Pseudo-state feedback stabilization of commensurate fractional order systems

Abstract

This paper addresses the problem of pseudo-state feedback stabilization of commensurate fractional order systems (FOS). In the proposed approach, Linear Matrix Inequalities (LMI) formalism is used to check if the pseudo-state matrix eigenvalues belong to the FOS stability region of the complex plane. A review of LMI stability conditions is first proposed for fractional order 0 < ν < 1 and 1 < ν < 2 . The paper then focuses particularly on the case 0 < ν < 1 as the stability region is non-convex and associated LMI condition is not as straightforward to obtain as in the case 1 < ν < 2 . A new LMI stability condition is thus proposed. Based on this condition, a necessary and sufficient LMI method for the design of stabilizing controllers is given. This method paves the way for extension to FOS of various LMI-based results. Among these possible extensions, a first result on robust control of polytopic fractional order systems is given in this paper.