Robustness of linear periodically-time-varying closed-loop systems

The paper is concerned with the development of quantitative results that lead to a characterisation of the uncertainty types to which linear, periodically time-varying (LPTV), closed-loop systems are insensitive. The results are expressed in terms of the gap metric, which is a measure of the distance between the graphs of two systems. Towards establishing the main results, a formula for the directed gap is derived for LPTV systems. This formula is essentially a, generalisation of that given in Georgiou (1983), for linear, time-invariant (LTI) systems. In fact, all of the results obtained in this paper are analogous to those known to hold for LTI systems (although they do not follow directly from these). Importantly, the results presented in this paper apply to a class of sampled-data (SD) control systems as a special case and are a step towards characterising the quality of SD approximations of LTI, continuous-time controllers