Stability analysis and stabilisation in linear continuous-time periodic systems by complex scaling

By scaling in the complex domain (namely, complex scaling) the return difference relations of linear continuous-time periodic (LCP) feedback systems, we generalise the 2-regularised stability criteria for asymptotic stability in this paper. The stability conditions of the generalised criterion are necessary and sufficient, and involve neither contour and locus orientation specification, nor open-loop Floquet factorisation and its eigenvalues distribution. Finite-dimensional implementation of the suggested criteria is considered via a two-step truncation approach. The finite-dimensional criteria are implementable either graphically with locus plotting, or numerically without locus plotting, besides retaining the aforementioned technical advantages. Furthermore, also exploiting the complex scaling technique, stabilisation of LCP systems with static state feedback is worked out in the internal or external stability sense, whose alternative interpretations in terms of the small-gain theorem and the Gronwall inequality are explicated. To illustrate the main results, the lossy Mathieu differential equation is investigated.