Further Studies on Frozen-Time Eigenvalues in the Stability Analysis for Periodic Linear Systems

In this paper we present some enlightening results to show why and how stability assessment for Linear Time-Varying (LTV) systems based solely on the location of the "frozen-time eigenvalues (FTE)" fails to be sufficient or necessary, using two classes of parametrized periodic LTV systems derived from two examples given by Markus-Yamabe [6] and Wu [11]. Exact domain of stability in the parameter space obtained using analytical or numerical solutions of the Floquet characteristic Exponents are presented, and compared to that predicted by FTEs. The results are useful in the study of robustness and stabilization of Linear Time Invariant (LTI) systems, as will be shown in this paper that an unstable LTI system maybe stabilized or destabilized by periodic structural perturbations (pumping) without any control input.