Existence of right and left representations of the graph for linear periodically time-varying systems

The graph representation of a system (the set of all input-output pairs) ha s gained considerable attention in the control literature in view of its us efulness for the analysis of feedback systems. In this paper it is shown th at the graph of any stabilizable, linear, periodically time-varying (LPTV), continuous-time system can be expressed as the range and kernel of bounded , causal, LPTV systems that are, respectively, left and right invertible by bounded, causal, LPTV systems. These so-called strong-right and strong-lef t representations are closely related to the perhaps more common notion of coprime factor representations. As an example of their usefulness, a neat c haracterization of closed-loop stability is obtained in terms of strong-rig ht and strong-left representations of the plant and controller graphs. This in turn leads to a Youla-style parametrization of stabilizing controllers. All of the results obtained accommodate possibly infinite-dimensional inpu t and output spaces and apply, as a special case, to sampled-data control-s ystems. Furthermore, they are particularly useful for robustness analysis i n terms of the gap metric.