Adaptive backstepping control for fractional recursive systems

This paper investigates adaptive backstepping control for a class of fractional nonlinear recursive systems. A novel fractional Mittag–Leffler boundedness theorem and Mittag–Leffler stability theorem are introduced via a dual-functional fractional differential inequality technique. These theorems introduce a free Lyapunov function term and reduce the conservatism of the fractional Lyapunov stability theory. Two significant properties of positive-definite Lyapunov functions are established, indicating that the fractional derivative of a positive-definite Lyapunov function cannot be strictly less than zero. These properties demonstrate that the existing fractional differential inequalities require more rigorous proofs. Moreover, a multivariate adaptive backstepping control strategy for fractional nonlinear recursive systems is derived. Fuzzy approximation techniques are employed to address the nonlinear dynamics of the controlled object, while a fuzzy state observer is developed to estimate the unmeasured states. It is then proved that the output tracking reference error of the controlled system is Mittag–Leffler bounded. The validity of the proposed control scheme is substantiated by two simulation examples.