Abstract
To tackle several limitations recently highlighted in the field of fractional differentiation and fractional models, some authors have proposed new kernels for the definition of fractional integration/differentiation operators. Some limitations still remain, however, with these kernels, whereas solutions prior to the introduction of fractional models exist in the literature. This paper shows that the fractional pseudo state space description, a fractional model widely used in the literature, is a special case of the Volterra equations, equations introduced nearly a century ago. Volterra equations can thus be viewed as a serious alternative to fractional pseudo state space descriptions for modelling power law type long memory behaviours. This paper thus presents a new class of model involving a Volterra equation and several kernels associated with this equation capable of generating power law behaviours of various kinds. One is particularly interesting as it permits a power law behaviour in a given frequency band and, thus, a limited memory effect on a given time range (as the memory length is finite, the description does not exhibit infinitely slow and infinitely fast time constants as for pseudo state space descriptions).
Highlights
For 30 years, research on fractional differentiation and integration operators has steadily increased
Dynamic systems with power law type long memory behaviours are generally modelled in the literatDuryenuasminigc fsryasctteiomnsalwmitohdpeolsw
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Summary
For 30 years, research on fractional differentiation and integration operators has steadily increased. A fractional pseudo state space description (1)—which it is better to denote a pseudo state space description, since the variable x(t) does not meet the definition of a state as shown in [9,12]—is a simple “fractionalisation” of the classical state space description, without physical justification and resulting from the need for models that fit power law type long memory behaviours.
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