This article addresses the robust exponential stabilization problem of underactuated mechanical systems in the presence of bounded external disturbances using sliding mode control. Based on the Lyapunov method, a sufficient condition for the existence of the smallest possible ball which bounds the reduced-order sliding mode dynamics, is first derived in terms of linear matrix inequality (LMI). A sliding mode controller is then synthesized to guarantee that system state trajectories are exponentially convergent to another ball with a prespecified  onvergence rate.

This paper addresses the problem of nonlinear discrete-time flatness-based controller design for a Permanent Magnet-Excited Synchronous Motor (PMSM). To eliminate the static errors of the system state variables and consider the nonlinear characteristics of a PMSM, a cascaded flatnessbased control scheme is proposed. Simulation results are provided to illustrate the effectiveness of the proposed control structures, in terms of better performance.