AI 中文总结
本文针对输入符号不确定的线性正系统，将极小极大最优双重控制问题转化为零和动态博弈，通过随机控制输入得到精确解，提出兼具成本与ℓ₁增益最优性的隐式双重控制策略，填补了正系统双重控制相关结果的空白。
AI 中文摘要
尽管极小极大双重控制的最新进展已为不确定的一般线性时不变系统以及（次）最优双重控制器提供了精确解，但线性正系统的相应结果仍然缺失。本文旨在填补这一空白，从而为可扩展的双重控制算法铺平道路。我们研究了具有未知动力学的线性正系统的一般极小极大最优双重控制问题，并将其重新表述为标准零和动态博弈。通过允许随机控制输入，我们针对输入存在符号不确定性的标量情况精确求解了相应的贝尔曼方程，得到了一个在成本和ℓ₁增益方面均最优的隐式双重控制策略。该最优双重策略在超状态空间的特定区域利用探索进行最优探测；在该探索区域之外，控制器简化为确定性的确定性等价策略，表明已获得足够信息以识别正确的输入方向。此外，这些结果使我们能够分析正系统极小极大双重控制的基本局限性，并为正系统更一般的双重控制问题的未来工作提供基础。
英文摘要
While recent advances in minimax dual control have led to exact solutions for uncertain general linear time-invariant systems as well as (sub)optimal dual controllers, corresponding results for linear positive systems are still lacking. This paper aims to fill this gap and thereby pave the way toward scalable dual control algorithms. We study the general minimax optimal dual control problem for positive linear systems with unknown dynamics and reformulate it as a standard zero-sum dynamic game. By allowing randomized control inputs, we solve the corresponding Bellman equation exactly for the scalar case with sign uncertainty in the input. This yields an implicit dual control policy that is optimal both in terms of cost and $\ell_1$-gain. The optimal dual policy uses exploration in a specific region of the hyperstate space to conduct optimal probing. Outside this exploration regime, the controller reduces to a deterministic certainty equivalence policy, indicating that sufficient information has been obtained to identify the correct input direction. In addition, these results allow us to analyze fundamental limitations of minimax dual control for positive systems and provide a foundation for more general dual control problems for positive systems for future work.
