Extending the Concept of Analog Butterworth Filter for Fractional Order Systems

TL;DR

This paper introduces a novel design of Fractional Order Butterworth filters in the complex w-plane, expanding traditional filter design methods to fractional systems with diverse damping characteristics.

Contribution

It is the first to formulate and demonstrate fractional Butterworth filter design directly in the complex w-plane for fractional order systems.

Findings

Successful simulation examples validate the design approach.

Practical frequency domain filter design demonstrates applicability.

The method accommodates various damping conditions in fractional systems.

Abstract

This paper proposes the design of Fractional Order (FO) Butterworth filter in complex w-plane (w=sq; q being any real number) considering the presence of under-damped, hyper-damped, ultra-damped poles. This is the first attempt to design such fractional Butterworth filters in complex w-plane instead of complex s-plane, as conventionally done for integer order filters. Firstly, the concept of fractional derivatives and w-plane stability of linear fractional order systems are discussed. Detailed mathematical formulation for the design of fractional Butterworth-like filter (FBWF) in w-plane is then presented. Simulation examples are given along with a practical example to design the FO Butterworth filter with given specifications in frequency domain to show the practicability of the proposed formulation.