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

Abstract

AbstractThe impulse response of a fractional-order system with the transfer function $$s^{\delta }/{[(s^{\alpha }-a)^2+b^2]^n}$$ s δ / [ ( s α - a ) 2 + b 2 ] n , where $$n \in \mathbb {N}$$ n ∈ N , $$a \in {\mathbb {R}}$$ a ∈ R , $$b \in {\mathbb {R}}^+$$ b ∈ R + , $$\alpha \in {\mathbb {R}}^+$$ α ∈ R + , $$\delta \in {\mathbb {R}}$$ δ ∈ 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.