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• The uncertainty is a mismatched type without known upper bounds. • Three triggering mechanisms are proposed to reduce the redundant control signal transmission. • The corresponding triggering controllers containing triggering thresholds bounds are constructed to compensate the uncertainties and ensure the required H ∞ performance. This paper investigates the H ∞ control of uncertain linear systems under the event-triggered control framework. The uncertainty is a mismatched type without known upper bounds. To reduce the redundant transmission of control signals, three triggering strategies are developed, namely, the absolute threshold triggering manner is built firstly to deal with the small control input variation. By revisiting the first one, the large control input is monitored by the second triggering scheme. Lastly, a trade-off between the above two triggering measures is made to cover all possible inputs without sacrificing system performance largely. Based on the proposed triggering strategies, different adaptive controllers containing the triggering threshold bound are constructed to eliminate the unknown uncertainty and ensure uniformly bounded with the required H ∞ performance level, which is verified by simulation studies. [ABSTRACT FROM AUTHOR]