Optimal Sliding Mode Design For Uncertain Nonlinear Systems With Time-delay Based On SAA

For uncertain time-delay systems and nonlinear systems with uncertainties, the problems of optimal control, optimal sliding mode design, global robust optimal sliding mode control and physically realizable problem are considered in this dissertation.Firstly, for time-delay systems and nonlinear systems without uncertainties, this study is to find an approximate solution method and to combine it with SMC theory and to achieve the optimal sliding modes. Secondly, for time-delay systems and nonlinear systems with uncertainties, optimal sliding mode control is introduced. Thirdly, optimal sliding mode control for time-delay nonlinear systems with uncertainties is discussed. The study addresses the following topics:1. The history of the development in SMC theory is reviewed. The latest research tendency in this field is introduced, and the main methods to design sliding surfaces are addressed specifically. The existing problems and research objectives are presented. The research subject and significance of this dissertation are also given.2. The optimal control problem for a class of time-delay linear systems with quadratic performance indexes is considered. By treating some state variables as virtual control, a quadratic performance index is given. An equation about all of the state variables is obtained through the virtual optimal control law, and then we construct a novel optimal sliding surface with this equation. For the two-point boundary value (TPBV) problem with both time-delay and time-advance terms deduced from the maximum principle, by using the successive approximation approach (SAA), two iterative sequences of differential equations with known initial and terminal conditions are first constructed, respectively. Their solution sequences are then proven to uniformly converge to the solution of the original TPBV problem. For the case of infinite time horizon, the condition of existence and uniqueness of the optimal solution is given and proven. Furthermore, by finite iteration of the solution sequences, approximate solutions for the optimal control are obtained, which consists of an analytical state feedforward and feedback term and a time-delay compensation term in the limit of adjoint vector sequence. Sufficient asymptotic stability conditions of the closed-loop systems are derived both for linear and nonlinear systems with time-delay. This part provides the foundation for the later study.3. The problem of designing optimal sliding manifolds with a quadratic performance index for a class of uncertain systems with state time-delay is considered. The optimal control technique is employed to construct sliding manifold, so the sliding motion is optimized. The conditions of robustness to matched parameter uncertainties and external disturbances are discussed. Disturbances observer is designed to solve the physically realizable problem. The stability of the optimal sliding motion is analyzed. Simulation results demonstrate the efficiency of the proposed method.4. The problem of designing nonlinear sliding manifolds with a quadratic performance index for a class of nonlinear uncertainty systems is considered. The solving optimal sliding mode problem is transformed into solving nonlinear TPBV problem, and then transformed into a sequence of iteration. Then an adjoint vector is introduced to compensate the nonlinear part in state equation. Using SAA, the original nonlinear TPBV problem is transformed into a sequence of nonhomogeneous linear TPBV problems without nonlinear terms and then an approximate optimal control law is obtained. Furthermore, the optimal control theory is adopted to construct nonlinear sliding manifold. The dynamics of sliding motion could minimize the given quadratic performance index and is robust to uncertainties with known upper bounds. Simulation results demonstrate the efficiency of the proposed method.5. For a class of uncertain nonlinear systems with time delays, the problem of designing optimal sliding mode controllers is discussed. The conditions of robustness to matched parameter uncertainties and external disturbances are discussed. A design strategy which can robustify the optimal controllers is presented. Sufficient asymptotic stability conditions of the closed-loop systems are derived. Simulations are given to show the effectiveness. 6. The conclusions and the directions for the future research work are given in the end of the paper.