Observer-based fuzzy adaptive control for non-linear time-varying delay systems with unknown control direction

A novel fuzzy state observer-based adaptive control is proposed for a class of strict-feedback non-linear time-varying delay systems with unknown control direction. A state observer with fuzzy approximators is established to estimate the system states. Based on the Nussbaum-type function in detecting the control direction and backstepping method, adaptive output feedback controller is constructed without requiring a priori knowledge of the signs of the unknown control directions, which can achieve the output tracking. It is rigourously proven that the proposed Lyapunov-Krasovskii functionals and the adaptive backstepping method are able to guarantee semi-globally uniform ultimate boundedness of all the signals in the closed-loop systems, whereas the tracking error converges to a small neighbourhood of the origin. A main advantage of the proposed controller is that it contains only one novel adaptive parameter that needs to be updated online, which can improve the transient and steady-state performances, and the other one is that the proposed scheme can be applied to the systems that do not satisfy the matching conditions. Finally, the simulations had been provided to verify the effectiveness of the proposed scheme.