Quadratic Lyapunov functions for stability analysis in fractional-order systems with not necessarily differentiable solutions

Abstract

Solutions of fractional-order differintegral equations are generally not necessarily integer-order differentiable, neither in the strong nor in the weak sense, thus limiting the stability analysis in systems based on the most conventional fractional-order operators. In this paper, a consistent and well-posed definition for fractional-order systems is performed based on the study of alternative fractional-order operators that preserve the most interesting and useful properties of differintegrals, even in the case of not necessarily integer-order (weakly) differentiable functions. In addition, it is shown that these operators comply to a recently verified well-known inequality, which allows us to demonstrate Mittag-Leffler stability in a more general class of fractional-order systems, considering quadratic Lyapunov functions, by demonstrating a generalization of the Lyapunov direct method for a class of fractional-order nonlinear systems. Illustrative examples are given to highlight the feasibility of the proposed method, and a multivariable fractional integral sliding mode control application is presented.