Optimal design of fractional order low pass Butterworth filter with accurate magnitude response

Abstract

The design of (1+α) order, 0<α<1, low pass Butterworth filter approximated in terms of an integer order continuous-time transfer function using a nature-inspired optimization technique called Gravitational Search Algorithm (GSA) is presented in this paper. While approximations of the non-integer order Laplacian operator sα in terms of second order rational function using the continued fraction expansion method for the design of fractional order low pass Butterworth filters (FOLBFs) is recently reported in literature, however, such a design technique is non-optimal. In this work, the metaheuristic global search process of GSA efficiently explores and intensely exploits the nonlinear, non-uniform, multidimensional, and multimodal FOLBF design problem error landscape. At the end of the iterative search routine of GSA, the optimal values of the coefficients in terms of the third order rational approximations are achieved which accurately approximate the magnitude response of the ideal FOLBF. The proposed GSA based FOLBFs consistently achieve the best solution quality with the fastest convergence rate as compared with the designs based on Real coded Genetic Algorithm (RGA) and Particle Swarm Optimization (PSO). Comparison with the recent literature also demonstrates the superiority of the proposed designs. SPICE simulations justify the design feasibility of the proposed FOLBF models.