Pointwise frequency responses framework for stability analysis in periodically time-varying systems

This paper explicates a pointwise frequency-domain approach for stability analysis in periodically time-varying continuous systems, by employing piecewise linear time-invariant (PLTI) models defined via piecewise-constant approximation and their frequency responses. The PLTI models are piecewise LTI state-space expressions, which provide theoretical and numerical conveniences in the frequency-domain analysis and synthesis. More precisely, stability, controllability and positive realness of periodically time-varying continuous systems are examined by means of PLTI models; then their pointwise frequency responses (PFR) are connected to stability analysis. Finally, Nyquist-like and circle-like criteria are claimed in terms of PFR's for asymptotic stability, finite-gain Lp-stability and uniformly boundedness, respectively, in linear feedbacks and nonlinear Luré connections. The suggested stability conditions have explicit and direct matrix expressions, where neither Floquet factorisations of transition matrices nor open-loop unstable poles are involved, and their implementation can be graphical and numerical. Illustrative studies are sketched to show applications of the main results.