  In this paper, we consider the robust beamforming design in a reconfigurable
intelligent surface (RIS)-aided cell-free (CF) system considering the channel
state information (CSI) uncertainties of both the direct channels and cascaded
channels at the transmitter with capacity-limited backhaul. We jointly optimize
the precoding at the access points (APs) and the phase shifts at multiple RISs
to maximize the worst-case sum rate of the CF system subject to the constraints
of maximum transmit power of APs, unit-modulus phase shifts, limited backhaul
capacity, and bounded CSI errors. By applying a series of transformations, the
non-smoothness and semi-infinite constraints are tackled in a low-complexity
manner that facilitates the design of an alternating optimization (AO)-based
iterative algorithm. The proposed algorithm divides the considered problem into
two subproblems. For the RIS phase shifts optimization subproblem, we exploit
the penalty convex-concave procedure (P-CCP) to obtain a stationary solution
and achieve effective initialization. For precoding optimization subproblem,
successive convex approximation (SCA) is adopted with a convergence guarantee
to a Karush-Kuhn-Tucker (KKT) solution. Numerical results demonstrate the
effectiveness of the proposed robust beamforming design, which achieves
superior performance with low complexity. Moreover, the importance of RIS phase
shift optimization for robustness and the advantages of distributed RISs in the
CF system are further highlighted.
