This paper considers the linear-quadratic dual control problem where the system parameters need to be identified and the control objective needs to be optimized in the meantime. Contrary to existing works on data-driven linear-quadratic regulation, which typically provide error or regret bounds within a certain probability, we propose an online algorithm that guarantees the asymptotic optimality of the controller in the almost sure sense. Our dual control strategy consists of two parts: a switched controller with time-decaying exploration noise and Markov parameter inference based on the cross-correlation between the exploration noise and system output. Central to the almost sure performance guarantee is a safe switched control strategy that falls back to a known conservative but stable controller when the actual state deviates significantly from the target state. We prove that this switching strategy rules out any potential destabilizing controllers from being applied, while the performance gap between our switching strategy and the optimal linear state feedback is exponentially small. Under our dual control scheme, the parameter inference error scales as $O(T^{-1/4+\epsilon})$, while the suboptimality gap of control performance scales as $O(T^{-1/2+\epsilon})$, where $T$ is the number of time steps, and $\epsilon$ is an arbitrarily small positive number. Simulation results on an industrial process example are provided to illustrate the effectiveness of our proposed strategy.

翻译：本文考虑了线性水下双重控制问题, 需要确定系统参数, 同时也需要优化控制目标。 与目前关于数据驱动线性水下监管的工程相反, 通常在某种可能性范围内提供错误或遗憾界限。 我们提议了一个在线算法, 保证控制器在几乎肯定的意义上的无症状最佳性。 我们的双重控制战略由两部分组成: 一个交换控制器, 带有时间淡化的勘探噪音和 Markov 参数推推, 以勘探噪音和系统输出之间的交叉关系为基础。 几乎可以肯定的绩效保障的核心是安全调换控制战略, 当实际状态明显偏离目标状态时, 它会回到已知的保守但稳定的控制者手中。 我们证明, 这一转换战略排除了任何潜在的破坏控制器在应用中的不稳性优化, 而我们调换战略与最佳线性状态反馈之间的性差则非常小。 在我们的双重控制计划下, 参数误差比值为 $O-1/4 和 epsilon}, 而其次偏差值则提供了 美元 性能 的 度 度 度 度 度 步骤 。