When the direction of a potential mean shift can be anticipated, the one-sided exponentially weighted moving average (EWMA) X¯¯¯ control chart using the truncation method (namely, the one-sided TEWMA X¯¯¯ chart) is more efficient than those conventional one- and two-sided EWMA X¯¯¯ schemes for process monitoring. Although attractive, there are no studies on designing the one-sided TEWMA X¯¯¯ chart by taking measurement errors into account. In this context, we investigate the effect of measurement errors on the performance of the one-sided TEWMA X¯¯¯ chart based on the linear covariate error model. Additionally, a Markov chain model is established to evaluate the run length properties of the scheme in the presence of measurement errors. Then, an optimal design procedure is developed for searching the optimal design parameters of the scheme. Based on these mentioned studies, several tables and figures are presented to evaluate the detecting performance of the scheme under different parameters of the linear covariate error model, and then a conventional one-sided EWMA X¯¯¯ chart with reflecting boundary is introduced to further study the effect of the presence and absence of measurement errors on control chart comparison studies. Simulation results show that although the detecting performance of the proposed scheme is significantly affected by measurement errors, its performance is still superior to the classic competing chart under the same comparison conditions. Finally, an illustrative example is given to show the implementation of the recommended scheme.