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The authors consider a class of non-linear systems preceded by unknown backlash-like hysteresis. Unlike other schemes dealing with hysteresis, the non-linear systems of this study not only involve unknown parameters but also take non-linear terms linear in the unmeasured states into account. An adaptive output feedback controller by utilising a modified high-gain observer and backstepping approach is proposed, which is able to mitigate the effects of the hysteresis, and guarantee the global stability of the closed-loop system and the arbitrarily small steady-state tracking error. Moreover, it is proved that by introducing an initialisation technique, the ℒ∞ tracking performance can be achieved.