Optimal variable-order fractional PID controllers for dynamical systems

This paper studies the design of variable-order fractional proportional-integral-derivative (VFPID) controllers for linear dynamical systems. For this purpose, a technique to discretize fractional differential equations with variable-order operators is proposed. The resulting model and a particle swarm optimization algorithm are used to search for the optimal parameters of the VFPID controllers. Three examples illustrate the performance of the closed-loop system under the action of the VFPID and compare it with the results obtained by means of the PID and fractional PID (FPID). Furthermore, time-dependent parameters are also tested, showing that the VFPID yields a superior performance.