On stability preservation under sampling and approximation of feedbacks for retarded systems

It is standard worldwide, in both industrial and academic activities, to design in continuous time a stabilizing state feedback for a nonlinear system and then implement it by digital devices using sampling and zero-order holding. It is well known, from practice, that stability is rather preserved if sampling is performed at suitable high frequency. This fact has been proved in the literature, also from a theoretical point of view, for delay-free nonlinear systems. An analogous result has not been proved, from the theoretical point of view, for nonlinear retarded systems. Moreover, in the case of retarded systems, implementation by means of digital devices often requires some further approximation due to non-availability in the buffer of the value of the system variables at some past times. In order to cope with this problem, we make use here of the well-known approximation scheme based on first order splines. We introduce the standard hypothesis that there exists a stabilizing continuous-time state feedback and that there exists a related suitable Lyapunov– Krasovskii functional (in a large class already introduced in the literature) for the continuous-time, globally asymptotically stable, closed-loop system. We show, for fully nonlinear retarded systems, that, by sampling at suitable high frequency the system (finite dimensional) variable, stabilization in the sample-and-hold sense is guaranteed, when the held input is obtained as a feedback of the (first order) spline approximation of the (infinite dimensional) system state, whose entries are available at sampling times. In case of non-uniform sampling, the current sampling period (time elapsed between the latest two samplings) has to be measured (for instance by a clock device set to zero at each sampling time). A sampled-data, event-based control law is also studied.