Non-linear Mittag–Leffler stabilisation of commensurate fractional-order non-linear systems

Mittag–Leffler stability is a property of fractional-order dynamical systems, also called fractional Lyapunov stability, requiring the evolution of the positive-definite functions to be Mittag–Leffler, rather than the exponential meaning in Lyapunov stability theory. Similarly, fractional Lyapunov function plays an important role in the study of Mittag–Leffler stability. The aim of this study is to create closed-loop systems for commensurate fractional-order non-linear systems (FONSs) with Mittag–Leffler stability. We extend the classical backstepping to fractional-order backstepping for stabilising (uncertain) FONSs. For this purpose, several conditions of control fractional Lyapunov functions for FONSs are investigated in terms of Mittag–Leffler stability. Within this framework, (uncertain) FONSs Mittag–Leffler stabilisation is solved via fractional-order backstepping and the global convergence of closed-loop systems is guaranteed. Finally, the efficiency and applicability of the proposed fractional-order backstepping are demonstrated in several examples.

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