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This study presents an analytical approach for proper selection of output functions to be regulated for the Vienna rectifier such that the resultant closed-loop systems are minimum phase. Specifically, two different adaptive control methodologies based on the input-output linearisation are developed and categorised. In the first category, three output functions are introduced and imposed to be zero by using three dynamic feedback laws. On the basis of states to be regulated, four different cases are studied and it is shown that only one of these cases results in a one-dimensional zero dynamics with an asymptotically stable equilibrium point. In the second category, two output functions are defined and output zeroing problem is solved with two control inputs. In addition, the remaining control is used to feedback linearise the resultant two-dimensional zero dynamics. Eighteen different cases are studied to demonstrate that only in one of these cases, the corresponding zero dynamics is feedback linearisable. Stability properties of the adaptive systems are investigated to show that the proposed adaptive controllers are capable of DC output voltage regulation and power factor correction in the presence of parametric and non-parametric uncertainties. Finally, simulation results are presented to confirm the validity of the developed approaches.