The Infinite State Representation of Fractional Order Differential Systems: A Survey - Part 1

Abstract

Since the first fractional integration operator dedicated to simulation 25 years ago, the Infinite State representation has evolved into a general methodology for the analysis of fractional order system dynamics. Based on the frequency-distributed model of the fractional integrator, a fractional system is decomposed into an infinite dimensional set of ODEs, whose state variables are the frequency-distributed state variables of the fractional system. Thus, the Infinite State approach provides a conventional integer-order solution to the initialization problem, a new interpretation of the initial conditions of fractional derivatives, and a definition of fractional energy, preliminary to the analysis of system stability based on the Lyapunov approach. Its finite dimensional approximation enables the study of fractional transients thanks to numerical simulation and the discovery of hidden properties of fractional systems. Moreover, it is a helpful and necessary technique for the analysis of nonlinear and chaotic systems and represents a powerful characterization tool.