Biquadratic approximation of fractional-order Laplacian operators

This paper introduces a Biquadratic approximation of the fractional-order Laplacian operator of order α; s^α, 0 <; α ≤ 1. The significance of this approach lies in developing finite-order transfer functions that approximate infinite-order differential (integral) Laplacian operators. A special form of a Biquadratic transfer function is designed to approximate s^±α over a narrowband spectrum that enjoys an exact gain and flat phase frequency response. A modular structure can easily be designed by cascading several Biquadratic transfer functions centered at different corner frequencies to widen the frequency spectrum. Such approximation simplifies the design of fractional-order proportional-integral-derivative (FoPID) controllers. The effectiveness and the simplicity of the proposed method are demonstrated via several numerical examples.