In this paper, an arbitrary finite-time tracking\ncontrol (AFTC) method is developed for magnetic\nlevitation systems with uncertain dynamics and external\ndisturbances. By introducing a novel augmented slidingmode\nmanifold function, the proposed method can\neliminate the singular problem in traditional terminal\nsliding-mode control, as well as the reaching-phase\nproblem. Moreover, the tracking errors can reach the\nreference value with faster convergence and better\ntracking precision in arbitrarily determined finite time. In\naddition, a fuzzy-arbitrary finite-time tracking control (FAFTC)\nscheme that combines a fuzzy technique with\nAFTC to enhance the robustness and sliding performance\nis also proposed. A fuzzy logic system is used to replace\nthe discontinuous control term. Thus, the chattering\nphenomenon is resolved without degrading the tracking\nperformance. The stability of the closed-loop system is\nguaranteed by the Lyapunov theory. Finally, the\neffectiveness of the proposed methods is illustrated by\nsimulation and experimental study in a real magnetic\nlevitation system.
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