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) has gained considerable attention in the control literature in view of its usefulness for the analysis of feedback systems. In this paper it is shown that 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-left representations are closely related to the perhaps more common notion of coprime factor representations. As an example of their usefulness, a neat characterization of closed-loop stability is obtained in terms of strong-right 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 input and output spaces and apply, as a special case, to sampled-data control-systems. Furthermore, they are particularly useful for robustness analysis in terms of the gap metric.