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Most of smart material-based actuators, such as magnetostrictive actuators and shape memory alloy (SMA) actuators, exhibit asymmetric hysteresis effect. This non-smooth nonlinearity can result in system inaccuracy and instability. In order to overcome this undesirable effect, a class of nonlinear systems with asymmetric hysteresis actuators described by an Asymmetric Shifted Prandtl-Ishlinskii (ASPI) model are treated as systems to be controlled, and the paper attempts to develop a robust adaptive control scheme to stabilize such systems and achieve precise tracking. With the proposed control scheme, the global stability of the closed-loop systems is analyzed and simulation results confirm the the proposed robust approach.