Stability regions of fractional systems in the space of perturbed orders

Abstract

© 2019 The Institution of Engineering and Technology. When dealing with fractional order systems, perturbations in differentiation orders arise frequently due to issues with floating point arithmetics, or due to imprecisions of various order estimation algorithms. This study establishes new results regarding stability/instability of fractional systems with perturbed differentiation orders, knowing the related properties of their unperturbed counterparts. First of all, starting from a point in the space of differentiation orders, sufficient stability/instability conditions of all systems with differentiation orders varying along a line segment with a prescribed direction are established. Then, a continuation procedure is developed allowing computation of the maximum perturbation (along some given direction) which guarantees that the number of zeros in the closed right-half plane of the characteristic function remain unchanged. Finally, sufficient conditions are established guaranteeing stability/instability of all systems having differentiation orders within a domain. The established results allow concluding on the stability of incommensurate fractional transfer functions. They are illustrated by a number of examples, including an experimental one.