Application of diffusive representations to discretized fractional systems: Stability, positivity and simulations issues

Continuous-time fractional transfer functions such as s^-α have been studied from many viewpoints. Diffusive representations have given an infinite-dimensional realization for these operators; this is a new framework that reveals dissipativity and proves useful for numerical approximations. Similarly, discrete-time fractional filters have already been studied, and can be recast in the new framework of discrete-time diffusive representations. The aim of this paper is to show how such diffusive representations can be used to analyse two different discretizations (backward difference and impulse invariance) of the same continuous-time system, namely a standard oscillator damped by a fractional operator. - Internal stability stems from LaSalle invariance principle with a global Lyapunov functional defined as the sum of the mechanical energy of the oscillator and a storage function built from the diffusive representation of the fractional damping. - External stability (in the energy sense) can be proved using strong positivity (i.e. coercivity) of the fractional damping and a passivity theorem. When the coupling parameter is replaced by a memoryless nonlinearity, the small gain theorem and the strong positivity of an appropriate filter prove that energy-stability remains true for the global nonlinear filter, hence displaying some robustness.