Further results on inheriting the monotonicity of frequency responses by fractional-order systems from their integer-order counterparts

Recently, it has been proved that the monotonicity of magnitude-frequency response of an integer-order system, described by a minimum-phase stable rational transfer function G ( s ), is preserved in its fractional-order counterparts, given by G ( s α ), where α ∈ (0, 1). In this paper, it is shown that the aforementioned proposition is also valid for the phase-frequency responses, that is, if the minimum-phase stable rational transfer function G ( s ) possesses a monotonic phase-frequency response, the monotonicity of magnitude-frequency response of G ( s α ) with α ∈ (0, 1) is ensured. Moreover, further results on monotonicity of magnitude/phase-frequency response with respect to order, when the monotonicity of phase/magnitude-frequency response with respect to frequency holds, are revealed. Furthermore, relevant results on preservation of the monotonicity of frequency response of integer-order systems in the infinite ranges beginning with a non-zero frequency in their fractional-order counterparts are obtained. Different examples are provided to verify the obtained analytical achievements.