In this paper, we study the recently proposed fractional-order operators with general analytic kernels. The kernel of these operators is a locally uniformly convergent power series that can be chosen adequately to obtain a family of fractional operators and, in particular, the main existing fractional derivatives. Based on the conditions for the Laplace transform of these operators, in this paper, some new results are obtained—for example, relationships between Riemann–Liouville and Caputo derivatives and inverse operators. Later, employing a representation for the product of two functions, we determine a form of calculating its fractional derivative; this result is essential due to its connection to the fractional derivative of Lyapunov functions. In addition, some other new results are developed, leading to Lyapunov-like theorems and a Lyapunov direct method that serves to prove asymptotic stability in the sense of the operators with general analytic kernels. The FOB-stability concept is introduced, which generalizes the classical Mittag–Leffler stability for a wide class of systems. Some inequalities are established for operators with general analytic kernels, which generalize others in the literature. Finally, some new stability results via convex Lyapunov functions are presented, whose importance lies in avoiding the calculation of fractional derivatives for the stability analysis of dynamical systems. Some illustrative examples are given.

This work shows the voltage regulation of a DC–DC buck converter by applying sliding mode control using three different cases of sliding surfaces. The DC–DC buck converter is modeled by ordinary differential equations (ODEs) that are solved by applying numerical methods. The ODEs describe two state variables that are associated to the capacitor voltage and the inductor current. The state variable associated to voltage is regulated by applying two well-known sliding surfaces and a third one that is introduced herein to improve the response of the sliding mode control. The stability of the proposed sliding surface is verified by using a Lyapunov theorem to guarantee closed-loop stability. Finally, simulation results show the improvement of voltage regulation when applying the proposed sliding surface compared to already reported approaches.

A fault-tolerant control scheme is proposed for a class of commensurate-order fractional nonlinear systems that consists of two fractional-order observers (hybrid scheme). The diagnosis of the faults is performed by means of a model-free fractional proportional integral reduced-order observer that uses the fractional algebraic observability property. A fractional dynamical controller obtained in a natural way from the dynamics of a fractional high-gain observer is designed, which is constructed from a fractional generalised observability canonical form; the controller performs output tracking, thus eliminating the effects of the faults. A stability analysis on the overall system demonstrates that the origin is Mittag–Leffler stable. The proposed methodology is assessed by means of simulations on the fractional models of the Van der Pol oscillator and a DC motor.