平面线性二次调节器梯度流中的有限时间边界碰撞

中文摘要 AI 辅助

针对线性二次调节器的策略梯度方法利用无限时域上的代价来优化反馈增益。当该代价在某个固定初始状态处评估时，它不必在稳定边界的每个部分附近发散。我们研究欧几里得梯度流能否在有限优化时间内到达该边界。对于具有单输入和正定二次权重的可控平面系统，代价的精确表示给出了存在此类碰撞的充要条件。该条件归结为标量根计算，并包含临界情形，其中双重方向根吸引稳定增益的开集。对于三角族，该准则给出了显式的代数参数区域，并且每条轨迹要么收敛到Riccati增益，要么与稳定边界碰撞。紧凑代价子水平的尖锐阈值提供了收敛证书。沿每条碰撞轨迹，稳定裕度和累积状态格兰姆矩阵的最小特征值线性消失，尽管格兰姆矩阵在碰撞前保持正定。数值实验考察了初始化依赖性、临界方向附近的慢通过，以及在第二状态方向添加激励的影响。

英文摘要

Policy gradient methods for the linear quadratic regulator optimize feedback gains using a cost over an infinite horizon. When this cost is evaluated at one fixed initial state, it need not diverge near every part of the stability boundary. We study whether the Euclidean gradient flow can reach this boundary in finite optimization time. For controllable planar systems with one input and positive definite quadratic weights, an exact representation of the cost yields a necessary and sufficient condition for the existence of such a collision. The condition reduces to a scalar root calculation and includes the critical case, where a double direction root attracts an open set of stabilizing gains. For a triangular family, the criterion gives an explicit algebraic parameter region, and every trajectory either converges to the Riccati gain or collides with the stability boundary. A sharp threshold for compact cost sublevels provides a convergence certificate. Along every colliding trajectory, the stability margin and the smallest eigenvalue of the accumulated state Gramian vanish linearly, although the Gramian remains positive definite before collision. Numerical experiments examine the dependence on initialization, slow passage near a critical direction, and the effect of adding excitation in a second state direction.

发表机构

• School of Future Technology, Xi’an Jiaotong University（西安交通大学未来技术学院）

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