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In this paper, we consider IRS-assisted transmissions from a multi-antenna access point (AP) to a receiver with uncertain channel information. By adjusting the magnitude of reflecting coefficients, the IRS can sustain its operations by harvesting energy from the AP's signal beamforming. Considering channel estimation errors, we model both the AP-IRS channel and the AP-IRS-receiver as a cascaded channel by norm-based uncertainty sets. This allows us to formulate a robust optimization problem to minimize the AP's transmit power, subject to the user's worst-case data rate requirement and the IRS's worst-case power budget constraint. Instead of using the alternating optimization method, we firstly propose a heuristic scheme to decompose the IRS's {phase shift} optimization and the AP's active beamforming. Based on semidefinite relaxations of the worst-case constraints, we further devise an iterative algorithm to optimize the AP's transmit beamforming and the magnitude of the IRS's reflecting coefficients efficiently by solving a set of semidefinite programs. Simulation results reveal that the AP requires a higher transmit power to deal with the channel uncertainty. Moreover, the negative effect of channel uncertainty can be alleviated by using a larger-size IRS.