安全数据重建：一种直接数据驱动的方法
Secure Data Reconstruction: A Direct Data-Driven Approach
AI总结：
本文针对未知系统在数据可能被恶意篡改情况下的安全数据重建问题，提出基于行为语言和LASSO近似的直接数据驱动方法，并给出恢复真实轨迹的充分条件，为后续数据驱动控制提供安全保障。
摘要：
本文研究未知系统的安全数据重建问题，其中从系统采集的数据容易受到恶意篡改。我们旨在在缺乏系统模型先验知识的情况下恢复真实轨迹。为此，采用行为语言来表示系统，利用输入/输出轨迹而非状态空间模型来描述系统。我们考虑两种攻击场景。在第一种场景中，采集数据中最多有 $k$ 个条目是恶意的。另一方面，第二种场景假设传感器或执行器中最多有 $k$ 个通道可能被攻破，这意味着从这些通道采集的任何数据都可能被伪造。针对这两种场景，我们将轨迹恢复问题形式化为一个优化问题，并引入确保成功恢复真实数据的充分条件。由于寻找这些问题的精确解在计算上可能效率低下，我们进一步使用 $\ell_1$ 范数和组最小绝对收缩与选择算子（LASSO）对其进行近似。我们证明，在某些条件下，这些近似问题也能找到真实轨迹，同时保持较低的计算复杂度。最后，我们将所提出的算法扩展到含噪声数据。通过重建安全轨迹，本工作为后续的数据驱动控制方法提供了一种保障机制。
英文摘要：
This paper addresses the problem of secure data reconstruction for unknown systems, where data collected from the system are susceptible to malicious manipulation. We aim to recover the real trajectory without prior knowledge of the system model. To achieve this, a behavioral language is used to represent the system, describing it using input/output trajectories instead of state-space models. We consider two attack scenarios. In the first scenario, up to $k$ entries of the collected data are malicious. On the other hand, the second scenario assumes that at most $k$ channels from sensors or actuators can be compromised, implying that any data collected from these channels might be falsified. For both scenarios, we formulate the trajectory recovery problem as an optimization problem and introduce sufficient conditions to ensure successful recovery of the true data. Since finding exact solutions to these problems can be computationally inefficient, we further approximate them using an $\ell_1$-norm and group Least Absolute Shrinkage and Selection Operator (LASSO). We demonstrate that under certain conditions, these approximation problems also find the true trajectory while maintaining low computation complexity. Finally, we extend the proposed algorithms to noisy data. By reconstructing the secure trajectory, this work serves as a safeguard mechanism for subsequent data-driven control methods.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
