Special Session 20: Dynamics with fractional and time scale derivatives
Contents
In this paper we formulate a coherent approach to signals and systems theory on regular time scales. The two derivatives - nabla (forward) and delta (backward) are used and the corresponding eigenfunctions computed. They are the so-called nabla and delta exponentials. With these exponentials the nabla and delta Laplace transforms are deduced and their properties studied. These transforms are back compatible with the current Laplace and Z transforms. They are used to study the nabla linear systems defined by differential equations. These equations mimmic the usual continuous-time equations that are uniformly approximated when the jump interval becomes small. Impulse response and transfer function notions are introduced and obtained. This implies a unified mathematical framework that allows us to approximate the classic continuous-time case when the sampling rate is high or obtain the current discrete-time case based on difference equation when the jump becomes constant.