Impulse response of a generalized fractional second order filter

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