Robust fractional PID controller synthesis approach for the permanent magnetic synchronous motor

In this work, we propose the synthesis method of different robust fractional order PID controllers, which are synthesized from solving the constrained optimization algorithm for the permanent magnet synchronous motor (PMSM) speed control problem. This problem is formulated through the time and frequency-domain specifications where the obtained fractional controller should simultaneously satisfy some conflicted goals, such as good tracking dynamic behavior of the imposed set-point reference, good attenuation of the plant uncertainties, good suppression of the sensor noise effect and ensure an enough trade-off between the nominal performances and the robust stability of the feedback control system. The previous properties are satisfied not only for the nominal plant-model used in synthesis controller step, but also for a set of the neighboring plant uncertainties. Moreover, the integral time absolute error criterion should be minimized to satisfy the imposed time-domain requirements. However, the weighted-mixed sensitivity problem based upon the different proposed adjustable weights is minimized to ensure the imposed frequency-domain specifications where the load disturbances, sensor noises and the neglected nonlinear and fast PMSM dynamics are considered. The PMSM speed drive is controlled where its dynamic behavior is modeled by the unstructured-multiplicative uncertainty. The obtained simulation results are compared in time and frequency domains by those given with the conventional robust $$\mathcal {H}_\infty$$H∞ controller to demonstrate the effectiveness of the proposed fractional controllers.