This paper addresses the optimal control problem known as the linear quadratic regulator in the case when the dynamics are unknown. We propose a multistage procedure, called Coarse-ID control, that estimates a model from a few experimental trials, estimates the error in that model with respect to the truth, and then designs a controller using both the model and uncertainty estimate. Our technique uses contemporary tools from random matrix theory to bound the error in the estimation procedure. We also employ a recently developed approach to control synthesis called System Level Synthesis that enables robust control design by solving a quasi-convex optimization problem. We provide end-to-end bounds on the relative error in control cost that are optimal in the number of parameters and that highlight salient properties of the system to be controlled such as closed-loop sensitivity and optimal control magnitude. We show experimentally that the Coarse-ID approach enables efficient computation of a stabilizing controller in regimes where simple control schemes that do not take the model uncertainty into account fail to stabilize the true system.

中文翻译：

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线性二次调节器的样本复杂度

本文解决了在动力学未知的情况下称为线性二次调节器的最优控制问题。我们提出了一个称为Coarse-ID控制的多阶段程序，该程序从一些实验试验中估算出一个模型，根据真实性估算出该模型中的误差，然后使用该模型和不确定性估算来设计控制器。我们的技术使用随机矩阵理论中的现代工具来限制估计过程中的误差。我们还采用了最近开发的控制综合方法，称为系统级综合通过解决准凸优化问题来实现鲁棒的控制设计。我们提供了控制成本相对误差的端到端界限，这些界限在参数数量上是最佳的，并且突出了要控制的系统的显着特性，例如闭环灵敏度和最佳控制幅度。我们通过实验表明，粗略ID方法能够在不考虑模型不确定性的简单控制方案无法稳定真实系统的情况下有效地计算稳定控制器。