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.