Impulse response of commensurate fractional-order systems: multiple complex poles

Abstract

The impulse response of a fractional-order system with the transfer function s^{delta }/{[(s^{alpha }-a)^2+b^2]^n}, where n in mathbb {N}, a in {mathbb {R}}, b in {mathbb {R}}^+, alpha in {mathbb {R}}^+, delta in {mathbb {R}}, is derived via real and imaginary parts of two-parameter Mittag-Leffler functions and their derivatives. With the aid of a robust algorithm for evaluating these derivatives, the analytic formulas can be used for an effective transient analysis of fractional-order systems with multiple complex poles. By some numerical experiments it is shown that this approach works well also when the popular SPICE-family simulating programs fail to converge to a correct solution.