Time-delayed Nonlinear Feedback Controllers to Suppress The Principal Parameter Excitation

Abstract

Six different time-delayed controllers are introduced within this article to explore theirefficiencies in suppressing the nonlinear oscillations of a parametrically excited system. The applied controltechniques are the linear and nonlinear versions of the position, velocity, and acceleration of the consideredsystem. The time-delay of the closed-loop control system is included in the proposed model. As the modelunder consideration is a nonlinear time-delayed dynamical system, the multiple scales homotopy method isutilized to derive two nonlinear algebraic equations that govern the vibration amplitude and the correspondingphase angle of the controlled system. Based on the obtained algebraic equations, the stability charts of theloop-delays are plotted. The influence of both the control gains and loop-delays on the steady-state vibrationamplitude is examined. The obtained results illustrated that the loop-delays can play a dominant role in eitherimproving the control efficiency or destabilizing the controlled system. Accordingly, two simple objectivefunctions are introduced in order to design the optimum values of the control gains and loop-delays in sucha way that improves the controllers’ efficiency and increases the system robustness against instability. Theefficiency of the proposed six controllers in mitigating the system vibrations is compared. It is found thatthe cubic-acceleration feedback controller is the most efficient in suppressing the system vibrations, whilethe cubic-velocity feedback controller is the best in bifurcation control when the loop-delay is neglected.However, the analytical and numerical investigations confirmed that the cubic-acceleration controller is thebest either in vibration suppression or bifurcation control when the optimal time-delay is considered. It isworth mentioning that this may be the first article that has been dedicated to introducing an objective functionto optimize the control gains and loop-delays of nonlinear time-delayed feedback controllers.