Optimal Sliding Mode Control For Time-delay Nonlinear Systems With Uncertainty

For uncertain time-delay nonlinear systems, the problems of optimal control, optimal sliding mode design and global robust optimal sliding mode control are considered in this dissertation.One of the most distinguished features of the sliding mode control (SMC) is that after reaching the sliding mode, the system has robustness to matched parameter uncertainties and external disturbances. The system dynamics performance on sliding mode can be pre-designed. And the control algorithm is simple and easy to be realized. All these features of SMC show a great superiority on controlling uncertain time-delay and nonlinear systems. Obviously, the optimization of sliding mode is a significant problem for a SMC system. Linear quadratic (LQ) performance index can synthetically express the requirement of the system performance, and the optimal LQ Regular theory is one of the most important contents of optimal control theory. Nevertheless, the design of optimal controllers is based on exact known models. Therefore it is necessary to take the robustness into account in the study of optimal controller design. In addition, the solution of the optimal control law for time-delay nonlinear systems is difficult to obtain.Considering the problems above, the basic idea of this study is to find an approximate solution method for time-delay nonlinear systems first, and then to combine it with SMC theory to achieve the following objectives. Firstly, the sliding motion is optimized by applying optimal control technique to designing sliding mode. Secondly, SMC theory is introduced to optimal controller design to put the robustness of sliding mode into conventional optimal systems. This is a very significant subject. About uncertain time-delay nonlinear systems, such research has hardly been found in the literature. The study addresses the following topics: 1. The history of the development in sliding mode control theory is reviewed. The main methods and the latest research tendency are also summarized. The existing problems and research objective is presented.2. The optimal control problem for a class of time-delay linear systems and time-delay nonlinear systems with quadratic performance indexes is considered, respectively. According to the necessary optimality conditions, the linear and nonlinear two-point boundary value (TPBV) problems with both time-delay and time-advance terms are induced, respectively. A sensitivity approach is adopted to convert the original TPBV problems into that of solving a series of linear TPBV iterative formulas without time-delay and time-advance variable terms. By the finite-step iteration, approximate solutions for the optimal control are obtained. 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 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 systems is considered. As we know, it is difficult to choose an appropriate nonlinear sliding manifold on which the sliding motion has desired performance. To solve this problem, firstly, a nonsingular diffeomorphism transformation is used to convert the nonlinear system into a regular form. Then by viewing the some state variables as virtual control, the problem of optimal sliding mode design is transformed into optimal control problem for nonlinear systems. 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.5. For a class of uncertain linear systems and nonlinear systems with time delays, the problem of designing global robust optimal sliding mode controllers (GROSMC) is discussed. Inspired by the concept of integral sliding mode(ISM), which can ensure the global sliding mode during the entire response of the system, a design strategy which can robustify the optimal controllers is presented. First, the optimal controller is obtained for the nominal system ignoring uncertainties using optimal theory. Then based on ISM, a discontinuous compensation control law (also called robustification control law) is designed to suppress the uncertainties from the initial time moment. As a result, not only the optimal performance can be obtained but the global robustness is guaranteed also. Simulations are given to compare the proposed GROSMC with the conventional optimal control. Results show the effectiveness and superiority in robustness of the proposed method.6. The conclusions and the directions for the future research work are given in the end of the paper.