Impulse response of a generalized fractional second order filter

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.