Robust Fractional-order PID Tuning Method for a Plant with an Uncertain Parameter

The robust design of fractional-order proportional-integral-differential (FOPID) controllers for controlled plants with uncertainty is a popular research topic. The well-studied “flat phase” condition is effective for the gain variation but not for variations in other parameters. This paper addresses the problem of tuning a robust FOPID controller for a plant with a known structure and an uncertain parameter (a coefficient or order in the plant transfer function). The method is based on preserving the phase margin of the open-loop system when the plant parameter varies around the nominal value. First, the partial derivatives of the gain crossover frequency with respect to the plant parameters are calculated. Then, the partial derivatives of the phase margin with respect to the plant parameters are obtained as the robust performance indexes. In addition, the equations needed to compute FOPID parameters that meet the specifications in the frequency domain are obtained and used as nonlinear constraints. Finally, the FOPID parameters can be obtained by optimizing the robust performance indexes under these constraints. Simulation experiments are carried out on examples with different types of uncertain parameters to verify the effectiveness of the tuning method. The results show that the requirements are fulfilled and that the system with the proposed FOPID controller is stable and robust to variations in the uncertain parameters. Comparisons clearly show that the controllers designed by the proposed method provide relatively robust performance.