The robust beamforming design in multi-functional reconfigurable intelligent surface (MF-RIS) assisted wireless networks is investigated in this work, where the MF-RIS supports signal reflection, refraction, and amplification to address the double-fading attenuation and half-space coverage issues faced by traditional RISs. Specifically, we aim to maximize the system energy efficiency by jointly optimizing the transmit beamforming vector and MF-RIS coefficients in the case of imperfect channel state information (CSI). We first leverage the S-procedure and Bernstein-Type Inequality approaches to transform the formulated problem into tractable forms in the bounded and statistical CSI error cases, respectively. Then, we optimize the MF-RIS coefficients and the transmit beamforming vector alternately by adopting an alternating optimization framework, under the quality of service constraint for the bounded CSI error model and the rate outage probability constraint for the statistical CSI error model. Simulation results demonstrate the significant performance improvement of MF-RIS compared to benchmark schemes.In addition, it is revealed that the cumulative CSI error caused by increasing the number of RIS elements is larger than that caused by increasing the number of transmit antennas.

翻译：本文研究了多功能可重构智能表面辅助无线网络中的鲁棒波束成形设计，其中MF-RIS支持信号反射、折射与放大，以解决传统RIS所面临的双重衰落衰减与半空间覆盖问题。具体而言，我们旨在非完美信道状态信息条件下，通过联合优化发射波束成形向量与MF-RIS系数来最大化系统能量效率。我们首先利用S-过程与伯恩斯坦型不等式方法，分别在有界CSI误差与统计CSI误差情形下将原问题转化为可处理形式。随后，在针对有界CSI误差模型的服务质量约束与针对统计CSI误差模型的速率中断概率约束下，采用交替优化框架交替优化MF-RIS系数与发射波束成形向量。仿真结果表明，与基准方案相比，MF-RIS能带来显著的性能提升。此外，研究揭示：增加RIS单元数量所产生的累积CSI误差大于增加发射天线数量所产生的累积误差。