Optimal sliding mode control for linear time-delay systems with sinusoidal disturbances

This paper develops a successive approximation approach (SAA) of optimal sliding mode control (SMC) for linear time-delay systems with sinusoidal disturbances. A sequence of two-point boundary value (TPBV) problems with both time-delay and time-advance terms is derived from the optimal sliding mode design. According to the SAA, the original TPBV problems are transformed into a sequence of linear TPBV problems without delay and advance terms. The obtained SMC ensures that the state trajectories reach the sliding surface in finite time and remain on it thereafter. The stability of the sliding mode is proved. A numerical simulation is employed to verify the effectiveness of the proposed approach.