  In reconfigurable intelligent surface (RIS) aided millimeter-wave (mmWave)
communication systems, in order to overcome the limitation of the conventional
channel state information (CSI) acquisition techniques, this paper proposes a
location information assisted beamforming design without the requirement of the
conventional channel training process. First, we establish the geometrical
relation between the channel model and the user location, based on which we
derive an approximate CSI error bound based on the user location error by means
of Taylor approximation, triangle and power mean inequalities, and semidefinite
relaxation (SDR). Second, for combating the uncertainty of the location error,
we formulate a worst-case robust beamforming optimization problem. To solve the
problem efficiently, we develop a novel iterative algorithm by utilizing
various optimization tools such as Lagrange multiplier, matrix inversion lemma,
SDR, as well as branch-and-bound (BnB). Additionally, we provide sufficient
conditions for the SDR to output rank-one solutions, and modify the BnB
algorithm to acquire the phase shift solution under an arbitrary constraint of
possible phase shift values. Finally, we analyse the algorithm convergence and
complexity, and carry out simulations to validate the theoretical derivation of
the CSI error bound and the robustness of the proposed algorithm. Compared with
the existing non-robust approach and the robust beamforming techniques based on
S-procedure and penalty convex-concave procedure (CCP), our method can converge
more quickly and achieve better performance in terms of the worst-case
signal-to-noise ratio (SNR) at the receiver.
