Fractional Fliess operators: Two approaches

A useful representation of an input-output map in a nonlinear control system is the Chen-Fliess functional series or Fliess operator. It can be viewed as a noncommutative generalization of a Taylor series, and its algebraic nature is especially well suited for a number of important applications. The objective of this paper is to describe a generalization of this class of operators, so called fractional Fliess operators. These are functional series whose coefficients have a certain fractional growth rate (Gevrey series) and whose iterated integrals are defined in terms of fractional integrals. The motivation for this idea is two-fold. First, fractional system theory is a well developed area with a variety of applications, so this concept is a natural generalization in its own right. But even in the classical case it has been observed that the cascade interconnection of two Fliess operators can result in a composite system that has a certain fractional nature. Hence, developing this generalization may also provide some insight into this issue.