A general class of optimal nonlinear resonant controllers of fractional order with time-delay for active vibration control – theory and experiment

Abstract

The present article generalizes the classical integer-order resonant controllers for active vibration control. The proposed controller utilizes a second-order linear filter with response feedback and a control input in the form of a nonlinear, time-delayed function of the fractional derivative of the filter variable. Thereby, the proposed controller encompasses the integer-order resonant controllers, like positive position feedback, acceleration feedback, negative velocity feedback, etc. as special cases and allows an extra degree of freedom in terms of choosing the optimum form of the control input function. Theoretical analysis of the system is performed by the method of multiple time scales and finally, the results are verified by numerical simulations and experiments. Two new numerical methods of parameters optimization are discussed. Detailed parametric studies are performed to reveal the effects of different design parameters, viz. the control gain, time-delay, and the fractional-order of the input function on the system performance. The theoretical and experimental studies demonstrate the supremacy of the fractional-order control over integer-order resonant control, especially for higher control gain and delay. The existence of an optimal value of the fractional order of the control function is established.