Adaptive fuzzy controller for non-affine systems with zero dynamics

In this article, the direct adaptive fuzzy control problem is investigated for a class of general non-linear systems with zero dynamics. The direct adaptive fuzzy controller is developed based on a unified observer which is used to estimate the time derivatives of the output. The corrective term of the proposed observer involves a well-defined design function which is shown to be satisfied by the commonly used high-gain-based observers, namely for the usual high-gain observers and the sliding-mode observers together with their implementable versions. By using a general error function, and without resorting to the famous strictly positive real condition or the filtering of the observation error, a general proportional-integral (PI) law for updating the fuzzy parameters is proposed. Ultimately boundedness of the error signals is shown through Lyapunov's direct method. Theoretical results are illustrated through two simulation examples.