  Ultrasonic motors (USMs) are commonly used in aerospace, robotics, and
medical devices, where fast and precise motion is needed. Remarkably, sliding
mode controller (SMC) is an effective controller to achieve precision motion
control of the USMs. To improve the tracking accuracy and lower the chattering
in the SMC, the fractional-order calculus is introduced in the design of an
adaptive SMC in this paper, namely, adaptive fractional-order SMC (AFOSMC), in
which the bound of the uncertainty existing in the USMs is estimated by a
designed adaptive law. Additionally, a short memory principle is employed to
overcome the difficulty of implementing the fractional-order calculus on a
practical system in real-time. Here, the short memory principle may increase
the tracking errors because some information is lost during its operation.
Thus, a compensator according to the framework of Bellman's optimal control
theory is proposed so that the residual errors caused by the short memory
principle can be attenuated. Lastly, experiments on a USM are conducted, which
comparative results verify the performance of the designed controller.
