Robust Beamforming Design for Active Sub-Connected RIS Assisted Cell-Free MIMO Systems: A Two-Stage Distributed Approach

Reconfigurable intelligent surface (RIS) assisted cell-free multiple-input multiple-output (MIMO) systems have emerged as a promising paradigm for future wireless communications. To overcome the multiplicative fading effect inherent in the passive RIS, a novel active RIS equipped with reflection type power amplifiers has been proposed, which however is confronted with high hardware cost and power consumption under the fully-connected architecture. To address this issue, we consider a cost and power-efficient sub-connected active RIS assisted cell-free system in this paper, where the whole active RIS is divided into multiple sub-RISs, each being connected to a dedicated power amplifier. We aim to maximize the system sum rate under imperfect channel state information (CSI) by jointly optimizing the AP transmit beamforming matrices and the RIS reflection coefficient matrix. Since the traditional centralized beamforming scheme may lead to a high computational burden at the central processing unit (CPU), we propose a two-stage distributed iterative algorithm to efficiently find high quality suboptimal solutions. Specifically, in stage 1, users apply the classical weighted minimum mean-square error (WMMSE) method to optimize their local variables in parallel. Then in stage 2, APs optimize their respective transmit beamforming matrices, the RIS reflection phase shift vector and the RIS reflection amplification vector sequentially. The corresponding semi-closed-form optimal solutions are available by jointly leveraging the Lagrange duality theory, majorization minimization (MM) and symmetric alternating direction method of multiplier (S-ADMM) techniques. Moreover, we develop a simplified distributed algorithm to further reduce system communication overhead. Numerical results demonstrate that the two proposed distributed algorithms can achieve comparable sum rate performance to the centralized scheme while attaining lower computational overhead.