This paper investigates fractional- and distributed-order proportional-integral-derivative (FOPID and DOPID) controllers for permanent magnet synchronous motors (PMSM). The design of FOPID controllers requires the setting of five parameters, i.e. the gains of the three different actions and the non-integer orders of integration and differentiation. An efficient generalized particle swarm optimization (GPSO) sets the parameters in two steps: firstly for an inner loop, secondly for an outer loop. The DOPID controller is characterized by seven different orders of integration or differentiation. In this second case, the tuning is also performed by GPSO in two steps. In both cases, the feedback transfer function of the inner loop is a non-integer order plant for the outer loop, then the outer controller should be of fractional order to obtain the best performance/robustness indexes, as shown by simulation.

本文研究了永磁同步电动机（PMSM）的分数阶和分布阶比例积分微分（FOPID和DOPID）控制器。FOPID控制器的设计要求设置五个参数，即三个不同动作的增益以及积分和求和的非整数阶。有效的广义粒子群优化（GPSO）通过两个步骤设置参数：首先用于内循环，其次用于外循环。DOPID控制器具有七个不同的集成或区分顺序。在第二种情况下，GPSO也会分两步执行调整。在这两种情况下，内循环的反馈传递函数都是外循环的非整数阶工厂，