A note on time-domain simulation of feedback fractional-order systems

The study of feedback fractional-order systems has been receiving considerable attention due to the facts that many physical systems are well characterized by fractional-order models, and that fractional-order controllers are used in feedback systems with the intention of breaking through the performance limitation of integer-order controllers. Owing to the lack of effective analytic methods for the time-domain analysis and simulation of linear feedback fractional-order systems, we suggest in this paper two reliable and accurate numerical methods for inverting fractional-order Laplace transforms. One is based on computing Bromwich´s integral with a numerical integration scheme capable of accuracy control, and the other is based on expanding the time response function in a B-spline series. In order to demonstrate the superiority in solution accuracy and computational complexity of these two numerical methods over the Grunwald-Letniknov approximation method and Podlubny´s analytic formulas, which are in a form of double infinite series, the time-domain simulations of the feedback control of a fractional-order process with a PD^μ-controller and a fractional-order band-limited lead compensator are worked out. The simulation results indicate that a convergence problem indeed occurs in using Podlubny´s infinite series expressions, and that the problem could not be overcome by a series acceleration scheme