An evaluation of integer- and fractional-order controllers for small satellites

Fractional calculus (FC) represents the generalization of the integer calculus, in which the order of the exponent, as well as the differential and integral operators are restricted to integer values. This paper focuses on the existing fractional derivatives commonly used, while highlighting the advantages and disadvantages of each one of them. Furthermore, a mathematical interpretation of the fractional derivative is presented. The superiority of FC and its relative merits are also addressed through a specific example related to control engineering. This example deals with the implementation of a proportional integral derivative (PID) controller, where the objective is to evaluate a fractional- and integer-order PID controllers for the purpose of correcting the error in a dynamic process based on specific performance measures (e.g., overshoots, undershoots, transient response, and steady-state error). The simulation results indicates that the fractional-order PID provides a significant improvements compared to the integer-order PID, as well as a more flexible structure to adapt optimally to the process at hand.