IPMSM drive systems which are driven by matrix converters have many advantages, including one-stage power conversion, high efficiency, bidirectional power transfer, near unity input power factor, and low input current harmonics. However, the DC-link voltage of the matrix-converter is varied from 0.866 to because a large electrolytic capacitor is not used. These voltage variations deteriorate the dynamic current and speed responses and reduce the adjustable speed range of the IPMSM drive systems. In addition, the variations of the external load and inertia are very large for matrix-converter based drive systems. To solve this problem, two adaptive controllers are investigated in this paper for matrix-converter PMSM drive systems. From the experimental results, the proposed two adaptive speed controllers are superior to the PI speed controller, including faster transient responses, better tracking abilities, and greater robustness. As a result, the proposed matrix-converter based IPMSM drive system is suitable in many industrial applications.