Gap-metric robustness analysis of linear periodically time-varying feedback systems

Quantitative robust stability results are established in this paper for fee dback systems that evolve in continuous-time and exhibit linear, periodical ly time-varying (LPTV) behavior. The results presented are analogous to res ults known to hold for linear, time-invariant (LTI) systems, although they do not follow directly from these. System uncertainty is measured using the gap metric, which quanti es the distance between systems in terms of the a perture between their graphs (the subspaces corresponding to all input-outp ut pairs for each system). The main robustness result characterizes the lar gest gap-ball of LPTV plants stabilized by a nominal LPTV feedback controll er known to stabilize a nominal LPTV plant at the center of this ball. A ke y step in the proof of this result makes use of a formula derived for the d irected gap between LPTV systems. This formula is essentially a generalizat ion of Georgiou's for LTI systems. Importantly, all of the results presente d apply to a class of sampled-data control systems, as a special case.