Frequency-domain analysis of linear systems with periodic jumps: definition of hybrid transfer function, pole and zero

A frequency-domain analysis is carried out for a class of linear systems in the presence of periodic time-driven jumps. The result is achieved by introducing the notion of transfer function in this context, and by relying on a suitably defined transform for hybrid arcs, whose properties are comprehensively discussed. It is firstly shown that knowledge of the transfer function permits the computation in closed-form of the complete (state and output) response of the hybrid plant, which may be otherwise intractable from the computational point of view. Moreover, the hybrid transfer function resulting from series interconnection of two plants is derived and exploited to introduce and discuss, in the single-input single-output case, the concepts of zero and pole of the hybrid transfer function. Differently from the classical purely continuous or discrete time cases, only a subset of the latter is related to the stability properties of the hybrid system.