Optimal fractional order PI controller design for time-delayed processes

The usage of fractional calculus enables additional flexibility and precision regarding the control parameters. This study introduces a fully analytical design for a fractional-order Proportional-Integral (FOPI) controller, eliminating the need for predefined parameters or iterative optimization. The Åström recursive algorithm, previously applied to integer-order controllers, is adapted for the first time to optimize FOPI controllers based on integral of squared time error (ISTE), integral of squared time-squared error (IST2E), and integral of squared time-cubed error (IST3E) performance criteria. This study differs from others since instead of specific types, it combines three types of first-order plus time-delay processes: stable (SFOPTD), integrating (IFOPTD), and unstable (UFOPTD) processes. Analytical formulas have been derived for optimal parameter selection, while separate formulas provided for gain margin (GM), phase margin (PM), and maximum sensitivity (Ms) enable the pre-determination of system robustness. The controller's performance is validated using simulations of the step response, disturbance rejection, control effort, and perturbation response. In addition, real-time experiments on an inverted pendulum illustrate its utility in dynamic processes. This provides a comprehensive framework aimed to further the development of fractional-order control by providing a systematic solution to a large scope of industrial applications.