Pseudo‐Lyapunov methods for Grünwald‐Letnikov and initialized fractional systems

Abstract

This paper presents reduced‐order methods to study the stability of initialized or Grünwald‐Letnikov fractional nonlinear systems. It is shown that the initialization procedure must be formalized by introducing a class of systems, and the corresponding stability analysis must be established for each element of that class. The main features obtained using this novel approach are (a) the requirements for stability are imposed directly on the equations of the system and involve only finite‐dimensional variables; (b) the conclusions are asserted on the variables of interest; (c) the method can be extended in several ways, including multi‐order systems. Illustrative examples, including an application in adaptive control, are finally presented to convey the usefulness of our approach.