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

Abstract

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.

Vibration suppression of a nonlinear magnetic levitation system via time delayed nonlinear saturation controller

“…In [6][7][8][9][10][11][12][13][14][15], the authors studied different techniques to apply the saturation controller on different nonlinear dynamical systems. Zhao and Xu [16][17][18] applied the delayed feedback control to suppress the vibration of nonlinear systems to external excitation. Their delayed feedback controller reduced the vibration at a ratio about 65% for the positive feedback and 86% for the negative one comparing to the nonlinear vibration absorber without delayed feedback control.…”

“…Warminski et al [15] studied active suppression of nonlinear composite beam vibrations by selected control algorithms and one of them is nonlinear saturation controller (NSC) which exhibited good results for reducing the vibration amplitude of the composite beam. Zhao and Xu [16][17][18] applied the delayed feedback control to suppress the vibration in nonlinear dynamical systems to external excitations. With the time delay varying for a fixed gain, it is seen that the vibration can be suppressed at some values of the delay called vibration suppression region.…”

“…18) shows the time histories for both the main system and the controller amplitudes can be noticed from figures(18) and(19) that the transient time (time for initial unwanted oscillations) has been decreased by increasing the time delays but the steady state amplitudes are still the same. So from figures(17) to(19), it can be seen that the main system steady state amplitude has been reduced from about 0.4 to about 0.02.…”

“…The proposed controller has an effectiveness a E which is about 20 ( a E = System amplitude without controller / System amplitude with controller)are outside the stable solutions region discussed in figure(4). Figures(18) to(20) are plotted for the initial conditions (…”

Experiment on Parameter identification of a Time Delayed Vibration Absorber

“…For the situation that the excitation frequency varies, active control (Beltran-Carbajal, F., SilvaNavarro, G., & Vazquez-Gonzalez, B., 2011; Liao, G. J., Gong, X. L., Kang, C. J., & Xuan, S. H., 2011; Xu, Z., Gong, X., Liao, G., & Chen, X., 2010) is introduced in to tune the absorber's resonant frequency and damping, which brings in time delay at the same time. Extensive studies show that the time delay affects the system dynamic stability a lot (Cermak, J., & Jansky, J., 2014;Zhang, X. Y., & Sun, J. Q., 2014) and can be used as a frequency and damping tuner as well (Elmali, H., Renzulli, M., & Olgac, N., 2000;Vyhlí dal, T., Olgac, N., & Kučera, V., 2014;Zhao, Y. Y., & Xu, J., 2012). Generally, the time delay in the control loop caused by calculating and filtering is called the inherent delay; the delay in the control algorithm set by programmer is called the artificial delay.…”