Fractional order control of the two-dimensional wave equation. (September 2015) - Sirota, Lea; Halevi, Yoram
Abstract: Control of systems governed by the two-dimensional linear wave equation in finite spatial domain is considered and presented through vibrating rectangular membranes. The membranes are modeled by modal decomposition in one spatial axis and infinite dimensional transfer functions in the other. The transfer functions are built of fractional order exponents, regarded as non-pure delays, which are shown to represent traveling waves whose shape changes during motion. The membranes are controlled in closed loop to achieve position profile tracking and attenuation of disturbances. The actuation is along two opposite boundaries, which controls the entire wave motion between them. The control algorithm stops the wave propagation in the control axis by creating active non-reflecting boundaries. In addition, it compensates the remaining non-pure delay by extending the classical dead time compensation principle. As a result, despite the infinite dimension of the system and its fractional order transfer functions, the closed loop transfer function is given by a rational first order lag with a pure time delay. The resulting controllers are also of fractional order and their implementation is obtained by dedicated approximations. The system stability with the approximated controllers is investigated formally using robustness tools. The control algorithm is demonstrated by means of several examples.