Symplectic Numerical Method For Computational Optimal Control And Its Application In The Control Of Spacecraft Near The Libration Point

From the middle of20th century, optimal control is viewed as the one core of modern control theory, and it has attracted attentions with successful applications in many science and engineering fields, especially the aeronautics and astronautics. With the increasing of complexity of real physical system, the complexity of analysis and design for controlled system is also increasing much. So the theory analysis and controller design can not be obtained only by analytical method. As a result, computational optimal control has been paid adequate attention by researchers and engineers. However, most of the traditional mathematical methods care for the approximate numerical results for precise results. They have no time to attend to the inherent quality of the optimal control problem expressed by the mathematical model. In general, the numerical algorithms can be structured without consideration of the inherent quality of optimal control problem. Symplectic numerical methods compared with the traditional numerical methods can keep the inherent quality of the original continuous system. Meanwhile, the numerical accuracy, efficiency and the stability of the numerical methods can be ensured. Consequently, the investigations of symplectic numerical methods for optimal control problem are very importamt!Therefore, the symplectic numerical methods for computational optimal control are researched in this dissertation. From the special periodic optimal control problem to general nonlinear optimal control problem and to real guidance problem, this dissertation aims at contributing some useful researches and trials on the construction of symplectic numerical algorithm for optimal control. This part of work is the basis of the whole dissertation. On the other hand, the rest work of this dissertation is about the application of symplectic algorithms in control mission of spacecraft near the libration. From the station-keeping and formation keeping of spacecrafts on periodic orbits, trajectory optimization for spacecrafts formation reconfiguration, and real guidance for orbit transfer, also the rest work of this dissertation aims at contributing some useful theory and technology for libration mission in the near future. Accordingly, the following studies are carried out in this dissertation:1. The research and application of the symplectic algorithm for periodic optimal control.(1) For the optimal control requirement of spacecrafts and their formation on periodic orbits near the libration, the symplectic algorithm for periodic optimal control is proposed and is viewed as the core content. It contains the symplectic algorithms for the periodic Lyapunov and Riccati differentia] equation. Based on the above symplectic algorithms, symplectic algorithms for computation of the H2norm and H∞norm of periodic system are proposed. The "overflowing" problem in large periodic and unsteady periodic system has been avoided by the proposed symplectic algorithm. Through the comparisons with the periodic generator method and the fourth-order symplectic Runge-Kutta method, the proposed symplectic algorithms for periodic optimal control have obvious advantage on accuracy and efficiency.(2) Furthermore, the improved time-varying periodic optimal controller based on mixed parameter optimization is proposed and the station keeping mission of a single spacecraft on Halo orbit is achieved successfully; also the formation keeping mission of spacecrafts on Halo orbit is implemented by the distributed cooperative periodic optimal controller. The simulation results of application of the proposed symplectic algorithms show that the periodic optimal controller conquers the limitation of time-invariant controller in low Halo orbit. Besides, the accuracy of follower spacecrafts formation can keep as the level of millimeter, and it meets the requirement of NASA.2. The research and application of the symplectic algorithm for nonlinear optimal control.(1) For the nonlinear trajectory optimization requirement of spacecrafts formation reconfiguration near the libration, the symplectic algorithms for nonlinear optimal control are proposed. In detail, four different high accuracy and high efficiency symplectic algorithms are proposed based on dual variational principle and generating function. Furthermore, a three-variable symplectic algorithm is proposed for the nonlinear optimal control with free terminal time. On one hand, these symplectic algorithms are constructed based on variational principle, so the necessary condition of optimal control can be satisfied. Besides, they need no precise guess of the initial costate variables, and the costate variables on all time point can be provided for validating the optimality of the numerical solutions. On the other hand, the nonlinear optimal control is transformed into nonlinear equations, and the low-efficiency problem of large scale nonlinear program has been avoided. Through the comparisons with other symplectic numerical methods (Discrete Mechanics and Optimal Control method) and non-symplectic numerical methods (Gauss pseudospectral method), the proposed symplectic numerical methods for nonlinear optimal control have1-2orders advantage on accuracy and efficiency.(2) Furthermore, based on the leader-follower model and the virtual structure model, the balanced-energy reconfiguration of spacecrafts formation near the libration point is proposed and solved. The simulation results of application of symplectic algorithms show that the symplectic algorithm can online quickly solve the trajectory programming of balanced-energy reconfiguration. Beside, with the increasing time of reconfiguration process, the fuel consumption can be decreased a lot. 3. The research and application of the nonlinear optimal control problems with control inequality constraints.(1) For the nonlinear trajectory optimization requirement of the spacecraft with control inequality constraints near the libration point, the numerical method based on the parametric variational principle is proposed for linear-quadratic and nonlinear system optimal control problem. The proposed numerical method avoids the so-called Gibbs phenomenon due to a lot of increasing integration points for non-smooth optimal control problem. Through the comparisons with the Euler, Trapezoidal collocation methods and Radau pseudospectral method, the proposed numerical method has obvious advantage on accuracy and efficiency.(2) Furthermore, the proposed numerical method has been applied into the solving of orbit transfer between the periodic orbits near the libration for finite low-thrust spacecraft. Numerical simulations show that the thrust amplitude of constrained spacecraft is higher than the thrust amplitude of constrained spacecraft. Besides, the consumption of energy of constrained spacecraft is larger than the consumption of energy of unconstrained spacecraft.4. The research and application of the real time guidance symplectic algorithm.(1) For the real guidance requirement of orbit transfer between the periodic orbit and from the orbit near the Earth to the periodic orbit near the libration point, receding horizon control is taken as a reference trajectory guidance method. The symplectic algorithms based on generating function and interval mixed energy method are constructed for solving receding horizon control. The efficiency of the receding horizon control is increased much by avoiding online integration of Riccati differential equation. Through the comparisons with the traditional backward sweep method, the efficiency of the proposed symplectic algorithm has increased1-2orders and the symplectic algorithm is insensitive to large time step.(2) Furthermore, the proposed symplectic algorithm has been applied into the solving of orbit rendezvous guidance and orbit transfer guidance from the Earth orbit to the libration orbit orbit. Numerical simulations show that the spacecraft with the symplectic guidance algorithm can follow the reference orbit precisely under the deterministic velocity errors or the random velocity errors, and obtain the satisfying terminal control accuracy. Through the comparisons with the backward sweep method, symplectic guidance algorithm has the strong robustness on random control parameters, and it has the small trajectory error.