Inverse optimal control for averaged cost per stage linear quadratic regulators

Inverse Optimal Control (IOC) is a powerful framework for learning a behaviour from observations of experts. The framework aims to identify the underlying cost function that the observed optimal trajectories (the experts' behaviour) are optimal with respect to. In this work, we considered the case of identifying the cost and the feedback law from observed trajectories generated by an ``average cost per stage" linear quadratic regulator. We show that identifying the cost is in general an ill-posed problem, and give necessary and sufficient conditions for non-identifiability. Moreover, despite the fact that the problem is in general ill-posed, we construct an estimator for the cost function and show that the control gain corresponding to this estimator is a statistically consistent estimator for the true underlying control gain. In fact, the constructed estimator is based on convex optimization, and hence the proved statistical consistency is also observed in practice. We illustrate the latter by applying the method on a simulation example from rehabilitation robotics.