Symplectic Numerical Method For Constrained Optimal Control Problem And Its Application

Optimal control problems are encountered in various disciplines. The study of computational optimal control is of significant value because the analytical solutions are not available for most optimal control problems. In fact, aeronautics and astronautics have keen needs for numerical methods with perfect performances with respect to the computational efficiency, accuracy, reliability, and robustness. Most engineering optimal control problems will be subjected to all kinds of constraints, which result in various constrained optimal control problems. Mathematically speaking, various constraints can be roughly categorized as inequality constraints and equality ones. This dissertation develops symplectic numerical methods for constrained optimal control problems, and then focuses on the application of the proposed methods to the optimal control of spacecraft formations around libration points. The specific works of this dissertation are listed as follows1.A symplectic sequence iteration approach for nonlinear optimal control with inequality constraints is proposed. Nonlinear optimal control problems are firstly converted into a sequence of constrained linear quadratic optimal control problems by quasilinearization methods. Then, based on least action principles and symplectic discretizations, constrained linear optimal control problems are reduced to standard linear complementary problems which can be solved easily. Algorithm analysis and numerical simulations show that the proposed method has several advantages including fast convergence, well robustness and high computational efficiency.2.With the introduction of Lagrange multipliers and penalty functions, two symplectic methods are established for optimal control problems with equality constraints. In symplectic methods based on Lagrange multipliers, Lagrange multipliers are regarded as fundamental variables like state variables. Original optimal control problems are transferred into nonlinear algebraic equations with explicit Jacobian matrixes by symplectic discretization to the mixed action principles including Lagrange multiplier, state and costate variables. Then optimal solutions can be obtained by Newton's methods. On the other hand, with the introduction of proper penalty functions and control strategies of penalty factors, constrained optimal control problems are reduced to optimal control problems without constraints in symplectic methods based on penalty functions. And then the obtained unconstrained optimal control problems can be solved by well developed symplectic algorithms. Analytical analysis and numerical experiments demonstrate symplectic methods based on Lagrange multipliers are more sensitive to the forms of constraints compared with ones based on penalty functions. Therefore, symplectic methods with Lagrange multiplers can tackle optimal control problems with linear equality constraints but fail to obtain convergent solutions in the case of nonlinear equality constraints. By contrast, symplectic methods with penalty functions can solve optimal control problems with linear or nonlinear equality constraints effectively.3.The proposed symplectic methods are successfully applied to the optimal control of loose spacecraft formations around libration points. In previous studies, all constraints were rigid in spacecraft formations, and the minimization of energy cost was not highlighted enough. Thus, this study proposed a novel formation named loose sphere leader-follwing formations and then focused on the optimal control of the formation reconfigurations and keepings with a goal of the minimization of energy cost. Collision avoidances among follower spacecrafts are taken into consideration since there are no constraints between followers and then the formation for all followers is loose. With normalization techniques and controllable parameters, an effective penalty function to avoid collision is developed. And then symplectic methods based on penalty functions are utilized to solve the optimal control problems efficiently. Numerical results show that errors of formations can be reduced to millimeters or smaller only by few control input and the collision can be avoided successfully by the developed penalty function.