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.
Categories: cs.SY math.OC