Stability Analysis Techniques for Commensurate Fractional-Order Systems with Caputo Derivatives

Dynamics of fractional-order systems has presently become a very extensively studied research area. This can be attributed to the various benefits of fractional-order derivatives such as hereditary properties, more degrees of freedom, and other advantages over the traditional integer-order derivatives. The standard Routh- Hurwitz criterion provides necessary and sufficient conditions for the stability of systems of integer-order differential equations, however, this criterion is only a sufficient condition for the Matignon-stability theorem for fractional-order derivatives. Therefore, devising methods for determining the stability of fractional-order systems has become necessary. In this paper, we study techniques such as the Sylvester criterion, Schur complement, Routh-Hurwitz criterion, and the Root locus technique for stability analysis of commensurate fractional-order systems in the Caputo sense. We demonstrate these methods on the fractional Lorenz and reverse butterfly-shaped systems in exploring their performance on these systems. The Root Locus technique explicitly identifies stability conditions and allows for bifurcation analysis of these fractional systems. These findings enhance modeling in various scientific and engineering fields.