带约束的高超声速飞行器最优弹道的数值算法

Numerical Solution of Optimal Trajectory for Hypersonic Vehicle with Constraints

为了准确地求解高超声速飞行器再入巡航段节省燃料的最优弹道,在直接法得到次优解的基础上,提出了全程协态估算方法。该方法不仅能精确地估算出协态变量的初值,而且能得到协态变量的曲线,特别是对于存在内点约束的问题,协态变量不再连续时,能估算出各协态的跳变量。针对带约束的高超声速飞行器最省燃料弹道的特点,利用序列二次规划(SQP)算法得到其次优弹道,并应用所提的方法估算出协态变量的初值和跳变量,由极大值原理构造两点边值问题(TPBVP),以估算结果作为初值求解得到最优弹道。仿真结果表明:在300W/cm2的热流约束下,每个周期上最优解比稳态解节省约2.4%的燃料;另外,所提的全程协态估算方法能精确地估算出协态变量的初值和跳变量,以其作为初值有利于降低求解TPBVP的难度,使算法快速收敛到问题的最优解。

In order to obtain the precise optimal reentry cruise trajectory for a hypersonic vehicle with constraints, a co-state curve estimation method is developed based on the suboptimal results of the optimal control problem computed by the direct method. Using this technique, not only the initial values but also the time history of the co-states can be precisely estimated. Furthermore, when path constraints existed in the system and the co-state curves are not continuous, the jumps of the co-states can be worked out. According to the characteristics of the hypersonic vehicle minimum fuel trajectory, the suboptimal results are calculated by sequential quadratic programming（SQP）, and then the co-state initial values and jumps are guessed. After the two point boundary value problem（TPBVP） of the system is established based on the maximum principle, the optimal results are obtained. Simulation results indicate that under the constraint of a heat current of 300 W/cm^2 the optimal results save about 2.4 % fuel over the steady solutions. Meanwhile, the difficulties of the TPVBP are reduced with the accurate co-state initial values and jumps provided by this method, so that the optimal results can be gained rapidly.