PI Control of Loudspeakers Based on Linear Fractional Order Model

Abstract

This paper aims at the proportional-integral (PI) control of the cone vibration of the electrodynamic loudspeakers system recently described using a linear fractional order model. After introducing the fractional order model of the circuit of these loudspeakers, firstly, a new method is developed to design a fractional order PI controller to place the poles of the system in a desired area of the complex plane which is called D-stabilizing. The design parameters of the method depend directly on the speed of the system output response, the cone vibration. Moreover, the offered fractional order controller avoids any non-minimum phase zero, which causes undesired undershoots in the output, for the closed-loop control system. Secondly, considering uncertainties in the coefficients of the model, a methodology is presented to determine up to how much the uncertainties can increase such that the controller is still able to maintain both D-stability and the absence of non-minimum phase zeros for the control system. Finally, the merit of the presented results and the superiority of the designed fractional order controller over its conventional integer order counterpart are illustrated through numerical simulations.