Does sample-time emulation preserve exponential stability?

Whereas classical control theory provides many methods for designing continuous-time feedback controllers, nowadays control algorithms are implemented on digital platforms and have to be designed in sampled time. Approaches to sampled-time control design are based on either discretization of the plant enabling discrete-time controller synthesis, or various redesign methods converting a continuous-time controller into a sampled-time approximation, providing comparable closed-loop system properties. The simplest of redesign approaches, typically used in practice, is the emulation of continuous-time feedback by sufficiently fast sampling. In spite of its simplicity, emulation gives rise to an important problem: does emulation at a sufficiently high rate (or, equivalently, with a small sampling time) preserve the stability of the closed-loop system? In this paper, we address this problem for the case of exponential stability (local or global). Even for linear systems, the problem of stability preservation becomes non-trivial when sampling is aperiodic. For nonlinear systems, viability of emulation approach is usually proved only under quite restrictive assumptions on the plant and the controller, which, as will be shown, in fact be discarded.