On the biquadratic approximation of fractional-order Laplacian operators

Abstract

This paper introduces new biquadratic approximation algorithms to approximate fractional-order Laplacian operators of order $$\alpha \,;\,s^{ \pm \alpha } , 0 < \alpha \le 1$$ ? ? s ± ? , 0 < ? ≤ 1 , by finite-order rational transfer functions. The significance of this approach lies in developing an algorithm that depends only on $$\alpha$$ ? , which enables one to synthesize both fractional-order inductors and capacitors. The fundamental biquadratic transfer function enjoys a flat phase characteristics at its corner frequency. The bandwidth of approximation can be extended by cascading several biquadratic modules, each centered at calculated corner frequency. The proposed approach is straightforward and outperforms similar approximation algorithms. The equal orders of the biquadratic structure allow one to synthesize both fractional-order inductors and fractional-order capacitors using passive circuits. The same approach can easily be extended to design active filters, or fractional-order proportional-integral-derivative controllers. The main ideas of the proposed method are demonstrated using several numerical examples.