Optimal Sliding Mode Design for a Class of Uncertain Systems with Time-delay

The problem of designing optimal sliding manifolds with a quadratic performance index is presented for a class of linear uncertain systems with state time-delay.First,based on the regular form of the state equation,the problem of designing optimal sliding manifolds is transformed into an optimal control problem for linear time-delay systems.Then according to the necessary conditions of optimal control,the two-point boundary value(TPBV) problems with both time-delay and time-advance terms are induced.A sensitivity approach getting the optimal control law for linear time-delay system is adopted to convert the original TPBV problems into that of solving a series of TPBV iterative formulas without time-delay and time-advance variable terms.By the finite-step recursion,an approximate solution for the optimal sliding mode is obtained.Finally,a sufficient asymptotic stability condition of the optimal sliding motion is derived and proved. Simulation results validate the efficiency of the proposed method.