Transients of fractional-order integrator and derivatives

Abstract

Although fractional-order differentiation is generallyconsidered as the basis of fractional calculus, we demonstrate in this paper that the real basis is in fact fractional-order integration, mainly because definition and properties of fractional differentiation rely deeply on fractional integration. A second objective of this paper is to demonstrate that mastery of initial conditions as an infinite dimensional state variable allows analysis of the transients (or free responses) of the fractional integrator and derivatives. Thus, this paper is focused on fractional integration and particularly on the fractional integrator, which is an infinite dimension integer-order differential system. Consequently, the properties of this integrator are linked to its infinite dimension state variable and the mastery of its transients rely on the knowledge of its initial conditions. Its frequency distributed model is introduced, and its transients are analyzed; a finite dimension approximation of the fractional integrator is defined and validated. Numerical simulations exhibit the essential role played by the initial state vector. The definitions of the Caputo and Riemann-Liouville fractional derivatives exhibit two fundamental operations: integer-order differentiation and fractional-order integration. Thus, the transients of these two derivatives rely on the mastery of the infinite state variable of the associated fractional integrator. The Laplace transform equations of the explicit derivatives are revisited: the main result is that usual relations are wrong because the initial conditions of the associated fractional integrator are not taken into account. Finally, with the help of numerical simulations, we show that practical initialization and transients analysis of the explicit derivatives depend essentially on the mastery of the integrator initial conditions.