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In this paper, the speed regulation problem for permanent magnet synchronous motor (PMSM) system under vector control framework is studied. First, a speed regulation scheme based on standard internal model control (IMC) method is designed. For the speed loop, a standard internal model controller is first designed based on a first-order model of PMSM by analyzing the relationship between reference quadrature axis current and speed. For the two current loops, PI algorithms are employed respectively. Second, considering the disadvantages that the standard IMC method is sensitive to control input saturation and may lead to poor speed tracking and load disturbance rejection performances, a modified IMC scheme is developed based on a two-port IMC method, where a feedback control term is added to form a composite control structure. Third, considering the case of large variations of load inertia, two adaptive IMC schemes with two different adaptive laws are proposed. A method based on disturbance observer is adopted to identify the inertia of PMSM and its load. Then a linear adaptive law is developed by analyzing the relationship between the internal model and identified inertia. Considering the control input saturation in practical applications, a fuzzy adaptive law based IMC scheme is developed based on apriori experimental tests and experiences, where a fuzzy inferencer based supervisor is designed to automatically tune the parameter of speed controller according to the identified inertia. The effectiveness of the proposed methods have been verified by Matlab simulation and TMS320F2808 DSP experimental results.