On the construction of self-similar rational models for power-law processes

The rational modelling of power-law processes is equivalent to finding a linear time-invariant system described by a rational function H(s) with a power-law magnitude response. The construction of such systems is based on the notion of scale-invariance for a specific scale change formalized by the γ-homogeneity principle. The scale-invariant nature of these systems is reflected not only in their magnitude response, but also in their pole-zero patterns. Based on these properties, rational models for power-law processes can be constructed by cascading frequency scaled replicas of rational prototype functions. A construction theorem is provided and the implications of conditions imposed on the prototypes are discussed. The simulations illustrating dynamics exhibited by self-similar rational systems are shown