A geometric framework for distributed frequency models

Geometric control theory, developed by Basile and Marro, and independently, by Wonham and Morse in the 1970s revolves around characterizing the properties of finite dimensional, linear and time-invariant systems using geometry. Some examples of these properties are invariance, controllability and observability. The task addressed in this paper is to develop the geometric tools for fractional systems using the diffusive representation (also known as the distributed frequency) model. The mathematics involved in this approach is different from the classical case as fractional systems are inherently infinite dimensional. Unlike the integer order derivative, the fractional derivative is a non-local operator, and so the evolution of the so-called pseudo-state depends on not just its current value but also its past history. Thus, the notion of an initial condition for a fractional system can come in different forms. This leads to different kinds of fractional derivative operators. With some of these operators, the semigroup property is lost. With the distributed frequency model, the initial condition comes in the form of an initialization function. This takes care of the infinite dimensional nature of fractional systems. Furthermore, the distributed frequency model retains the semigroup property. This is useful in developing invariance and controlled invariance for fractional systems. Moreover, these properties of fractional systems are verified numerically using a higher order finite-dimensional approximation, which retains all the geometric properties of the distributed frequency model.