Impulse response of a generalized fractional second order filter

TL;DR

This paper derives the impulse response of a generalized fractional second order filter, analyzing its properties and stability conditions, supported by numerical examples.

Contribution

It introduces a generalized fractional second order filter form and analyzes its impulse response and stability characteristics.

Findings

Impulse response derived for the generalized filter.

Stability conditions identified for different parameter values.

Numerical examples verify theoretical results.

Abstract

The impulse response of a generalized fractional second order filter of the form ${{({{s}^{2\alpha}}+a{{s}^{\alpha}}+b)}^{-\gamma}}$ is derived, where $0<\alpha \le 1$, $0<\gamma <2$. The asymptotic properties of the impulse responses are obtained for two cases, and the two cases show the similar properties for the changing of $\gamma$ values. It is shown that only when ${{({{s}^{2\alpha}}+a{{s}^{\alpha}}+b)}^{-1}}$ has the critical stability, the generalized fractional second order filter ${{({{s}^{2\alpha}}+a{{s}^{\alpha}}+b)}^{-\gamma}}$ has different properties as we change the value of $\gamma$. Finally, numerical examples to illustrate the impulse response are provided to verify the proposed concepts.