Minimum variance controllers have been employed in a wide-range of industrial applications. A key challenge experienced by many adaptive controllers is their poor empirical performance in the initial stages of learning. In this paper, we address the problem of initializing them so that they provide acceptable transients, and also provide an accompanying finite-time regret analysis, for adaptive minimum variance control of an auto-regressive system with exogenous inputs (ARX). Following [3], we consider a modified version of the Certainty Equivalence (CE) adaptive controller, which we call PIECE, that utilizes probing inputs for exploration. We show that it has a $C \log T$ bound on the regret after $T$ time-steps for bounded noise, and $C\log^2 T$ in the case of sub-Gaussian noise. The simulation results demonstrate the advantage of PIECE over the algorithm proposed in [3] as well as the standard Certainty Equivalence controller especially in the initial learning phase. To the best of our knowledge, this is the first work that provides finite-time regret bounds for an adaptive minimum variance controller.

翻译：最小方差控制器已广泛应用于各类工业场景。许多自适应控制器面临的关键挑战在于其学习初期的经验性能较差。本文针对外生输入自回归系统的自适应最小方差控制问题，研究如何初始化控制器以获得可接受的暂态响应，并给出相应的有限时间遗憾分析。借鉴文献[3]，我们提出了一种确信等价自适应控制器的改进版本——PIECE，通过引入探测输入实现探索行为。研究表明，在有界噪声条件下，该算法在T个时间步后的遗憾界为$C \log T$；在次高斯噪声条件下，遗憾界为$C\log^2 T$。仿真结果验证了PIECE相较于文献[3]所提算法以及标准确信等价控制器的优势，尤其在初始学习阶段表现更为突出。据我们所知，这是首次为自适应最小方差控制器建立有限时间遗憾界的研究工作。