Frequency specifications regions of fractional-order PI controllers for first order plus time delay processes

This work presents a comparative analysis on the stability of several fractional-order PI controllers. The stability analysis is developed by assuming a frequency domain tuning of the regulators to control a first order plus time delay plant. In order to analyze the stability that controllers provide, a new concept, region of feasible frequency specifications is presented. The fractional order controllers analyzed are I^α, PI ^α, II^β and I^αD ^1-α, and all the results obtained are compared to the conventional PI controller. The stability analysis demonstrates that each controller has a different region of feasible frequency specifications and, in all cases, they depend of their non-integer integral or derivative actions. A methodology of the robustness analysis is also proposed. This methodology provides a controllers tuning method based on the region of feasible frequency specifications concept. Finally, an application example, the speed control of a DC motor with variant parameters, is presented. Simulations and experiments illustrate that the region of feasible frequency specifications concept allows to obtain optimal controllers under the point of view of frequency requirements fulfilment.