The incorporation of non-orthogonal multiple access (NOMA) and reconfigurable intelligent surface (RIS) provides a new solution to improve spectral efficiency. Unfortunately, the channel estimation error of RIS is noteworthy due to the absent active radio frequency link. Meanwhile, the transceivers at base stations (BS) and users also exist hardware impairments (HWI). The imperfect channel state information (CSI) and HWI will degrade the communication performance such as outage probability or data rate. To alleviate such performance deterioration, we propose a robust design scheme to minimize the BS transmit power when the target rate of users is satisfied in RIS-assisted single-beam and multi-beam NOMA networks with transceiver HWI and imperfect CSI. Further, the perfect successive interference cancellation (SIC) condition in single-beam and multi-beam NOMA networks is investigated, and the optimization problem is proposed based on perfect SIC conditions. To tackle the non-convex problem, we provide an effective algorithm combining Bernstein-type inequality and semi-definite relaxation. Numerical results demonstrate that the proposed scheme outperforms non-robust RIS-assisted NOMA scheme that ignores HWI and imperfect CSI. Moreover, the multi-beam NOMA network obtains higher beam gain compared to the single-beam NOMA network, and thus there is a superior performance in multi-beam NOMA networks.

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