Stability of fractional variable order difference systems

Abstract

Abstract The problem of stability of the Grünwald-Letnikov-type linear fractional variable order discrete-time systems is discussed. As a definition of the Grünwald-Letnikov difference is a convolution type, the 𝓩-transform is used as an effective tool for the stability analysis. The conditions for asymptotic stability and for instability are presented. In the case of a scalar system we state conditions that guarantee asymptotic stability in inequalities for a coefficient that appears on the right hand side of the equation defined the system}. We describe regions of the stability for systems accordingly to locus of eigenvalues of a matrix associated to the considered system. In the general case of the linear difference systems one can determine the regions of location of eigenvalues of matrices associated to the systems in order to guarantee the asymptotic stability of the considered systems. Some of the frames of these regions are illustrated in the examples.