Abstract: The aim of the present work is to generalize the contraction theory for the analysis of the convergence of fractional order systems for both continuous-time and discrete-time systems. Contraction theory is a methodology for assessing the stability of trajectories of a dynamical system with respect to one another. The result of this study is a generalization of the Lyapunov matrix equation and linear eigenvalue analysis. The proposed approach gives a necessary and sufficient condition for exponential and global convergence of nonlinear fractional order systems. The examples elucidate that the theory is very straightforward and exact.

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