Optimal Sliding Mode Control for Linear Systems with Time-delay

The sliding-mode control for linear systems with time-delay Is concerned. By treating some state variables as virtual control, a quadratic performance index is given. According to the necessary conditions for the optimality, the two-point boundary value (TPBV) problems with both time-delay and time-advance terms are obtained. By developing the successive approximation approach (SAA) of differential equation into infinite horizon, the original TPBV problem which is derived from the optimal switching manifold design is transformed into a sequence of linear TPBV problems without delay and advance terms. The solution sequence of the linear TPBV problems uniformly converges to the solution of the original problem.. By using a finite term of the adjoint vector sequence, a suboptimal switching manifold is obtained. The switching manifold designed assures that the state trajectories of the closed-loop system converge to zero as fast as possible on ideal sliding surface. The convergence velocity of every state trajectory on the ideal sliding surface can be adjusted through choosing quadratic performance index. Numerical simulation is given to show the effectiveness of the proposed design approach.