The vortex-driven magnetization process of micron-sized, exchange-coupled square elements with composition of $Ni_{80}Fe_{20}$ (12 nm)/$Ir_{20}Mn_{80}$ (5 nm) is investigated. The exchange-bias is introduced by field-cooling through the blocking temperature (TB) of the system, whereby Landau-shaped vortex states of the $Ni_{80}Fe_{20}$ layer are imprinted into the $Ir_{20}Mn_{80}$. In the case of zero-field cooling, the exchange-coupling at the ferromagnetic/antiferromagnetic interface significantly enhances the vortex stability by increasing the nucleation and annihilation fields, while reducing coercivity and remanence. For the field-cooled elements, the hysteresis loops are shifted along the cooling field axis. The loop shift is attributed to the imprinting of displaced vortex state of $Ni_{80}Fe_{20}$ into $Ir_{20}Mn_{80}$, which leads to asymmetric effective local pinning fields at the interface. The asymmetry of the hysteresis loop and the strength of the exchange-bias field can be tuned by varying the strength of cooling field. Micromagnetic modeling reproduces the experimentally observed vortex-driven magnetization process if the local pinning fields induced by exchange-coupling of the ferromagnetic and antiferromagnetic layers are taken into account.