Title : State variables , initial conditions and transients of fractional order derivatives and systems

The sub-title of this presentation could be “The fractional order integrator approach”. Although fractional order differentiation is commonly considered as the basis of fractional calculus, its effective basis is in fact fractional order integration, mainly because definitions, calculation and properties of fractional derivatives and Fractional Differential Systems (FDS) rely deeply on fractional integration. Thus, this speech will be focussed on fractional integration and on the fractional integrator which is a frequency distributed differential system. The properties of this integrator are linked to its infinite dimension state vector and the mastery of its transients rely on the knowledge of its initial conditions (i.e. initial state vector). Three definitions of fractional derivatives will be considered: implicit fractional differentiation and explicit Caputo and Riemann-Liouville differentiation. The explicit derivatives exhibit two fundamental operations: integer differentiation and fractional integration. Consequently, the transients of these two derivatives rely necessarily on the initial state vector of the associated fractional integrator. So, commonly used Laplace transform equations of the explicit derivatives are wrong because they do not include the initial state vector of this integrator. The explicit derivatives are usually considered as the foundation of FDS theory, especially the Caputo derivative for physical reasons. In fact, their complete Laplace transform equations show that they are inappropriate to derive FDS free responses. On the contrary, the implicit derivative provides a nice framework for the derivation and the analysis of FDS free responses and transients, with physical interpretation of initial conditions. Moreover, the explicit fractional integrators associated to these derivatives allow a rigorous definition of the state variables of a FDS, with a distinction between pseudo state variables and infinite dimension state variables. These integrator state variables can be estimated by an observer based technique either for fractional explicit derivatives or for FDS, so their initialisation problem can be solved using the past history of the derivatives or of the system. Numerical simulations will illustrate the efficiency of the fractional integrator approach to the mastery of transients, with a particular attention to the visualization of the internal state variables.