基于微分几何理论的高超声速目标拦截末制导方法研究

Given the problem that the traditional guidance law can hardly intercept the hypersonic vehicles, the novel guidance methods with the kinematic kill capability are in urgent demand. This project intends to carry out researches on the terminal guidance against hypersonic vehicles with the differential geometric curve theory and surface theory. By comprehensively considering the rapidly changing targets dynamics, the effects of the earth rotation and radius of curvature on the relative kinematics, the inherent principles of the trajectory and the changes of space curves and surfaces are explored. The more accurate interception model is reestablished with the differential geometric curve theory in non decoupling comndition, and its inherent relations with the traditional interception model are revealed. With these contributions serving as the basis, the optimal control theory, sliding mode control theory and fractional calculus theory are introduced into the guidance area. The differential geometric guidance algorithm is derived based on the idea of direct collision, which broadens the differential geometric theory application and develops the novel guidance law. The capture states of the differential geometric guidance law are analyzed with the mathematical set theory, and the implicit factors that cause the changes of the capture states are analyzed, and the accomplishments have the merits of universal application and give the solution to the terminal guidance states selection and capture ability analysis. The project is of great theoretical significance and practical value, which will offer theoretical support and technological reservation for the interception against hypersonic target.

针对传统制导律难以有效拦截高超声速目标的问题，亟需研究具有直接碰撞能力的新型制导方法。本项目拟以微分几何曲线论和曲面论为基础，研究高超声速目标拦截的制导问题。综合考虑目标的高动态特性、地球自转和曲率半径对相对运动学关系的影响，探索飞行器运动轨迹与空间曲线和曲面变化的内在规律，建立非解耦条件下更加精确的弹目相对运动学模型，揭示传统弹目相对运动建模与微分几何建模的内在关系。以此为基础，将最优化理论、滑模理论、分数阶微积分理论等引入制导领域，基于直接碰撞的思想提出多约束条件下微分几何制导算法，拓展微分几何理论在制导领域的应用，发展制导新理论。结合集合理论研究微分几何制导律的捕获状态数学模型，分析不确定测量信息对捕获状态影响规律，提出具有普适特性的捕获状态研究方法，解决末制导的初始状态选取和捕获能力分析问题。本项目的开展为高超声速目标拦截提供理论支撑和技术储备，具有重要的理论意义和实用价值。

针对高超声速目标飞行速度快、机动范围大、突防能力强、打击范围广等特点，采用传统的制导规律难以保证制导性能，项目以微分几何理论为基础，结合现代控制理论，研究了反高超声速目标新型末制导方法。研究中，利用微分几何理论建立了非解耦条件下高超声速目标拦截的微分几何模型，克服了传统双平面解耦建模导致的制导信息丢失问题，研究了传统时域中三维拦截建模和弧长域中微分几何建模内在联系，为反临近空间目标末制导方法设计奠定模型基础。借助于最优化理论、分数阶理论以及滑模控制理论分别设计了基于零化视线角速率的微分几何制导律、分数阶微分几何制导律、非线性微分几何制导律和二阶滑模控制的微分几何制导律，结果表明，相对于传统末制导方法，所设计的微分几何制导方法具有制导精度提高、过载变化平稳等特点，更加有利于对高超声速目标拦截。借助于集合理论和奇异点理论，研究了所设计制导方法的捕获状态，分析了不同捕获状态对制导性能的影响，提出了捕获状态研究的基本思路，为拦截器末制导初始状态选取提供了理论依据。项目的研究成果为反高超声速目标制导方法研究提供了理论铺垫和技术积累，具有重要的理论意义。