In this paper, we consider the robust beamforming design for a reconfigurable intelligent surface (RIS) aided cell-free (CF) system by taking into account the channel state information (CSI) uncertainties of both the direct channel and cascaded channel at the transmitter. We jointly optimize the precoding at the access points (APs) and the phase shifts at the multiple RISs to maximize the worst-case system sum rate adopting a bounded CSI error model. By applying a series of transformations and approximations, the considered problem is decomposed into several subproblems via the block coordinate descent (BCD) algorithm. To address the nonconvexity arised in the subproblem of RIS phase shift optimization, we exploit the alternating direction method of multipliers (ADMM) to obtain a suboptimal solution. Furthermore, to reduce the computational complexity, we propose a new scheme that applies the ADMM framework to replace the BCD algorithm. Numerical results demonstrate the effectiveness of the proposed robust beamforming design and the importance of accurate cascaded channel estimation.
 

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