Routh–Hurwitz stability criterion, we employ linear control term of washout filter-aided dynamic feedback controller to stabilize bifurcation of the HH model. As a result, we can deduce linear control gain based on the criterion, and simulation shows the method is effective for making the HH model stable. The controller designs described here are achieved by electrical stimulus, so it may have potential applications in the diagnosis and therapy of dynamical diseases.     

Analysis of Duffing oscillator with time-delayed fractional-order PID controller

The free vibration of Duffing oscillator with time-delayed fractional-order Proportional-Integral-Derivative (FOPID) controller based on displacement feedback is studied. The second-order approximate analytical solution is obtained by KBM asymptotic method. The effects of the parameters in FOPID controller on the dynamical properties are characterized by some equivalent parameters. The correctness of the approximate analytical results is verified by the numerical results. The effects of the time-delayed FOPID controller with displacement feedback on control performances of Duffing oscillator are analyzed in detail by time response, and the stability conditions of zero solution and periodic motions are also presented. Finally, the control performances on Duffing oscillator with large damping are further analyzed. And the results show that one could take the advantage of time delay, when the parameters of time-delayed FOPID controller are chosen reasonably.     

Stabilizing the unstable periodic orbits of a chaotic system using model independent adaptive time-delayed controller

In this paper we deal with the control of chaotic systems. Knowing that a chaotic attractor contains a myriad of unstable periodic orbits (UPO’s), the aim of our work is to stabilize some of the UPO’s embedded in the chaotic attractor and which have interesting characteristics. First, using the input-to-state linearization method in conjunction with a time-delayed state feedback, we design a control signal that can achieve stabilization. Next, an adaptive time-delayed state feedback is proposed which shows at once efficiency and simplicity and circumvents the construction complexity of the first controller. Finally, we propose a reduced order sliding mode observer to estimate the necessary states for the design of an adaptive time delayed state feedback controller. This last controller has one main advantage, it in fact achieves UPO stabilization without using the system model. The efficacy of the proposed methods is illustrated by numerical simulations onto Chua’s system.     

Stability analysis of a noise control system in a duct by using delay differential equation

The paper deals with the criteria for the closed- loop stability of a noise control system in a duct. To study the stability of the system, the model of delay differential equation is derived from the propagation of acoustic wave governed by a partial differential equation of hyperbolic type. Then, a simple feedback controller is designed, and its closed- loop stability is analyzed on the basis of the derived model of delay differential equation. The obtained criteria reveal the influence of the controller gain and the positions of a sensor and an actuator on the closed-loop stability. Finally, numerical simulations are presented to support the theoretical results.     

Delayed full-state feedback control of airfoil flutter using sliding mode control method

This paper studies the delayed feedback control of flutter of a two-dimensional airfoil using a sliding mode control (SMC) method. The dynamic equation of airfoil flutter is firstly established using the Lagrange method, in which the cubic hardening spring nonlinearity of pitch stiffness is considered. Then, the state equation with time delay is transformed into a standard state equation with implicit time delay by a special integral transformation. Next a nonlinear time-delay controller is designed using the SMC method. Finally the effectiveness of the proposed controller is verified through numerical simulations. Simulation results indicate that time delay in the control system has significant influence on the control performance. Control failure may happen if time delay is not considered in control design. The time-delay controller proposed is effective in suppressing the airfoil flutter with either small or large control time delay.     

Control of the nonlinear oscillator bifurcation under a superharmonic resonance

A weakly nonlinear oscillator is modeled by a differential equation. A superharmonic resonance system can have a saddle-node bifurcation, with a jumping transition from one state to another. To control the jumping phenomena and the unstable region of the nonlinear oscillator, a combination of feedback controllers is designed. Bifurcation control equations are derived by using the method of multiple scales. Furthermore, by performing numerical simulations and by comparing the responses of the uncontrolled system and the controlled system, we clarify that a good controller can be obtained by changing the feedback control gain. Also, it is found that the linear feedback gain can delay the occurrence of saddle-node bifurcations, while the nonlinear feedback gain can eliminate saddle-node bifurcations. Feasible ways of further research of saddle-node bifurcations are provided. Finally, we show that an appropriate nonlinear feedback control gain can suppress the amplitude of the steady-state response.     

A delay partitioning approach to output feedback control for uncertain discrete time-delay systems with actuator saturation

This paper is concerned with the problems of output feedback control for uncertain discrete time-delay systems with input saturation. The delay partitioning approach is proposed to obtain new stability criteria. The dynamic output feedback controller is designed based on a linear matrix inequality framework. A sufficient condition is developed, which guarantees the existence of dynamic output feedback controllers such that all trajectories of the closed-loop system starting from an admissible initial condition domain converge to a smaller ellipsoid. Simulation examples are provided to show the potential of the proposed techniques.