Energy-Efficient Robust Beamforming for Multi-Functional RIS-Aided Wireless Communication under Imperfect CSI

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.