数据驱动控制的基本局限：统计决策视角

中文摘要 AI 辅助

已有大量研究致力于数据驱动控制器的设计，但对其统计性能和基本局限的了解相对较少。本文提出了一种数据驱动控制的统计决策框架，其中控制器通过其风险（定义为相对于基于神谕模型的控制器的预期性能退化）及其在参数空间上的平均风险进行评估。在该框架内，我们提出了一系列数据驱动控制器的设计原则。我们进一步结合偏差-方差分解与克拉美罗不等式推导了风险的下界。特别地，通过变分法确定了达到平均风险下界的最优偏差，从而明确了数据驱动控制中的偏差-方差权衡。此外，推导的界揭示了数据驱动控制中的“水床”效应：在参数空间的一个区域内，相对于下界的任何风险改善，必须由其他区域的恶化来补偿。我们在两个典型的数据驱动控制问题（最优前馈控制和线性二次调节器基准）上说明了所提出的框架。通过将几种代表性数据驱动控制器与推导的下界进行比较，我们深化了对现有方法的统计解释，并揭示了任何控制器设计都无法避免的定量局限。

英文摘要

Substantial research efforts have been devoted to the design of data-driven controllers; however, comparatively less is known about their statistical performance and fundamental limitations. This contribution develops a statistical decision framework for data-driven control, in which a controller is evaluated by its risk, defined as the expected performance degradation relative to the oracle model-based controller, and by its average risk over the parameter space. Within this framework, we propose a collection of design principles for data-driven controllers. We further derive lower bounds on risks by combining the bias-variance decomposition with the Cramér-Rao inequality. In particular, the optimal bias that attains the lower bound for the average risk is determined by calculus of variations, thereby making the bias-variance tradeoff in data-driven control explicit. Moreover, the derived bound reveals a ``waterbed'' effect in data-driven control: any improvement in risk relative to the lower bound over one region of the parameter space must be compensated by deterioration elsewhere. We illustrate the proposed framework on two canonical data-driven control problems: optimal feedforward control and the linear quadratic regulator benchmark. By comparing several representative data-driven controllers with the derived lower bounds, we sharpen the statistical interpretation of existing methods and reveal quantitative limitations that no controller design can avoid.

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submitted to an IEEE journal