This study reports a novel adaptive non-singular fast terminal sliding mode controller for the tracking control and synchronization of a chaotic spur gear system. The proposed novel control law attenuates the chattering phenomena of the conventional sliding mode controller. In addition, a non-singular fast terminal sliding mode surface is employed to remove the singularity problem, increase the convergence rate, and guarantee finite-time convergence. An extreme learning machine (ELM) neural network is utilized to estimate the unknown dynamics of the spur gear system and the reaching law coefficients; hence, this control scheme is a combination of the direct and indirect adaptive control. The adaptation rules of the ELM are derived based on the Lyapunov stability theorem to ensure closed-looped stability. Finally, some different numerical simulations are considered to check the validity and efficiency of the proposed control strategy compared with other control methods.
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