In this paper, a robust extreme learning machine is proposed. In comparison with the original extreme learning machine and the regularized extreme learning machine, this robust algorithm minimizes both the mean and variance of modeling errors in the objective function to overcome the bias-variance dilemma. As a result, its generalization performance and robustness are enhanced, and these merits are further proved theoretically. In addition, this proposed algorithm can keep the same computational efficiency as the original extreme learning machine and the regularized extreme learning machine. Then, several benchmark data sets are used to test the effectiveness and soundness of the proposed algorithm. Finally, it is employed to model a real small-scale turbojet engine. This engine is fit well. Especially, on the idle phase, where the signal-to-noise ratio is low and it is very hard to model, the proposed algorithm performs well and its robustness is sufficiently showcased. All in all, the proposed algorithm provides a candidate technique for modeling real systems.