Time-Delayed Nonlinear Integral Resonant Controller to Eliminate the Nonlinear Oscillations of a Parametrically Excited System

In this work, a nonlinear integral resonant controller is utilized for the first time to suppress the principal parametric excitation of a nonlinear dynamical system. The whole system is modeled as a second-order nonlinear differential equation (i.e., main system) coupled to a nonlinear first-order differential equation (i.e., controller). The control loop time-delays are included in the studied model. The multiple scales homotopy approach is employed to obtain an approximate solution for the proposed time-delayed dynamical system. The nonlinear algebraic equation that governs the steady-state oscillation amplitude has been extracted. The effects of the time-delays, control gain, and feedback gains on the performance of the suggested controller have been investigated. The obtained results indicated that the controller performance depends on the product of the control and feedback signal gains as well as the sum of the time-delays in the control loop. Accordingly, two simple objective functions have been derived to design the optimum values of the loop-delays, control gain, and feedback gains in such a way that enhances the efficiency of the proposed controller. The analytical and numerical simulations illustrated that the proposed controller could eliminate the system vibrations effectively at specific values of the control and feedback signal gains. In addition, the selection method of the loop-delays that either enhances the control performance or destabilizes the system motion has been explained in detail.