Linear periodically time-varying scaling and its properties

Stimulated by a general necessary and sufficient condition for robust stability of sampled-data systems stated in an operator-theoretic framework, we introduce a novel technique called linear periodically time-varying (LPTV) scaling. We then give a simple example of sampled-data systems in which this new scaling allows exact robust stability analysis for static uncertainties while the conventional linear time-invariant (LTI) scaling fails to do so. This leads us to an interesting question whether LPTV scaling can be effective also in other situations, e.g., in the robust stability analysis of continuous-time feedback systems regarded as a special class of sampled-data systems, or for other classes of uncertainties. We thus study some basic properties of LPTV scaling by confining ourselves to the so-called D-scaling, and show that it provides no advantage over LTI scaling when it is applied to continuous-time LTI feedback systems, regardless of the class of uncertainties we take into consideration. This demonstrates that the LPTV scaling of the type we deal with in this paper is in some sense a special technique for sampled-data systems but is indeed an effective and more natural technique than the conventional LTI scaling as far as such systems are concerned. The technique can be further extended to include what is called noncausal LPTV scaling, and the implication of the present study in such a larger framework of LPTV scaling is also described.