利用广义Riccati递推中的微结构求解最优控制问题

中文摘要 AI 辅助

最优控制问题（Optimal Control Problems, OCPs）已应用于多种场景，这类问题常被转化为非线性规划（Nonlinear Program, NLP），其求解方法通常会解一个线性二次（Linear Quadratic, LQ）子问题。为高效求解该LQ子问题，Riccati递推会利用通过多射击法（Multiple Shooting method）离散OCP时产生的块对角结构。然而，将一般动力学约束重构为适配Riccati递推所需问题结构的形式时，会在这些块矩阵内引入额外的微结构，而该微结构尚未被利用。本研究展示了利用结构的求解器（如Fatrop）如何通过不仅利用块对角结构，还利用这些块内的微结构来获益。我们提出一种利用结构的LU分解算法，并将其集成到Riccati递推中。对随机线性系统和两个OCP示例的数值结果表明，搜索方向的计算时间最多减少20%，在相对零块大小约为0.25且存在大的阶段等式约束雅可比矩阵时，可观察到最大加速效果。

英文摘要

Optimal Control Problems (OCPs) have been applied to a variety of applications. Such problems are often transcribed into a Nonlinear Program (NLP) for which solution methods typically solve a Linear Quadratic (LQ) subproblem. To efficiently solve this LQ subproblem, the Riccati recursion exploits the block-diagonal structure that arises from discretizing an OCP using a Multiple Shooting method. However, reformulating general dynamics constraints to fit the required problem structure to use the Riccati recursion introduces additional micro-structure within these block matrices that remains unexploited. This work demonstrates how structure-exploiting solvers such as Fatrop can benefit from exploiting not only the block-diagonal structure but also the micro-structure within those blocks. We propose a structure-exploiting LU decomposition algorithm and integrate it within the Riccati recursion. Numerical results on randomized linear systems and two OCP examples demonstrate a reduction in computation time of the search direction of up to 20%, with the largest speedup observed for relative zero block sizes around 0.25 and large stage-wise equality constraint jacobians.

发表机构

• KU Leuven（荷语鲁汶大学）

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