This article develops a methodology that enables learning an objective function of an optimal control system from incomplete trajectory observations. The objective function is assumed to be a weighted sum of features (or basis functions) with unknown weights, and the observed data is a segment of a trajectory of system states and inputs. The proposed technique introduces the concept of the recovery matrix to establish the relationship between any available segment of the trajectory and the weights of given candidate features. The rank of the recovery matrix indicates whether a subset of relevant features can be found among the candidate features and the corresponding weights can be learned from the segment data. The recovery matrix can be obtained iteratively and its rank non-decreasing property shows that additional observations may contribute to the objective learning. Based on the recovery matrix, a method for using incomplete trajectory observations to learn the weights of selected features is established, and an incremental inverse optimal control algorithm is developed by automatically finding the minimal required observation. The effectiveness of the proposed method is demonstrated on a linear quadratic regulator system and a simulated robot manipulator.

本文开发了一种方法，该方法可以从不完整的轨迹观察中学习最佳控制系统的目标功能。假设目标函数是具有未知权重的特征（或基本函数）的加权总和，并且观察到的数据是系统状态和输入轨迹的一部分。所提出的技术引入了恢复矩阵的概念，以建立轨迹的任何可用段与给定候选特征的权重之间的关系。恢复矩阵的等级指示是否可以在候选特征中找到相关特征的子集，并且可以从分段数据中获知相应的权重。可以迭代获得恢复矩阵，并且恢复矩阵的等级不变性表明附加的观察结果可能有助于客观学习。基于恢复矩阵，建立了一种利用不完整轨迹观测值学习所选特征权重的方法，并通过自动寻找最小需求观测值来开发增量逆最优控制算法。在线性二次调节器系统和模拟机器人操纵器上证明了该方法的有效性。