In this paper, we study robust beamforming design for near-field physical-layer-security (PLS) systems, where a base station (BS) equipped with an extremely large-scale array (XL-array) serves multiple near-field legitimate users (Bobs) in the presence of multiple near-field eavesdroppers (Eves). Unlike existing works that mostly assume perfect channel state information (CSI) or location information of Eves, we consider a more practical and challenging scenario, where the locations of Bobs are perfectly known, while only imperfect location information of Eves is available at the BS. We first formulate a robust optimization problem to maximize the sum-rate of Bobs while guaranteeing a worst-case limit on the eavesdropping rate under location uncertainty. By transforming Cartesian position errors into the polar domain, we reveal an important near-field angular-error amplification effect: for the same location error, the closer the Eve, the larger the angle error, severely degrading the performance of conventional robust beamforming methods based on imperfect channel state information. To address this issue, we first establish the conditions for which the first-order Taylor approximation of the near-field channel steering vector under location uncertainty is largely accurate. Then, we propose a two-stage robust beamforming method, which first partitions the uncertainty region into multiple fan-shaped sub-regions, followed by the second stage to formulate and solve a refined linear-matrix-inequality (LMI)-based robust beamforming optimization problem. In addition, the proposed method is further extended to scenarios with multiple Bobs and multiple Eves. Finally, numerical results validate that the proposed method achieves a superior trade-off between rate performance and secrecy robustness, hence significantly outperforming existing benchmarks under Eve location uncertainty.

翻译：本文研究了近场物理层安全系统中的鲁棒波束成形设计，其中配备超大规模阵列的基站为多个近场合法用户提供服务，同时存在多个近场窃听者。与现有工作大多假设窃听者具有完美信道状态信息或位置信息不同，我们考虑了一个更实际且更具挑战性的场景：合法用户的位置完全已知，而基站仅能获取窃听者的不完美位置信息。我们首先构建了一个鲁棒优化问题，旨在最大化合法用户的和速率，同时确保在位置不确定条件下窃听速率满足最坏情况限制。通过将笛卡尔坐标位置误差转换到极坐标域，我们揭示了一个重要的近场角度误差放大效应：对于相同的位置误差，窃听者距离越近，其角度误差越大，这会严重降低基于不完美信道状态信息的传统鲁棒波束成形方法的性能。为解决此问题，我们首先建立了在位置不确定下近场信道导向矢量的一阶泰勒近似具有较高精度的条件。随后，我们提出了一种两阶段鲁棒波束成形方法：第一阶段将不确定区域划分为多个扇形子区域；第二阶段则构建并求解一个基于线性矩阵不等式的精细化鲁棒波束成形优化问题。此外，所提方法进一步扩展到多合法用户与多窃听者的场景。最后，数值结果表明，所提方法在速率性能与保密鲁棒性之间取得了更优的权衡，因此在窃听者位置不确定条件下显著优于现有基准方法。