Optimal guidance and control in space technology

Abstract

In this thesis, we deal with several optimal guidance and control problems of the spacecrafts arising from the study of lunar exploration. The research is composed of three parts: 1. Optimal guidance for the lunar module soft landing, 2. Spacecraft attitude control system design basing on double gimbal control moment gyroscopes (DGCMGs), and 3. Synchronization motion control for a class of nonlinear system.To achieve a precise pinpoint lunar module soft landing, we first derive a three dimensional dynamics to describe the motion of the module for the powered descent part by introducing three coordinate frames with consideration of the moon rotation. Then, we move on to construct an optimal guidance law to achieve the lunar module soft landing which is treated as a continuously powered descent process with a constraint on the angle of the module between its longitudinal axis and the moon surface. When the module reaches the landing target, the terminal attitude of the module should be within an allowable small deviation from being vertical with reference to lunar surface. The fuel consumption and the terminal time should also be minimized. The optimal descent trajectory of the lunar module is calculated by using the control parameterization technique in conjunction with a time scaling transform. By these two methods, the optimal control problem is approximated by a sequence of optimal parameter selection problems which can be solved by existing gradient-based optimization methods. MISER 3.3, a general purpose optimal control software package, was developed based on these methods. We make use of this optimal control software package to solve our problem. The optimal trajectory tracking problem, where a desired trajectory is to be tracked with the least fuel consumption in the minimum time, is also considered and solved.With the consideration of some unpredicted situations, such as initial point perturbations, we move on to construct a nonlinear optimal feedback control law for the powered deceleration phase of the lunar module soft landing. The motion of the lunar module is described in the three dimensional coordinate system. Based on the nonlinear dynamics of the module, we obtain the form of an optimal closed loop control law, where a feedback gain matrix is involved. It is then shown that this feedback gain matrix satisfies a Riccati-like matrix differential equation. The optimal control problem is first solved as an open loop optimal control problem by using the time scaling transform and the control parameterization method. By virtue of the relationship between the optimal open loop control and the optimal closed loop control along the optimal trajectory, we present a practical method to calculate an approximate optimal feedback gain matrix, without having to solve an optimal control problem involving the complex Riccati-like matrix differential equation coupled with the original system dynamics.To realize the spacecraft large angle attitude maneuvers, we derive an exact general mathematical description of spacecraft attitude motion driven by DGCMGs system. Then, a nonlinear control law is designed based on the second method of Lyapunov and the stability of the attitude control system is established during the design process. A singularity robustness plus null motion steering law is designed to realize the control law. Principle of DGCMGs’ singularity is proved, and the singularity analysis of the orthogonally mounted three DGCMGs system and that of the parallel mounted four DGCMGs system are presented.Finally, we consider a new class of nonlinear optimal tracking and synchronizing control problems subject to control constraints, where the motions of two distinct objects are required to achieve synchronization at the minimum time while achieving the optimal tracking of a reference target. We first provide a rigorous mathematical formulation for this class of optimal control problems. A new result ensuring the synchronization of the two distinct objects is obtained. On this basis, a computational method is developed for constructing an optimal switching control law under which the motions of the two distinct objects will achieve synchronization at the minimum time while achieving the optimal tracking of a reference target. This computational method is developed based on novel applications of the control parameterization method and a time scaling transform. A practical problem arising from the study of the angular velocity tracking and synchronization of two spacecrafts during their formation flight is formulated and solved by the method proposed.