求解复杂约束条件下最优控制问题的分段低阶Gauss伪谱法

针对含有复杂约束条件的最优控制问题,提出分段低阶Gauss伪谱法。以常规Gauss伪谱法为基础,划分时间区间,在子区间上利用低阶Gauss数值积分离散Bolza型性能指标,利用插值型数值积分的性质离散状态微分方程,利用低阶Gauss伪谱法处理复杂约束条件,得到对应的非线性规划。对状态轨线或控制函数较复杂的情形,该方法克服了传统Gauss伪谱法直接在时间区间上配置Gauss点,插值多项式阶数高、数值解不稳定的缺陷,并且数值解局部代数精度高、计算量小。最后将该方法应用于求解飞行器对地打击轨迹规划最优控制问题,结果表明算法有效可行。

The Solution of an Optimal Control Problem under Conditions of Complicated Restrictions by Using the Segmented Low-order Gauss Pseudospectral Method

As for the solution of optimal control issues under conditions of complex restrictive conditions, this paper proposes a segmented low-order Gauss pseudospectral method. In this paper, a nonlinear program issue is obtained by the basis of conventional Gauss pseudospectral method and the temporal intervals compartmentalization. During each subinterval, low-order Gauss numerical integration is utilized for discreteness performance of Bolza problem; the property of interpolated type numerical integration is utilized for discreteness of differential state equation; low-order Gauss pseudospectral method is utilized to process complicate constraints. The method overcomes the shortcomings of a higher order of interpolation polynomial and an unstable numerical solution caused by producing Gaussian points at the interval directly under the situation of complex state orbit and control function, thus, having high local algebraic precision and small computation in numerical solution. Ultimately, the method is applied to optimal trajectory planning for air-to-ground operations. The result shows that the algorithm is effective and feasible.